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A geometric villain that ruins our instinctive perception of nature
Vehbi Paksoy HCNSO
THE PLAN
1 Is obvious really obvious
2 Curvature of curves in 3D
3 Curvature of Surfaces in 3D
a
b
c
c
b
A
B
C
CURVES IN 3D
Given parametrically 119903 119905 = (119909 119905 119910 119905 119911 119905 )
119903(1199050)
119903(1199051)
119903(119905)
Parametrization by arc-length 119903 119904 = (119909 119904 119910 119904 119911 119904 )
119904 = 1199050
1199051
119903(119905) 119889119905 = 1199050
1199051
( 119909)2 + ( 119910)2 + ( 119911)2119889119905
HELIX
Standard parametrization 119903 119905 = (cos 119905 sin 119905 2119905)
Arc-length parametrization 119903 119904 = (cos119904
5 sin
119904
52119904
5)
119903(119904) = 1
P
P
WHAT IS THE DIFFERENCE
Curvature κ = 1
119877
R is the radius of the circle which gives the best approximation of the curve near
the point
Circle of radius R
R
κ =1R
Line
κ = 0
Helix
κ =
κ is the measure of the rate of change of tangent vector at a
point as we travel along the curve
κ(s)= 119903(119904)
COMPUTING THE CURVATURE
Arc-length parametrization can be tedious
Twisted Cubic
119903 119905 = (119905 1199052 1199053)
119904 = 01199051 + 41199062 + 91199064 119889119906 =
κ= 119903 119905 times 119903(119905)
119903(119905) 3
T
N
B = T x N
119879 = κ 119873
119873 = minusκ T + τ 119861
119861 = minus 120591 119873
SERRE-FRENET FRAME
119903(119904)
T
N
B
Pitch axis
Roll axisYaw axis
Cell Shape Dynamics From Waves to Migration
Meghan K Driscoll1 Colin McCann12 Rael Kopace1 Tess Homan1 John T Fourkas34 Carole Parent2 and Wolfgang Losert PLOS 2012
Curvature in the study of wave-like characteristics of
amoeba migration
PARAMETRIC SURFACES
A surface M in space is a 2 dimensional object usually given parametrically
119903 = (119909 119906 119907 119910 119906 119907 119911 119906 119907 )
u
v
r = (119940119952119956 120637 119956119946119951 120651 119956119946119951 120637 119956119946119951 120651 119940119952119956 120651 )
SPHERE
FROM NON-LINEAR TO LINEAR
119951
119955119958
119955119959
119955 = (119961 119958 119959 119962 119958 119959 119963 119958 119959 )
119955119958 = (119961119958 119962119958 119963119958)
119955119959 = (119961119959 119962119959 119963119959)
119951 =119955119958times119955119959
119955119958times119955119959
P
T119875119872M
119951(119927)
Gauss Map
119918119924 rarr 119930120784
119927 rarr 119951(119927)M
Shape operator is the negative of the derivative of the Gauss map
CURVATURE FOR SURFACES
Gauss Curvature K = 120639120783 sdot 120639120784
Mean Curvature H =120639120783+120639120784
2
Gauss and Mean curvatures
are determinant and half of
the trace of the shape
operator
120639120783 120639120784
120639120783 = 119950119946119951119946119950119958119950
120639120784 = 119950119938119961119946119950119958119950
M
Two dimensional creatures cannot compute 120639120783 and 120639120784 using infinitesimal
ruler and protractor BUT they can determine K = 120639120783 sdot 120639120784 This means 2D
creatures can determine the shape of their world without stepping out to
3rd dimension
GAUSSrsquoS THEOREM EGREGIUM
119870 = minus1
2 119864119866(120597
120597119907
119864119907
119864119866+
120597
120597119906
119866119906
119864119866)
119916 = 119955119958 ∙ 119955119958
G = 119955119959 ∙ 119955119959
Rotating moving or bending the surface does not change the Gauss
curvature but stretching or breaking does
Two surfaces with the same Gauss curvature are ldquolocallyrdquo the same but not
globally
K=0 K=0
VARIOUS SURFACES
CURVATURE IN ARCHITECTURE AND DESIGN
MINIMAL SURFACES
A surface M is minimal if 119815 =120639120783+120639120784
2= 0
Any planar surface is minimal (NOT INTERESTING)
A Gyroid (VERY INTERESTING)
Gyroid structures are found in certain
surfactant or lipid mesophases and block
copolymers
THEOREM Every soap film is a physical model of a minimal surface
The interpretation of the Costa-Hoffman-Meeks minimal surface as
insertion of multiple directional holes connecting the top to the water
and the water at the bottom to the sky provided a single gesture
combining all aspects -Tobias Walliser
Image LAVA
COSTA-HOFFMAN-MEEK SURFACE
CONCEPT OF A LINE
y=mx+b
slope-intercept
119955 119957 = (119957119950119957 + 119939)
parametric
In general a line in space is given by 119903 119905 = 119875 + 119905119906 So 119903 = 0
P119958
119951
V
120629119958V
119906
The covariant derivative 120629119958V of vector field V
is the projection of the change of vector field in
119906 direction onto the tangent plane
120572(119905)
A curve 120572(119905) on the surface is called a
ldquoGeodesicrdquo if 120571 prop prop = 0
Geodesics are the ldquolinesrdquo of curved spaces
V is parallel along a curve 120572(119905) if
120629 120630119829 = 120782
Flat
Curved
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
THE PLAN
1 Is obvious really obvious
2 Curvature of curves in 3D
3 Curvature of Surfaces in 3D
a
b
c
c
b
A
B
C
CURVES IN 3D
Given parametrically 119903 119905 = (119909 119905 119910 119905 119911 119905 )
119903(1199050)
119903(1199051)
119903(119905)
Parametrization by arc-length 119903 119904 = (119909 119904 119910 119904 119911 119904 )
119904 = 1199050
1199051
119903(119905) 119889119905 = 1199050
1199051
( 119909)2 + ( 119910)2 + ( 119911)2119889119905
HELIX
Standard parametrization 119903 119905 = (cos 119905 sin 119905 2119905)
Arc-length parametrization 119903 119904 = (cos119904
5 sin
119904
52119904
5)
119903(119904) = 1
P
P
WHAT IS THE DIFFERENCE
Curvature κ = 1
119877
R is the radius of the circle which gives the best approximation of the curve near
the point
Circle of radius R
R
κ =1R
Line
κ = 0
Helix
κ =
κ is the measure of the rate of change of tangent vector at a
point as we travel along the curve
κ(s)= 119903(119904)
COMPUTING THE CURVATURE
Arc-length parametrization can be tedious
Twisted Cubic
119903 119905 = (119905 1199052 1199053)
119904 = 01199051 + 41199062 + 91199064 119889119906 =
κ= 119903 119905 times 119903(119905)
119903(119905) 3
T
N
B = T x N
119879 = κ 119873
119873 = minusκ T + τ 119861
119861 = minus 120591 119873
SERRE-FRENET FRAME
119903(119904)
T
N
B
Pitch axis
Roll axisYaw axis
Cell Shape Dynamics From Waves to Migration
Meghan K Driscoll1 Colin McCann12 Rael Kopace1 Tess Homan1 John T Fourkas34 Carole Parent2 and Wolfgang Losert PLOS 2012
Curvature in the study of wave-like characteristics of
amoeba migration
PARAMETRIC SURFACES
A surface M in space is a 2 dimensional object usually given parametrically
119903 = (119909 119906 119907 119910 119906 119907 119911 119906 119907 )
u
v
r = (119940119952119956 120637 119956119946119951 120651 119956119946119951 120637 119956119946119951 120651 119940119952119956 120651 )
SPHERE
FROM NON-LINEAR TO LINEAR
119951
119955119958
119955119959
119955 = (119961 119958 119959 119962 119958 119959 119963 119958 119959 )
119955119958 = (119961119958 119962119958 119963119958)
119955119959 = (119961119959 119962119959 119963119959)
119951 =119955119958times119955119959
119955119958times119955119959
P
T119875119872M
119951(119927)
Gauss Map
119918119924 rarr 119930120784
119927 rarr 119951(119927)M
Shape operator is the negative of the derivative of the Gauss map
CURVATURE FOR SURFACES
Gauss Curvature K = 120639120783 sdot 120639120784
Mean Curvature H =120639120783+120639120784
2
Gauss and Mean curvatures
are determinant and half of
the trace of the shape
operator
120639120783 120639120784
120639120783 = 119950119946119951119946119950119958119950
120639120784 = 119950119938119961119946119950119958119950
M
Two dimensional creatures cannot compute 120639120783 and 120639120784 using infinitesimal
ruler and protractor BUT they can determine K = 120639120783 sdot 120639120784 This means 2D
creatures can determine the shape of their world without stepping out to
3rd dimension
GAUSSrsquoS THEOREM EGREGIUM
119870 = minus1
2 119864119866(120597
120597119907
119864119907
119864119866+
120597
120597119906
119866119906
119864119866)
119916 = 119955119958 ∙ 119955119958
G = 119955119959 ∙ 119955119959
Rotating moving or bending the surface does not change the Gauss
curvature but stretching or breaking does
Two surfaces with the same Gauss curvature are ldquolocallyrdquo the same but not
globally
K=0 K=0
VARIOUS SURFACES
CURVATURE IN ARCHITECTURE AND DESIGN
MINIMAL SURFACES
A surface M is minimal if 119815 =120639120783+120639120784
2= 0
Any planar surface is minimal (NOT INTERESTING)
A Gyroid (VERY INTERESTING)
Gyroid structures are found in certain
surfactant or lipid mesophases and block
copolymers
THEOREM Every soap film is a physical model of a minimal surface
The interpretation of the Costa-Hoffman-Meeks minimal surface as
insertion of multiple directional holes connecting the top to the water
and the water at the bottom to the sky provided a single gesture
combining all aspects -Tobias Walliser
Image LAVA
COSTA-HOFFMAN-MEEK SURFACE
CONCEPT OF A LINE
y=mx+b
slope-intercept
119955 119957 = (119957119950119957 + 119939)
parametric
In general a line in space is given by 119903 119905 = 119875 + 119905119906 So 119903 = 0
P119958
119951
V
120629119958V
119906
The covariant derivative 120629119958V of vector field V
is the projection of the change of vector field in
119906 direction onto the tangent plane
120572(119905)
A curve 120572(119905) on the surface is called a
ldquoGeodesicrdquo if 120571 prop prop = 0
Geodesics are the ldquolinesrdquo of curved spaces
V is parallel along a curve 120572(119905) if
120629 120630119829 = 120782
Flat
Curved
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
a
b
c
c
b
A
B
C
CURVES IN 3D
Given parametrically 119903 119905 = (119909 119905 119910 119905 119911 119905 )
119903(1199050)
119903(1199051)
119903(119905)
Parametrization by arc-length 119903 119904 = (119909 119904 119910 119904 119911 119904 )
119904 = 1199050
1199051
119903(119905) 119889119905 = 1199050
1199051
( 119909)2 + ( 119910)2 + ( 119911)2119889119905
HELIX
Standard parametrization 119903 119905 = (cos 119905 sin 119905 2119905)
Arc-length parametrization 119903 119904 = (cos119904
5 sin
119904
52119904
5)
119903(119904) = 1
P
P
WHAT IS THE DIFFERENCE
Curvature κ = 1
119877
R is the radius of the circle which gives the best approximation of the curve near
the point
Circle of radius R
R
κ =1R
Line
κ = 0
Helix
κ =
κ is the measure of the rate of change of tangent vector at a
point as we travel along the curve
κ(s)= 119903(119904)
COMPUTING THE CURVATURE
Arc-length parametrization can be tedious
Twisted Cubic
119903 119905 = (119905 1199052 1199053)
119904 = 01199051 + 41199062 + 91199064 119889119906 =
κ= 119903 119905 times 119903(119905)
119903(119905) 3
T
N
B = T x N
119879 = κ 119873
119873 = minusκ T + τ 119861
119861 = minus 120591 119873
SERRE-FRENET FRAME
119903(119904)
T
N
B
Pitch axis
Roll axisYaw axis
Cell Shape Dynamics From Waves to Migration
Meghan K Driscoll1 Colin McCann12 Rael Kopace1 Tess Homan1 John T Fourkas34 Carole Parent2 and Wolfgang Losert PLOS 2012
Curvature in the study of wave-like characteristics of
amoeba migration
PARAMETRIC SURFACES
A surface M in space is a 2 dimensional object usually given parametrically
119903 = (119909 119906 119907 119910 119906 119907 119911 119906 119907 )
u
v
r = (119940119952119956 120637 119956119946119951 120651 119956119946119951 120637 119956119946119951 120651 119940119952119956 120651 )
SPHERE
FROM NON-LINEAR TO LINEAR
119951
119955119958
119955119959
119955 = (119961 119958 119959 119962 119958 119959 119963 119958 119959 )
119955119958 = (119961119958 119962119958 119963119958)
119955119959 = (119961119959 119962119959 119963119959)
119951 =119955119958times119955119959
119955119958times119955119959
P
T119875119872M
119951(119927)
Gauss Map
119918119924 rarr 119930120784
119927 rarr 119951(119927)M
Shape operator is the negative of the derivative of the Gauss map
CURVATURE FOR SURFACES
Gauss Curvature K = 120639120783 sdot 120639120784
Mean Curvature H =120639120783+120639120784
2
Gauss and Mean curvatures
are determinant and half of
the trace of the shape
operator
120639120783 120639120784
120639120783 = 119950119946119951119946119950119958119950
120639120784 = 119950119938119961119946119950119958119950
M
Two dimensional creatures cannot compute 120639120783 and 120639120784 using infinitesimal
ruler and protractor BUT they can determine K = 120639120783 sdot 120639120784 This means 2D
creatures can determine the shape of their world without stepping out to
3rd dimension
GAUSSrsquoS THEOREM EGREGIUM
119870 = minus1
2 119864119866(120597
120597119907
119864119907
119864119866+
120597
120597119906
119866119906
119864119866)
119916 = 119955119958 ∙ 119955119958
G = 119955119959 ∙ 119955119959
Rotating moving or bending the surface does not change the Gauss
curvature but stretching or breaking does
Two surfaces with the same Gauss curvature are ldquolocallyrdquo the same but not
globally
K=0 K=0
VARIOUS SURFACES
CURVATURE IN ARCHITECTURE AND DESIGN
MINIMAL SURFACES
A surface M is minimal if 119815 =120639120783+120639120784
2= 0
Any planar surface is minimal (NOT INTERESTING)
A Gyroid (VERY INTERESTING)
Gyroid structures are found in certain
surfactant or lipid mesophases and block
copolymers
THEOREM Every soap film is a physical model of a minimal surface
The interpretation of the Costa-Hoffman-Meeks minimal surface as
insertion of multiple directional holes connecting the top to the water
and the water at the bottom to the sky provided a single gesture
combining all aspects -Tobias Walliser
Image LAVA
COSTA-HOFFMAN-MEEK SURFACE
CONCEPT OF A LINE
y=mx+b
slope-intercept
119955 119957 = (119957119950119957 + 119939)
parametric
In general a line in space is given by 119903 119905 = 119875 + 119905119906 So 119903 = 0
P119958
119951
V
120629119958V
119906
The covariant derivative 120629119958V of vector field V
is the projection of the change of vector field in
119906 direction onto the tangent plane
120572(119905)
A curve 120572(119905) on the surface is called a
ldquoGeodesicrdquo if 120571 prop prop = 0
Geodesics are the ldquolinesrdquo of curved spaces
V is parallel along a curve 120572(119905) if
120629 120630119829 = 120782
Flat
Curved
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
CURVES IN 3D
Given parametrically 119903 119905 = (119909 119905 119910 119905 119911 119905 )
119903(1199050)
119903(1199051)
119903(119905)
Parametrization by arc-length 119903 119904 = (119909 119904 119910 119904 119911 119904 )
119904 = 1199050
1199051
119903(119905) 119889119905 = 1199050
1199051
( 119909)2 + ( 119910)2 + ( 119911)2119889119905
HELIX
Standard parametrization 119903 119905 = (cos 119905 sin 119905 2119905)
Arc-length parametrization 119903 119904 = (cos119904
5 sin
119904
52119904
5)
119903(119904) = 1
P
P
WHAT IS THE DIFFERENCE
Curvature κ = 1
119877
R is the radius of the circle which gives the best approximation of the curve near
the point
Circle of radius R
R
κ =1R
Line
κ = 0
Helix
κ =
κ is the measure of the rate of change of tangent vector at a
point as we travel along the curve
κ(s)= 119903(119904)
COMPUTING THE CURVATURE
Arc-length parametrization can be tedious
Twisted Cubic
119903 119905 = (119905 1199052 1199053)
119904 = 01199051 + 41199062 + 91199064 119889119906 =
κ= 119903 119905 times 119903(119905)
119903(119905) 3
T
N
B = T x N
119879 = κ 119873
119873 = minusκ T + τ 119861
119861 = minus 120591 119873
SERRE-FRENET FRAME
119903(119904)
T
N
B
Pitch axis
Roll axisYaw axis
Cell Shape Dynamics From Waves to Migration
Meghan K Driscoll1 Colin McCann12 Rael Kopace1 Tess Homan1 John T Fourkas34 Carole Parent2 and Wolfgang Losert PLOS 2012
Curvature in the study of wave-like characteristics of
amoeba migration
PARAMETRIC SURFACES
A surface M in space is a 2 dimensional object usually given parametrically
119903 = (119909 119906 119907 119910 119906 119907 119911 119906 119907 )
u
v
r = (119940119952119956 120637 119956119946119951 120651 119956119946119951 120637 119956119946119951 120651 119940119952119956 120651 )
SPHERE
FROM NON-LINEAR TO LINEAR
119951
119955119958
119955119959
119955 = (119961 119958 119959 119962 119958 119959 119963 119958 119959 )
119955119958 = (119961119958 119962119958 119963119958)
119955119959 = (119961119959 119962119959 119963119959)
119951 =119955119958times119955119959
119955119958times119955119959
P
T119875119872M
119951(119927)
Gauss Map
119918119924 rarr 119930120784
119927 rarr 119951(119927)M
Shape operator is the negative of the derivative of the Gauss map
CURVATURE FOR SURFACES
Gauss Curvature K = 120639120783 sdot 120639120784
Mean Curvature H =120639120783+120639120784
2
Gauss and Mean curvatures
are determinant and half of
the trace of the shape
operator
120639120783 120639120784
120639120783 = 119950119946119951119946119950119958119950
120639120784 = 119950119938119961119946119950119958119950
M
Two dimensional creatures cannot compute 120639120783 and 120639120784 using infinitesimal
ruler and protractor BUT they can determine K = 120639120783 sdot 120639120784 This means 2D
creatures can determine the shape of their world without stepping out to
3rd dimension
GAUSSrsquoS THEOREM EGREGIUM
119870 = minus1
2 119864119866(120597
120597119907
119864119907
119864119866+
120597
120597119906
119866119906
119864119866)
119916 = 119955119958 ∙ 119955119958
G = 119955119959 ∙ 119955119959
Rotating moving or bending the surface does not change the Gauss
curvature but stretching or breaking does
Two surfaces with the same Gauss curvature are ldquolocallyrdquo the same but not
globally
K=0 K=0
VARIOUS SURFACES
CURVATURE IN ARCHITECTURE AND DESIGN
MINIMAL SURFACES
A surface M is minimal if 119815 =120639120783+120639120784
2= 0
Any planar surface is minimal (NOT INTERESTING)
A Gyroid (VERY INTERESTING)
Gyroid structures are found in certain
surfactant or lipid mesophases and block
copolymers
THEOREM Every soap film is a physical model of a minimal surface
The interpretation of the Costa-Hoffman-Meeks minimal surface as
insertion of multiple directional holes connecting the top to the water
and the water at the bottom to the sky provided a single gesture
combining all aspects -Tobias Walliser
Image LAVA
COSTA-HOFFMAN-MEEK SURFACE
CONCEPT OF A LINE
y=mx+b
slope-intercept
119955 119957 = (119957119950119957 + 119939)
parametric
In general a line in space is given by 119903 119905 = 119875 + 119905119906 So 119903 = 0
P119958
119951
V
120629119958V
119906
The covariant derivative 120629119958V of vector field V
is the projection of the change of vector field in
119906 direction onto the tangent plane
120572(119905)
A curve 120572(119905) on the surface is called a
ldquoGeodesicrdquo if 120571 prop prop = 0
Geodesics are the ldquolinesrdquo of curved spaces
V is parallel along a curve 120572(119905) if
120629 120630119829 = 120782
Flat
Curved
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
HELIX
Standard parametrization 119903 119905 = (cos 119905 sin 119905 2119905)
Arc-length parametrization 119903 119904 = (cos119904
5 sin
119904
52119904
5)
119903(119904) = 1
P
P
WHAT IS THE DIFFERENCE
Curvature κ = 1
119877
R is the radius of the circle which gives the best approximation of the curve near
the point
Circle of radius R
R
κ =1R
Line
κ = 0
Helix
κ =
κ is the measure of the rate of change of tangent vector at a
point as we travel along the curve
κ(s)= 119903(119904)
COMPUTING THE CURVATURE
Arc-length parametrization can be tedious
Twisted Cubic
119903 119905 = (119905 1199052 1199053)
119904 = 01199051 + 41199062 + 91199064 119889119906 =
κ= 119903 119905 times 119903(119905)
119903(119905) 3
T
N
B = T x N
119879 = κ 119873
119873 = minusκ T + τ 119861
119861 = minus 120591 119873
SERRE-FRENET FRAME
119903(119904)
T
N
B
Pitch axis
Roll axisYaw axis
Cell Shape Dynamics From Waves to Migration
Meghan K Driscoll1 Colin McCann12 Rael Kopace1 Tess Homan1 John T Fourkas34 Carole Parent2 and Wolfgang Losert PLOS 2012
Curvature in the study of wave-like characteristics of
amoeba migration
PARAMETRIC SURFACES
A surface M in space is a 2 dimensional object usually given parametrically
119903 = (119909 119906 119907 119910 119906 119907 119911 119906 119907 )
u
v
r = (119940119952119956 120637 119956119946119951 120651 119956119946119951 120637 119956119946119951 120651 119940119952119956 120651 )
SPHERE
FROM NON-LINEAR TO LINEAR
119951
119955119958
119955119959
119955 = (119961 119958 119959 119962 119958 119959 119963 119958 119959 )
119955119958 = (119961119958 119962119958 119963119958)
119955119959 = (119961119959 119962119959 119963119959)
119951 =119955119958times119955119959
119955119958times119955119959
P
T119875119872M
119951(119927)
Gauss Map
119918119924 rarr 119930120784
119927 rarr 119951(119927)M
Shape operator is the negative of the derivative of the Gauss map
CURVATURE FOR SURFACES
Gauss Curvature K = 120639120783 sdot 120639120784
Mean Curvature H =120639120783+120639120784
2
Gauss and Mean curvatures
are determinant and half of
the trace of the shape
operator
120639120783 120639120784
120639120783 = 119950119946119951119946119950119958119950
120639120784 = 119950119938119961119946119950119958119950
M
Two dimensional creatures cannot compute 120639120783 and 120639120784 using infinitesimal
ruler and protractor BUT they can determine K = 120639120783 sdot 120639120784 This means 2D
creatures can determine the shape of their world without stepping out to
3rd dimension
GAUSSrsquoS THEOREM EGREGIUM
119870 = minus1
2 119864119866(120597
120597119907
119864119907
119864119866+
120597
120597119906
119866119906
119864119866)
119916 = 119955119958 ∙ 119955119958
G = 119955119959 ∙ 119955119959
Rotating moving or bending the surface does not change the Gauss
curvature but stretching or breaking does
Two surfaces with the same Gauss curvature are ldquolocallyrdquo the same but not
globally
K=0 K=0
VARIOUS SURFACES
CURVATURE IN ARCHITECTURE AND DESIGN
MINIMAL SURFACES
A surface M is minimal if 119815 =120639120783+120639120784
2= 0
Any planar surface is minimal (NOT INTERESTING)
A Gyroid (VERY INTERESTING)
Gyroid structures are found in certain
surfactant or lipid mesophases and block
copolymers
THEOREM Every soap film is a physical model of a minimal surface
The interpretation of the Costa-Hoffman-Meeks minimal surface as
insertion of multiple directional holes connecting the top to the water
and the water at the bottom to the sky provided a single gesture
combining all aspects -Tobias Walliser
Image LAVA
COSTA-HOFFMAN-MEEK SURFACE
CONCEPT OF A LINE
y=mx+b
slope-intercept
119955 119957 = (119957119950119957 + 119939)
parametric
In general a line in space is given by 119903 119905 = 119875 + 119905119906 So 119903 = 0
P119958
119951
V
120629119958V
119906
The covariant derivative 120629119958V of vector field V
is the projection of the change of vector field in
119906 direction onto the tangent plane
120572(119905)
A curve 120572(119905) on the surface is called a
ldquoGeodesicrdquo if 120571 prop prop = 0
Geodesics are the ldquolinesrdquo of curved spaces
V is parallel along a curve 120572(119905) if
120629 120630119829 = 120782
Flat
Curved
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
P
P
WHAT IS THE DIFFERENCE
Curvature κ = 1
119877
R is the radius of the circle which gives the best approximation of the curve near
the point
Circle of radius R
R
κ =1R
Line
κ = 0
Helix
κ =
κ is the measure of the rate of change of tangent vector at a
point as we travel along the curve
κ(s)= 119903(119904)
COMPUTING THE CURVATURE
Arc-length parametrization can be tedious
Twisted Cubic
119903 119905 = (119905 1199052 1199053)
119904 = 01199051 + 41199062 + 91199064 119889119906 =
κ= 119903 119905 times 119903(119905)
119903(119905) 3
T
N
B = T x N
119879 = κ 119873
119873 = minusκ T + τ 119861
119861 = minus 120591 119873
SERRE-FRENET FRAME
119903(119904)
T
N
B
Pitch axis
Roll axisYaw axis
Cell Shape Dynamics From Waves to Migration
Meghan K Driscoll1 Colin McCann12 Rael Kopace1 Tess Homan1 John T Fourkas34 Carole Parent2 and Wolfgang Losert PLOS 2012
Curvature in the study of wave-like characteristics of
amoeba migration
PARAMETRIC SURFACES
A surface M in space is a 2 dimensional object usually given parametrically
119903 = (119909 119906 119907 119910 119906 119907 119911 119906 119907 )
u
v
r = (119940119952119956 120637 119956119946119951 120651 119956119946119951 120637 119956119946119951 120651 119940119952119956 120651 )
SPHERE
FROM NON-LINEAR TO LINEAR
119951
119955119958
119955119959
119955 = (119961 119958 119959 119962 119958 119959 119963 119958 119959 )
119955119958 = (119961119958 119962119958 119963119958)
119955119959 = (119961119959 119962119959 119963119959)
119951 =119955119958times119955119959
119955119958times119955119959
P
T119875119872M
119951(119927)
Gauss Map
119918119924 rarr 119930120784
119927 rarr 119951(119927)M
Shape operator is the negative of the derivative of the Gauss map
CURVATURE FOR SURFACES
Gauss Curvature K = 120639120783 sdot 120639120784
Mean Curvature H =120639120783+120639120784
2
Gauss and Mean curvatures
are determinant and half of
the trace of the shape
operator
120639120783 120639120784
120639120783 = 119950119946119951119946119950119958119950
120639120784 = 119950119938119961119946119950119958119950
M
Two dimensional creatures cannot compute 120639120783 and 120639120784 using infinitesimal
ruler and protractor BUT they can determine K = 120639120783 sdot 120639120784 This means 2D
creatures can determine the shape of their world without stepping out to
3rd dimension
GAUSSrsquoS THEOREM EGREGIUM
119870 = minus1
2 119864119866(120597
120597119907
119864119907
119864119866+
120597
120597119906
119866119906
119864119866)
119916 = 119955119958 ∙ 119955119958
G = 119955119959 ∙ 119955119959
Rotating moving or bending the surface does not change the Gauss
curvature but stretching or breaking does
Two surfaces with the same Gauss curvature are ldquolocallyrdquo the same but not
globally
K=0 K=0
VARIOUS SURFACES
CURVATURE IN ARCHITECTURE AND DESIGN
MINIMAL SURFACES
A surface M is minimal if 119815 =120639120783+120639120784
2= 0
Any planar surface is minimal (NOT INTERESTING)
A Gyroid (VERY INTERESTING)
Gyroid structures are found in certain
surfactant or lipid mesophases and block
copolymers
THEOREM Every soap film is a physical model of a minimal surface
The interpretation of the Costa-Hoffman-Meeks minimal surface as
insertion of multiple directional holes connecting the top to the water
and the water at the bottom to the sky provided a single gesture
combining all aspects -Tobias Walliser
Image LAVA
COSTA-HOFFMAN-MEEK SURFACE
CONCEPT OF A LINE
y=mx+b
slope-intercept
119955 119957 = (119957119950119957 + 119939)
parametric
In general a line in space is given by 119903 119905 = 119875 + 119905119906 So 119903 = 0
P119958
119951
V
120629119958V
119906
The covariant derivative 120629119958V of vector field V
is the projection of the change of vector field in
119906 direction onto the tangent plane
120572(119905)
A curve 120572(119905) on the surface is called a
ldquoGeodesicrdquo if 120571 prop prop = 0
Geodesics are the ldquolinesrdquo of curved spaces
V is parallel along a curve 120572(119905) if
120629 120630119829 = 120782
Flat
Curved
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
Circle of radius R
R
κ =1R
Line
κ = 0
Helix
κ =
κ is the measure of the rate of change of tangent vector at a
point as we travel along the curve
κ(s)= 119903(119904)
COMPUTING THE CURVATURE
Arc-length parametrization can be tedious
Twisted Cubic
119903 119905 = (119905 1199052 1199053)
119904 = 01199051 + 41199062 + 91199064 119889119906 =
κ= 119903 119905 times 119903(119905)
119903(119905) 3
T
N
B = T x N
119879 = κ 119873
119873 = minusκ T + τ 119861
119861 = minus 120591 119873
SERRE-FRENET FRAME
119903(119904)
T
N
B
Pitch axis
Roll axisYaw axis
Cell Shape Dynamics From Waves to Migration
Meghan K Driscoll1 Colin McCann12 Rael Kopace1 Tess Homan1 John T Fourkas34 Carole Parent2 and Wolfgang Losert PLOS 2012
Curvature in the study of wave-like characteristics of
amoeba migration
PARAMETRIC SURFACES
A surface M in space is a 2 dimensional object usually given parametrically
119903 = (119909 119906 119907 119910 119906 119907 119911 119906 119907 )
u
v
r = (119940119952119956 120637 119956119946119951 120651 119956119946119951 120637 119956119946119951 120651 119940119952119956 120651 )
SPHERE
FROM NON-LINEAR TO LINEAR
119951
119955119958
119955119959
119955 = (119961 119958 119959 119962 119958 119959 119963 119958 119959 )
119955119958 = (119961119958 119962119958 119963119958)
119955119959 = (119961119959 119962119959 119963119959)
119951 =119955119958times119955119959
119955119958times119955119959
P
T119875119872M
119951(119927)
Gauss Map
119918119924 rarr 119930120784
119927 rarr 119951(119927)M
Shape operator is the negative of the derivative of the Gauss map
CURVATURE FOR SURFACES
Gauss Curvature K = 120639120783 sdot 120639120784
Mean Curvature H =120639120783+120639120784
2
Gauss and Mean curvatures
are determinant and half of
the trace of the shape
operator
120639120783 120639120784
120639120783 = 119950119946119951119946119950119958119950
120639120784 = 119950119938119961119946119950119958119950
M
Two dimensional creatures cannot compute 120639120783 and 120639120784 using infinitesimal
ruler and protractor BUT they can determine K = 120639120783 sdot 120639120784 This means 2D
creatures can determine the shape of their world without stepping out to
3rd dimension
GAUSSrsquoS THEOREM EGREGIUM
119870 = minus1
2 119864119866(120597
120597119907
119864119907
119864119866+
120597
120597119906
119866119906
119864119866)
119916 = 119955119958 ∙ 119955119958
G = 119955119959 ∙ 119955119959
Rotating moving or bending the surface does not change the Gauss
curvature but stretching or breaking does
Two surfaces with the same Gauss curvature are ldquolocallyrdquo the same but not
globally
K=0 K=0
VARIOUS SURFACES
CURVATURE IN ARCHITECTURE AND DESIGN
MINIMAL SURFACES
A surface M is minimal if 119815 =120639120783+120639120784
2= 0
Any planar surface is minimal (NOT INTERESTING)
A Gyroid (VERY INTERESTING)
Gyroid structures are found in certain
surfactant or lipid mesophases and block
copolymers
THEOREM Every soap film is a physical model of a minimal surface
The interpretation of the Costa-Hoffman-Meeks minimal surface as
insertion of multiple directional holes connecting the top to the water
and the water at the bottom to the sky provided a single gesture
combining all aspects -Tobias Walliser
Image LAVA
COSTA-HOFFMAN-MEEK SURFACE
CONCEPT OF A LINE
y=mx+b
slope-intercept
119955 119957 = (119957119950119957 + 119939)
parametric
In general a line in space is given by 119903 119905 = 119875 + 119905119906 So 119903 = 0
P119958
119951
V
120629119958V
119906
The covariant derivative 120629119958V of vector field V
is the projection of the change of vector field in
119906 direction onto the tangent plane
120572(119905)
A curve 120572(119905) on the surface is called a
ldquoGeodesicrdquo if 120571 prop prop = 0
Geodesics are the ldquolinesrdquo of curved spaces
V is parallel along a curve 120572(119905) if
120629 120630119829 = 120782
Flat
Curved
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
COMPUTING THE CURVATURE
Arc-length parametrization can be tedious
Twisted Cubic
119903 119905 = (119905 1199052 1199053)
119904 = 01199051 + 41199062 + 91199064 119889119906 =
κ= 119903 119905 times 119903(119905)
119903(119905) 3
T
N
B = T x N
119879 = κ 119873
119873 = minusκ T + τ 119861
119861 = minus 120591 119873
SERRE-FRENET FRAME
119903(119904)
T
N
B
Pitch axis
Roll axisYaw axis
Cell Shape Dynamics From Waves to Migration
Meghan K Driscoll1 Colin McCann12 Rael Kopace1 Tess Homan1 John T Fourkas34 Carole Parent2 and Wolfgang Losert PLOS 2012
Curvature in the study of wave-like characteristics of
amoeba migration
PARAMETRIC SURFACES
A surface M in space is a 2 dimensional object usually given parametrically
119903 = (119909 119906 119907 119910 119906 119907 119911 119906 119907 )
u
v
r = (119940119952119956 120637 119956119946119951 120651 119956119946119951 120637 119956119946119951 120651 119940119952119956 120651 )
SPHERE
FROM NON-LINEAR TO LINEAR
119951
119955119958
119955119959
119955 = (119961 119958 119959 119962 119958 119959 119963 119958 119959 )
119955119958 = (119961119958 119962119958 119963119958)
119955119959 = (119961119959 119962119959 119963119959)
119951 =119955119958times119955119959
119955119958times119955119959
P
T119875119872M
119951(119927)
Gauss Map
119918119924 rarr 119930120784
119927 rarr 119951(119927)M
Shape operator is the negative of the derivative of the Gauss map
CURVATURE FOR SURFACES
Gauss Curvature K = 120639120783 sdot 120639120784
Mean Curvature H =120639120783+120639120784
2
Gauss and Mean curvatures
are determinant and half of
the trace of the shape
operator
120639120783 120639120784
120639120783 = 119950119946119951119946119950119958119950
120639120784 = 119950119938119961119946119950119958119950
M
Two dimensional creatures cannot compute 120639120783 and 120639120784 using infinitesimal
ruler and protractor BUT they can determine K = 120639120783 sdot 120639120784 This means 2D
creatures can determine the shape of their world without stepping out to
3rd dimension
GAUSSrsquoS THEOREM EGREGIUM
119870 = minus1
2 119864119866(120597
120597119907
119864119907
119864119866+
120597
120597119906
119866119906
119864119866)
119916 = 119955119958 ∙ 119955119958
G = 119955119959 ∙ 119955119959
Rotating moving or bending the surface does not change the Gauss
curvature but stretching or breaking does
Two surfaces with the same Gauss curvature are ldquolocallyrdquo the same but not
globally
K=0 K=0
VARIOUS SURFACES
CURVATURE IN ARCHITECTURE AND DESIGN
MINIMAL SURFACES
A surface M is minimal if 119815 =120639120783+120639120784
2= 0
Any planar surface is minimal (NOT INTERESTING)
A Gyroid (VERY INTERESTING)
Gyroid structures are found in certain
surfactant or lipid mesophases and block
copolymers
THEOREM Every soap film is a physical model of a minimal surface
The interpretation of the Costa-Hoffman-Meeks minimal surface as
insertion of multiple directional holes connecting the top to the water
and the water at the bottom to the sky provided a single gesture
combining all aspects -Tobias Walliser
Image LAVA
COSTA-HOFFMAN-MEEK SURFACE
CONCEPT OF A LINE
y=mx+b
slope-intercept
119955 119957 = (119957119950119957 + 119939)
parametric
In general a line in space is given by 119903 119905 = 119875 + 119905119906 So 119903 = 0
P119958
119951
V
120629119958V
119906
The covariant derivative 120629119958V of vector field V
is the projection of the change of vector field in
119906 direction onto the tangent plane
120572(119905)
A curve 120572(119905) on the surface is called a
ldquoGeodesicrdquo if 120571 prop prop = 0
Geodesics are the ldquolinesrdquo of curved spaces
V is parallel along a curve 120572(119905) if
120629 120630119829 = 120782
Flat
Curved
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
T
N
B = T x N
119879 = κ 119873
119873 = minusκ T + τ 119861
119861 = minus 120591 119873
SERRE-FRENET FRAME
119903(119904)
T
N
B
Pitch axis
Roll axisYaw axis
Cell Shape Dynamics From Waves to Migration
Meghan K Driscoll1 Colin McCann12 Rael Kopace1 Tess Homan1 John T Fourkas34 Carole Parent2 and Wolfgang Losert PLOS 2012
Curvature in the study of wave-like characteristics of
amoeba migration
PARAMETRIC SURFACES
A surface M in space is a 2 dimensional object usually given parametrically
119903 = (119909 119906 119907 119910 119906 119907 119911 119906 119907 )
u
v
r = (119940119952119956 120637 119956119946119951 120651 119956119946119951 120637 119956119946119951 120651 119940119952119956 120651 )
SPHERE
FROM NON-LINEAR TO LINEAR
119951
119955119958
119955119959
119955 = (119961 119958 119959 119962 119958 119959 119963 119958 119959 )
119955119958 = (119961119958 119962119958 119963119958)
119955119959 = (119961119959 119962119959 119963119959)
119951 =119955119958times119955119959
119955119958times119955119959
P
T119875119872M
119951(119927)
Gauss Map
119918119924 rarr 119930120784
119927 rarr 119951(119927)M
Shape operator is the negative of the derivative of the Gauss map
CURVATURE FOR SURFACES
Gauss Curvature K = 120639120783 sdot 120639120784
Mean Curvature H =120639120783+120639120784
2
Gauss and Mean curvatures
are determinant and half of
the trace of the shape
operator
120639120783 120639120784
120639120783 = 119950119946119951119946119950119958119950
120639120784 = 119950119938119961119946119950119958119950
M
Two dimensional creatures cannot compute 120639120783 and 120639120784 using infinitesimal
ruler and protractor BUT they can determine K = 120639120783 sdot 120639120784 This means 2D
creatures can determine the shape of their world without stepping out to
3rd dimension
GAUSSrsquoS THEOREM EGREGIUM
119870 = minus1
2 119864119866(120597
120597119907
119864119907
119864119866+
120597
120597119906
119866119906
119864119866)
119916 = 119955119958 ∙ 119955119958
G = 119955119959 ∙ 119955119959
Rotating moving or bending the surface does not change the Gauss
curvature but stretching or breaking does
Two surfaces with the same Gauss curvature are ldquolocallyrdquo the same but not
globally
K=0 K=0
VARIOUS SURFACES
CURVATURE IN ARCHITECTURE AND DESIGN
MINIMAL SURFACES
A surface M is minimal if 119815 =120639120783+120639120784
2= 0
Any planar surface is minimal (NOT INTERESTING)
A Gyroid (VERY INTERESTING)
Gyroid structures are found in certain
surfactant or lipid mesophases and block
copolymers
THEOREM Every soap film is a physical model of a minimal surface
The interpretation of the Costa-Hoffman-Meeks minimal surface as
insertion of multiple directional holes connecting the top to the water
and the water at the bottom to the sky provided a single gesture
combining all aspects -Tobias Walliser
Image LAVA
COSTA-HOFFMAN-MEEK SURFACE
CONCEPT OF A LINE
y=mx+b
slope-intercept
119955 119957 = (119957119950119957 + 119939)
parametric
In general a line in space is given by 119903 119905 = 119875 + 119905119906 So 119903 = 0
P119958
119951
V
120629119958V
119906
The covariant derivative 120629119958V of vector field V
is the projection of the change of vector field in
119906 direction onto the tangent plane
120572(119905)
A curve 120572(119905) on the surface is called a
ldquoGeodesicrdquo if 120571 prop prop = 0
Geodesics are the ldquolinesrdquo of curved spaces
V is parallel along a curve 120572(119905) if
120629 120630119829 = 120782
Flat
Curved
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
T
N
B
Pitch axis
Roll axisYaw axis
Cell Shape Dynamics From Waves to Migration
Meghan K Driscoll1 Colin McCann12 Rael Kopace1 Tess Homan1 John T Fourkas34 Carole Parent2 and Wolfgang Losert PLOS 2012
Curvature in the study of wave-like characteristics of
amoeba migration
PARAMETRIC SURFACES
A surface M in space is a 2 dimensional object usually given parametrically
119903 = (119909 119906 119907 119910 119906 119907 119911 119906 119907 )
u
v
r = (119940119952119956 120637 119956119946119951 120651 119956119946119951 120637 119956119946119951 120651 119940119952119956 120651 )
SPHERE
FROM NON-LINEAR TO LINEAR
119951
119955119958
119955119959
119955 = (119961 119958 119959 119962 119958 119959 119963 119958 119959 )
119955119958 = (119961119958 119962119958 119963119958)
119955119959 = (119961119959 119962119959 119963119959)
119951 =119955119958times119955119959
119955119958times119955119959
P
T119875119872M
119951(119927)
Gauss Map
119918119924 rarr 119930120784
119927 rarr 119951(119927)M
Shape operator is the negative of the derivative of the Gauss map
CURVATURE FOR SURFACES
Gauss Curvature K = 120639120783 sdot 120639120784
Mean Curvature H =120639120783+120639120784
2
Gauss and Mean curvatures
are determinant and half of
the trace of the shape
operator
120639120783 120639120784
120639120783 = 119950119946119951119946119950119958119950
120639120784 = 119950119938119961119946119950119958119950
M
Two dimensional creatures cannot compute 120639120783 and 120639120784 using infinitesimal
ruler and protractor BUT they can determine K = 120639120783 sdot 120639120784 This means 2D
creatures can determine the shape of their world without stepping out to
3rd dimension
GAUSSrsquoS THEOREM EGREGIUM
119870 = minus1
2 119864119866(120597
120597119907
119864119907
119864119866+
120597
120597119906
119866119906
119864119866)
119916 = 119955119958 ∙ 119955119958
G = 119955119959 ∙ 119955119959
Rotating moving or bending the surface does not change the Gauss
curvature but stretching or breaking does
Two surfaces with the same Gauss curvature are ldquolocallyrdquo the same but not
globally
K=0 K=0
VARIOUS SURFACES
CURVATURE IN ARCHITECTURE AND DESIGN
MINIMAL SURFACES
A surface M is minimal if 119815 =120639120783+120639120784
2= 0
Any planar surface is minimal (NOT INTERESTING)
A Gyroid (VERY INTERESTING)
Gyroid structures are found in certain
surfactant or lipid mesophases and block
copolymers
THEOREM Every soap film is a physical model of a minimal surface
The interpretation of the Costa-Hoffman-Meeks minimal surface as
insertion of multiple directional holes connecting the top to the water
and the water at the bottom to the sky provided a single gesture
combining all aspects -Tobias Walliser
Image LAVA
COSTA-HOFFMAN-MEEK SURFACE
CONCEPT OF A LINE
y=mx+b
slope-intercept
119955 119957 = (119957119950119957 + 119939)
parametric
In general a line in space is given by 119903 119905 = 119875 + 119905119906 So 119903 = 0
P119958
119951
V
120629119958V
119906
The covariant derivative 120629119958V of vector field V
is the projection of the change of vector field in
119906 direction onto the tangent plane
120572(119905)
A curve 120572(119905) on the surface is called a
ldquoGeodesicrdquo if 120571 prop prop = 0
Geodesics are the ldquolinesrdquo of curved spaces
V is parallel along a curve 120572(119905) if
120629 120630119829 = 120782
Flat
Curved
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
Cell Shape Dynamics From Waves to Migration
Meghan K Driscoll1 Colin McCann12 Rael Kopace1 Tess Homan1 John T Fourkas34 Carole Parent2 and Wolfgang Losert PLOS 2012
Curvature in the study of wave-like characteristics of
amoeba migration
PARAMETRIC SURFACES
A surface M in space is a 2 dimensional object usually given parametrically
119903 = (119909 119906 119907 119910 119906 119907 119911 119906 119907 )
u
v
r = (119940119952119956 120637 119956119946119951 120651 119956119946119951 120637 119956119946119951 120651 119940119952119956 120651 )
SPHERE
FROM NON-LINEAR TO LINEAR
119951
119955119958
119955119959
119955 = (119961 119958 119959 119962 119958 119959 119963 119958 119959 )
119955119958 = (119961119958 119962119958 119963119958)
119955119959 = (119961119959 119962119959 119963119959)
119951 =119955119958times119955119959
119955119958times119955119959
P
T119875119872M
119951(119927)
Gauss Map
119918119924 rarr 119930120784
119927 rarr 119951(119927)M
Shape operator is the negative of the derivative of the Gauss map
CURVATURE FOR SURFACES
Gauss Curvature K = 120639120783 sdot 120639120784
Mean Curvature H =120639120783+120639120784
2
Gauss and Mean curvatures
are determinant and half of
the trace of the shape
operator
120639120783 120639120784
120639120783 = 119950119946119951119946119950119958119950
120639120784 = 119950119938119961119946119950119958119950
M
Two dimensional creatures cannot compute 120639120783 and 120639120784 using infinitesimal
ruler and protractor BUT they can determine K = 120639120783 sdot 120639120784 This means 2D
creatures can determine the shape of their world without stepping out to
3rd dimension
GAUSSrsquoS THEOREM EGREGIUM
119870 = minus1
2 119864119866(120597
120597119907
119864119907
119864119866+
120597
120597119906
119866119906
119864119866)
119916 = 119955119958 ∙ 119955119958
G = 119955119959 ∙ 119955119959
Rotating moving or bending the surface does not change the Gauss
curvature but stretching or breaking does
Two surfaces with the same Gauss curvature are ldquolocallyrdquo the same but not
globally
K=0 K=0
VARIOUS SURFACES
CURVATURE IN ARCHITECTURE AND DESIGN
MINIMAL SURFACES
A surface M is minimal if 119815 =120639120783+120639120784
2= 0
Any planar surface is minimal (NOT INTERESTING)
A Gyroid (VERY INTERESTING)
Gyroid structures are found in certain
surfactant or lipid mesophases and block
copolymers
THEOREM Every soap film is a physical model of a minimal surface
The interpretation of the Costa-Hoffman-Meeks minimal surface as
insertion of multiple directional holes connecting the top to the water
and the water at the bottom to the sky provided a single gesture
combining all aspects -Tobias Walliser
Image LAVA
COSTA-HOFFMAN-MEEK SURFACE
CONCEPT OF A LINE
y=mx+b
slope-intercept
119955 119957 = (119957119950119957 + 119939)
parametric
In general a line in space is given by 119903 119905 = 119875 + 119905119906 So 119903 = 0
P119958
119951
V
120629119958V
119906
The covariant derivative 120629119958V of vector field V
is the projection of the change of vector field in
119906 direction onto the tangent plane
120572(119905)
A curve 120572(119905) on the surface is called a
ldquoGeodesicrdquo if 120571 prop prop = 0
Geodesics are the ldquolinesrdquo of curved spaces
V is parallel along a curve 120572(119905) if
120629 120630119829 = 120782
Flat
Curved
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
PARAMETRIC SURFACES
A surface M in space is a 2 dimensional object usually given parametrically
119903 = (119909 119906 119907 119910 119906 119907 119911 119906 119907 )
u
v
r = (119940119952119956 120637 119956119946119951 120651 119956119946119951 120637 119956119946119951 120651 119940119952119956 120651 )
SPHERE
FROM NON-LINEAR TO LINEAR
119951
119955119958
119955119959
119955 = (119961 119958 119959 119962 119958 119959 119963 119958 119959 )
119955119958 = (119961119958 119962119958 119963119958)
119955119959 = (119961119959 119962119959 119963119959)
119951 =119955119958times119955119959
119955119958times119955119959
P
T119875119872M
119951(119927)
Gauss Map
119918119924 rarr 119930120784
119927 rarr 119951(119927)M
Shape operator is the negative of the derivative of the Gauss map
CURVATURE FOR SURFACES
Gauss Curvature K = 120639120783 sdot 120639120784
Mean Curvature H =120639120783+120639120784
2
Gauss and Mean curvatures
are determinant and half of
the trace of the shape
operator
120639120783 120639120784
120639120783 = 119950119946119951119946119950119958119950
120639120784 = 119950119938119961119946119950119958119950
M
Two dimensional creatures cannot compute 120639120783 and 120639120784 using infinitesimal
ruler and protractor BUT they can determine K = 120639120783 sdot 120639120784 This means 2D
creatures can determine the shape of their world without stepping out to
3rd dimension
GAUSSrsquoS THEOREM EGREGIUM
119870 = minus1
2 119864119866(120597
120597119907
119864119907
119864119866+
120597
120597119906
119866119906
119864119866)
119916 = 119955119958 ∙ 119955119958
G = 119955119959 ∙ 119955119959
Rotating moving or bending the surface does not change the Gauss
curvature but stretching or breaking does
Two surfaces with the same Gauss curvature are ldquolocallyrdquo the same but not
globally
K=0 K=0
VARIOUS SURFACES
CURVATURE IN ARCHITECTURE AND DESIGN
MINIMAL SURFACES
A surface M is minimal if 119815 =120639120783+120639120784
2= 0
Any planar surface is minimal (NOT INTERESTING)
A Gyroid (VERY INTERESTING)
Gyroid structures are found in certain
surfactant or lipid mesophases and block
copolymers
THEOREM Every soap film is a physical model of a minimal surface
The interpretation of the Costa-Hoffman-Meeks minimal surface as
insertion of multiple directional holes connecting the top to the water
and the water at the bottom to the sky provided a single gesture
combining all aspects -Tobias Walliser
Image LAVA
COSTA-HOFFMAN-MEEK SURFACE
CONCEPT OF A LINE
y=mx+b
slope-intercept
119955 119957 = (119957119950119957 + 119939)
parametric
In general a line in space is given by 119903 119905 = 119875 + 119905119906 So 119903 = 0
P119958
119951
V
120629119958V
119906
The covariant derivative 120629119958V of vector field V
is the projection of the change of vector field in
119906 direction onto the tangent plane
120572(119905)
A curve 120572(119905) on the surface is called a
ldquoGeodesicrdquo if 120571 prop prop = 0
Geodesics are the ldquolinesrdquo of curved spaces
V is parallel along a curve 120572(119905) if
120629 120630119829 = 120782
Flat
Curved
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
r = (119940119952119956 120637 119956119946119951 120651 119956119946119951 120637 119956119946119951 120651 119940119952119956 120651 )
SPHERE
FROM NON-LINEAR TO LINEAR
119951
119955119958
119955119959
119955 = (119961 119958 119959 119962 119958 119959 119963 119958 119959 )
119955119958 = (119961119958 119962119958 119963119958)
119955119959 = (119961119959 119962119959 119963119959)
119951 =119955119958times119955119959
119955119958times119955119959
P
T119875119872M
119951(119927)
Gauss Map
119918119924 rarr 119930120784
119927 rarr 119951(119927)M
Shape operator is the negative of the derivative of the Gauss map
CURVATURE FOR SURFACES
Gauss Curvature K = 120639120783 sdot 120639120784
Mean Curvature H =120639120783+120639120784
2
Gauss and Mean curvatures
are determinant and half of
the trace of the shape
operator
120639120783 120639120784
120639120783 = 119950119946119951119946119950119958119950
120639120784 = 119950119938119961119946119950119958119950
M
Two dimensional creatures cannot compute 120639120783 and 120639120784 using infinitesimal
ruler and protractor BUT they can determine K = 120639120783 sdot 120639120784 This means 2D
creatures can determine the shape of their world without stepping out to
3rd dimension
GAUSSrsquoS THEOREM EGREGIUM
119870 = minus1
2 119864119866(120597
120597119907
119864119907
119864119866+
120597
120597119906
119866119906
119864119866)
119916 = 119955119958 ∙ 119955119958
G = 119955119959 ∙ 119955119959
Rotating moving or bending the surface does not change the Gauss
curvature but stretching or breaking does
Two surfaces with the same Gauss curvature are ldquolocallyrdquo the same but not
globally
K=0 K=0
VARIOUS SURFACES
CURVATURE IN ARCHITECTURE AND DESIGN
MINIMAL SURFACES
A surface M is minimal if 119815 =120639120783+120639120784
2= 0
Any planar surface is minimal (NOT INTERESTING)
A Gyroid (VERY INTERESTING)
Gyroid structures are found in certain
surfactant or lipid mesophases and block
copolymers
THEOREM Every soap film is a physical model of a minimal surface
The interpretation of the Costa-Hoffman-Meeks minimal surface as
insertion of multiple directional holes connecting the top to the water
and the water at the bottom to the sky provided a single gesture
combining all aspects -Tobias Walliser
Image LAVA
COSTA-HOFFMAN-MEEK SURFACE
CONCEPT OF A LINE
y=mx+b
slope-intercept
119955 119957 = (119957119950119957 + 119939)
parametric
In general a line in space is given by 119903 119905 = 119875 + 119905119906 So 119903 = 0
P119958
119951
V
120629119958V
119906
The covariant derivative 120629119958V of vector field V
is the projection of the change of vector field in
119906 direction onto the tangent plane
120572(119905)
A curve 120572(119905) on the surface is called a
ldquoGeodesicrdquo if 120571 prop prop = 0
Geodesics are the ldquolinesrdquo of curved spaces
V is parallel along a curve 120572(119905) if
120629 120630119829 = 120782
Flat
Curved
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
FROM NON-LINEAR TO LINEAR
119951
119955119958
119955119959
119955 = (119961 119958 119959 119962 119958 119959 119963 119958 119959 )
119955119958 = (119961119958 119962119958 119963119958)
119955119959 = (119961119959 119962119959 119963119959)
119951 =119955119958times119955119959
119955119958times119955119959
P
T119875119872M
119951(119927)
Gauss Map
119918119924 rarr 119930120784
119927 rarr 119951(119927)M
Shape operator is the negative of the derivative of the Gauss map
CURVATURE FOR SURFACES
Gauss Curvature K = 120639120783 sdot 120639120784
Mean Curvature H =120639120783+120639120784
2
Gauss and Mean curvatures
are determinant and half of
the trace of the shape
operator
120639120783 120639120784
120639120783 = 119950119946119951119946119950119958119950
120639120784 = 119950119938119961119946119950119958119950
M
Two dimensional creatures cannot compute 120639120783 and 120639120784 using infinitesimal
ruler and protractor BUT they can determine K = 120639120783 sdot 120639120784 This means 2D
creatures can determine the shape of their world without stepping out to
3rd dimension
GAUSSrsquoS THEOREM EGREGIUM
119870 = minus1
2 119864119866(120597
120597119907
119864119907
119864119866+
120597
120597119906
119866119906
119864119866)
119916 = 119955119958 ∙ 119955119958
G = 119955119959 ∙ 119955119959
Rotating moving or bending the surface does not change the Gauss
curvature but stretching or breaking does
Two surfaces with the same Gauss curvature are ldquolocallyrdquo the same but not
globally
K=0 K=0
VARIOUS SURFACES
CURVATURE IN ARCHITECTURE AND DESIGN
MINIMAL SURFACES
A surface M is minimal if 119815 =120639120783+120639120784
2= 0
Any planar surface is minimal (NOT INTERESTING)
A Gyroid (VERY INTERESTING)
Gyroid structures are found in certain
surfactant or lipid mesophases and block
copolymers
THEOREM Every soap film is a physical model of a minimal surface
The interpretation of the Costa-Hoffman-Meeks minimal surface as
insertion of multiple directional holes connecting the top to the water
and the water at the bottom to the sky provided a single gesture
combining all aspects -Tobias Walliser
Image LAVA
COSTA-HOFFMAN-MEEK SURFACE
CONCEPT OF A LINE
y=mx+b
slope-intercept
119955 119957 = (119957119950119957 + 119939)
parametric
In general a line in space is given by 119903 119905 = 119875 + 119905119906 So 119903 = 0
P119958
119951
V
120629119958V
119906
The covariant derivative 120629119958V of vector field V
is the projection of the change of vector field in
119906 direction onto the tangent plane
120572(119905)
A curve 120572(119905) on the surface is called a
ldquoGeodesicrdquo if 120571 prop prop = 0
Geodesics are the ldquolinesrdquo of curved spaces
V is parallel along a curve 120572(119905) if
120629 120630119829 = 120782
Flat
Curved
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
119951(119927)
Gauss Map
119918119924 rarr 119930120784
119927 rarr 119951(119927)M
Shape operator is the negative of the derivative of the Gauss map
CURVATURE FOR SURFACES
Gauss Curvature K = 120639120783 sdot 120639120784
Mean Curvature H =120639120783+120639120784
2
Gauss and Mean curvatures
are determinant and half of
the trace of the shape
operator
120639120783 120639120784
120639120783 = 119950119946119951119946119950119958119950
120639120784 = 119950119938119961119946119950119958119950
M
Two dimensional creatures cannot compute 120639120783 and 120639120784 using infinitesimal
ruler and protractor BUT they can determine K = 120639120783 sdot 120639120784 This means 2D
creatures can determine the shape of their world without stepping out to
3rd dimension
GAUSSrsquoS THEOREM EGREGIUM
119870 = minus1
2 119864119866(120597
120597119907
119864119907
119864119866+
120597
120597119906
119866119906
119864119866)
119916 = 119955119958 ∙ 119955119958
G = 119955119959 ∙ 119955119959
Rotating moving or bending the surface does not change the Gauss
curvature but stretching or breaking does
Two surfaces with the same Gauss curvature are ldquolocallyrdquo the same but not
globally
K=0 K=0
VARIOUS SURFACES
CURVATURE IN ARCHITECTURE AND DESIGN
MINIMAL SURFACES
A surface M is minimal if 119815 =120639120783+120639120784
2= 0
Any planar surface is minimal (NOT INTERESTING)
A Gyroid (VERY INTERESTING)
Gyroid structures are found in certain
surfactant or lipid mesophases and block
copolymers
THEOREM Every soap film is a physical model of a minimal surface
The interpretation of the Costa-Hoffman-Meeks minimal surface as
insertion of multiple directional holes connecting the top to the water
and the water at the bottom to the sky provided a single gesture
combining all aspects -Tobias Walliser
Image LAVA
COSTA-HOFFMAN-MEEK SURFACE
CONCEPT OF A LINE
y=mx+b
slope-intercept
119955 119957 = (119957119950119957 + 119939)
parametric
In general a line in space is given by 119903 119905 = 119875 + 119905119906 So 119903 = 0
P119958
119951
V
120629119958V
119906
The covariant derivative 120629119958V of vector field V
is the projection of the change of vector field in
119906 direction onto the tangent plane
120572(119905)
A curve 120572(119905) on the surface is called a
ldquoGeodesicrdquo if 120571 prop prop = 0
Geodesics are the ldquolinesrdquo of curved spaces
V is parallel along a curve 120572(119905) if
120629 120630119829 = 120782
Flat
Curved
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
CURVATURE FOR SURFACES
Gauss Curvature K = 120639120783 sdot 120639120784
Mean Curvature H =120639120783+120639120784
2
Gauss and Mean curvatures
are determinant and half of
the trace of the shape
operator
120639120783 120639120784
120639120783 = 119950119946119951119946119950119958119950
120639120784 = 119950119938119961119946119950119958119950
M
Two dimensional creatures cannot compute 120639120783 and 120639120784 using infinitesimal
ruler and protractor BUT they can determine K = 120639120783 sdot 120639120784 This means 2D
creatures can determine the shape of their world without stepping out to
3rd dimension
GAUSSrsquoS THEOREM EGREGIUM
119870 = minus1
2 119864119866(120597
120597119907
119864119907
119864119866+
120597
120597119906
119866119906
119864119866)
119916 = 119955119958 ∙ 119955119958
G = 119955119959 ∙ 119955119959
Rotating moving or bending the surface does not change the Gauss
curvature but stretching or breaking does
Two surfaces with the same Gauss curvature are ldquolocallyrdquo the same but not
globally
K=0 K=0
VARIOUS SURFACES
CURVATURE IN ARCHITECTURE AND DESIGN
MINIMAL SURFACES
A surface M is minimal if 119815 =120639120783+120639120784
2= 0
Any planar surface is minimal (NOT INTERESTING)
A Gyroid (VERY INTERESTING)
Gyroid structures are found in certain
surfactant or lipid mesophases and block
copolymers
THEOREM Every soap film is a physical model of a minimal surface
The interpretation of the Costa-Hoffman-Meeks minimal surface as
insertion of multiple directional holes connecting the top to the water
and the water at the bottom to the sky provided a single gesture
combining all aspects -Tobias Walliser
Image LAVA
COSTA-HOFFMAN-MEEK SURFACE
CONCEPT OF A LINE
y=mx+b
slope-intercept
119955 119957 = (119957119950119957 + 119939)
parametric
In general a line in space is given by 119903 119905 = 119875 + 119905119906 So 119903 = 0
P119958
119951
V
120629119958V
119906
The covariant derivative 120629119958V of vector field V
is the projection of the change of vector field in
119906 direction onto the tangent plane
120572(119905)
A curve 120572(119905) on the surface is called a
ldquoGeodesicrdquo if 120571 prop prop = 0
Geodesics are the ldquolinesrdquo of curved spaces
V is parallel along a curve 120572(119905) if
120629 120630119829 = 120782
Flat
Curved
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
Two dimensional creatures cannot compute 120639120783 and 120639120784 using infinitesimal
ruler and protractor BUT they can determine K = 120639120783 sdot 120639120784 This means 2D
creatures can determine the shape of their world without stepping out to
3rd dimension
GAUSSrsquoS THEOREM EGREGIUM
119870 = minus1
2 119864119866(120597
120597119907
119864119907
119864119866+
120597
120597119906
119866119906
119864119866)
119916 = 119955119958 ∙ 119955119958
G = 119955119959 ∙ 119955119959
Rotating moving or bending the surface does not change the Gauss
curvature but stretching or breaking does
Two surfaces with the same Gauss curvature are ldquolocallyrdquo the same but not
globally
K=0 K=0
VARIOUS SURFACES
CURVATURE IN ARCHITECTURE AND DESIGN
MINIMAL SURFACES
A surface M is minimal if 119815 =120639120783+120639120784
2= 0
Any planar surface is minimal (NOT INTERESTING)
A Gyroid (VERY INTERESTING)
Gyroid structures are found in certain
surfactant or lipid mesophases and block
copolymers
THEOREM Every soap film is a physical model of a minimal surface
The interpretation of the Costa-Hoffman-Meeks minimal surface as
insertion of multiple directional holes connecting the top to the water
and the water at the bottom to the sky provided a single gesture
combining all aspects -Tobias Walliser
Image LAVA
COSTA-HOFFMAN-MEEK SURFACE
CONCEPT OF A LINE
y=mx+b
slope-intercept
119955 119957 = (119957119950119957 + 119939)
parametric
In general a line in space is given by 119903 119905 = 119875 + 119905119906 So 119903 = 0
P119958
119951
V
120629119958V
119906
The covariant derivative 120629119958V of vector field V
is the projection of the change of vector field in
119906 direction onto the tangent plane
120572(119905)
A curve 120572(119905) on the surface is called a
ldquoGeodesicrdquo if 120571 prop prop = 0
Geodesics are the ldquolinesrdquo of curved spaces
V is parallel along a curve 120572(119905) if
120629 120630119829 = 120782
Flat
Curved
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
Rotating moving or bending the surface does not change the Gauss
curvature but stretching or breaking does
Two surfaces with the same Gauss curvature are ldquolocallyrdquo the same but not
globally
K=0 K=0
VARIOUS SURFACES
CURVATURE IN ARCHITECTURE AND DESIGN
MINIMAL SURFACES
A surface M is minimal if 119815 =120639120783+120639120784
2= 0
Any planar surface is minimal (NOT INTERESTING)
A Gyroid (VERY INTERESTING)
Gyroid structures are found in certain
surfactant or lipid mesophases and block
copolymers
THEOREM Every soap film is a physical model of a minimal surface
The interpretation of the Costa-Hoffman-Meeks minimal surface as
insertion of multiple directional holes connecting the top to the water
and the water at the bottom to the sky provided a single gesture
combining all aspects -Tobias Walliser
Image LAVA
COSTA-HOFFMAN-MEEK SURFACE
CONCEPT OF A LINE
y=mx+b
slope-intercept
119955 119957 = (119957119950119957 + 119939)
parametric
In general a line in space is given by 119903 119905 = 119875 + 119905119906 So 119903 = 0
P119958
119951
V
120629119958V
119906
The covariant derivative 120629119958V of vector field V
is the projection of the change of vector field in
119906 direction onto the tangent plane
120572(119905)
A curve 120572(119905) on the surface is called a
ldquoGeodesicrdquo if 120571 prop prop = 0
Geodesics are the ldquolinesrdquo of curved spaces
V is parallel along a curve 120572(119905) if
120629 120630119829 = 120782
Flat
Curved
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
VARIOUS SURFACES
CURVATURE IN ARCHITECTURE AND DESIGN
MINIMAL SURFACES
A surface M is minimal if 119815 =120639120783+120639120784
2= 0
Any planar surface is minimal (NOT INTERESTING)
A Gyroid (VERY INTERESTING)
Gyroid structures are found in certain
surfactant or lipid mesophases and block
copolymers
THEOREM Every soap film is a physical model of a minimal surface
The interpretation of the Costa-Hoffman-Meeks minimal surface as
insertion of multiple directional holes connecting the top to the water
and the water at the bottom to the sky provided a single gesture
combining all aspects -Tobias Walliser
Image LAVA
COSTA-HOFFMAN-MEEK SURFACE
CONCEPT OF A LINE
y=mx+b
slope-intercept
119955 119957 = (119957119950119957 + 119939)
parametric
In general a line in space is given by 119903 119905 = 119875 + 119905119906 So 119903 = 0
P119958
119951
V
120629119958V
119906
The covariant derivative 120629119958V of vector field V
is the projection of the change of vector field in
119906 direction onto the tangent plane
120572(119905)
A curve 120572(119905) on the surface is called a
ldquoGeodesicrdquo if 120571 prop prop = 0
Geodesics are the ldquolinesrdquo of curved spaces
V is parallel along a curve 120572(119905) if
120629 120630119829 = 120782
Flat
Curved
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
CURVATURE IN ARCHITECTURE AND DESIGN
MINIMAL SURFACES
A surface M is minimal if 119815 =120639120783+120639120784
2= 0
Any planar surface is minimal (NOT INTERESTING)
A Gyroid (VERY INTERESTING)
Gyroid structures are found in certain
surfactant or lipid mesophases and block
copolymers
THEOREM Every soap film is a physical model of a minimal surface
The interpretation of the Costa-Hoffman-Meeks minimal surface as
insertion of multiple directional holes connecting the top to the water
and the water at the bottom to the sky provided a single gesture
combining all aspects -Tobias Walliser
Image LAVA
COSTA-HOFFMAN-MEEK SURFACE
CONCEPT OF A LINE
y=mx+b
slope-intercept
119955 119957 = (119957119950119957 + 119939)
parametric
In general a line in space is given by 119903 119905 = 119875 + 119905119906 So 119903 = 0
P119958
119951
V
120629119958V
119906
The covariant derivative 120629119958V of vector field V
is the projection of the change of vector field in
119906 direction onto the tangent plane
120572(119905)
A curve 120572(119905) on the surface is called a
ldquoGeodesicrdquo if 120571 prop prop = 0
Geodesics are the ldquolinesrdquo of curved spaces
V is parallel along a curve 120572(119905) if
120629 120630119829 = 120782
Flat
Curved
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
MINIMAL SURFACES
A surface M is minimal if 119815 =120639120783+120639120784
2= 0
Any planar surface is minimal (NOT INTERESTING)
A Gyroid (VERY INTERESTING)
Gyroid structures are found in certain
surfactant or lipid mesophases and block
copolymers
THEOREM Every soap film is a physical model of a minimal surface
The interpretation of the Costa-Hoffman-Meeks minimal surface as
insertion of multiple directional holes connecting the top to the water
and the water at the bottom to the sky provided a single gesture
combining all aspects -Tobias Walliser
Image LAVA
COSTA-HOFFMAN-MEEK SURFACE
CONCEPT OF A LINE
y=mx+b
slope-intercept
119955 119957 = (119957119950119957 + 119939)
parametric
In general a line in space is given by 119903 119905 = 119875 + 119905119906 So 119903 = 0
P119958
119951
V
120629119958V
119906
The covariant derivative 120629119958V of vector field V
is the projection of the change of vector field in
119906 direction onto the tangent plane
120572(119905)
A curve 120572(119905) on the surface is called a
ldquoGeodesicrdquo if 120571 prop prop = 0
Geodesics are the ldquolinesrdquo of curved spaces
V is parallel along a curve 120572(119905) if
120629 120630119829 = 120782
Flat
Curved
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
The interpretation of the Costa-Hoffman-Meeks minimal surface as
insertion of multiple directional holes connecting the top to the water
and the water at the bottom to the sky provided a single gesture
combining all aspects -Tobias Walliser
Image LAVA
COSTA-HOFFMAN-MEEK SURFACE
CONCEPT OF A LINE
y=mx+b
slope-intercept
119955 119957 = (119957119950119957 + 119939)
parametric
In general a line in space is given by 119903 119905 = 119875 + 119905119906 So 119903 = 0
P119958
119951
V
120629119958V
119906
The covariant derivative 120629119958V of vector field V
is the projection of the change of vector field in
119906 direction onto the tangent plane
120572(119905)
A curve 120572(119905) on the surface is called a
ldquoGeodesicrdquo if 120571 prop prop = 0
Geodesics are the ldquolinesrdquo of curved spaces
V is parallel along a curve 120572(119905) if
120629 120630119829 = 120782
Flat
Curved
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
CONCEPT OF A LINE
y=mx+b
slope-intercept
119955 119957 = (119957119950119957 + 119939)
parametric
In general a line in space is given by 119903 119905 = 119875 + 119905119906 So 119903 = 0
P119958
119951
V
120629119958V
119906
The covariant derivative 120629119958V of vector field V
is the projection of the change of vector field in
119906 direction onto the tangent plane
120572(119905)
A curve 120572(119905) on the surface is called a
ldquoGeodesicrdquo if 120571 prop prop = 0
Geodesics are the ldquolinesrdquo of curved spaces
V is parallel along a curve 120572(119905) if
120629 120630119829 = 120782
Flat
Curved
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
119951
V
120629119958V
119906
The covariant derivative 120629119958V of vector field V
is the projection of the change of vector field in
119906 direction onto the tangent plane
120572(119905)
A curve 120572(119905) on the surface is called a
ldquoGeodesicrdquo if 120571 prop prop = 0
Geodesics are the ldquolinesrdquo of curved spaces
V is parallel along a curve 120572(119905) if
120629 120630119829 = 120782
Flat
Curved
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
Flat
Curved
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
119922 ne 120782 119922 = 120782
MERCATOR PROJECTION
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
ω
PARALLEL TRANSPORT AND HOLONOMY
The parallel vector V is rotated by ω
as it moved along the latitude 1199070
V
ω = - 2 π sin (1199070)
But 2D inhabitants of the sphere could not
observe the rotation since V is parallel For them
vector field moves ldquoparallelrdquo along the latitude
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
ldquoYOU ARE INVITED TO SEE THE EARTH IS SPINNINGrdquo
An iron ball of 28 kg is suspended
by a 67 meter ( about 220 ft ) wire
Using this experiment in 1851
Foucault proved that the earth
is spinning
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
V
By Cleon Teunissen
Rod is long So swings can be seen as
tangential to the sphere
Pendulum moves slowly around latitude
so we ignore centripetal force on it Only
gravitation acts on the pendulum
V is parallel along the latitude It has holonomy
ω = - 2 π sin (1199070)
THEOREM Earth rotates along its latitude circles
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
GAUSS CURVATURE MEAN CURVATURE
INTRINSIC DEPENDS ON HOW SURFACE IS PLACED
INVARIANT UNDER CERTAIN DEFORMATIONS NOT INVARIANT
STRENGHT RESISTANCE SURFACE TENSION AREA MINIMIZING
MOST FUNDAMENTAL
GEOMETRIC PROPERTY
GREAT TOOL FOR NOISE REDUCTION
IN DIGITAL IMAGING
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
CURVATURE IN HIGHER DIMENSIONS
119877120583120592 minus1
2119877119892120583120592 + Λ119892120583120592 =
8120587119866
1198884119879120583120592
Curvature Matter amp energy
