CSE590B Lecture 7

Cubic Curves

Implicit and Parametric

James F. Blinn JimBlinn.Com

http://courses.cs.washington.edu/courses/cse590b/13au/

http://courses.cs.washington.edu/courses/cse590b/13au/

Resultant of Two Cubics

C

D

CD

CD

216 r =

D D

D

C

C C

+ 9 + 9

C C

C

D

D D

CD

CD

CD

CD

CD

CD

CD

CD

CD

CD

CD

CD

-36r(C,D) = -C +b a D=CDa,b

x

w

Com

mon

root

Other Relations Between Cubics

11

11

22

1111

211

111

1

211

31

12

1

x

w

Com

mon

root

x

w

Com

mon

root

x

w

x

w

x

w

x

w

Com

mon

root

Common root

x

w

Com

mon

root

Common root Etc…

x

w

A B C D

P1 Space Visualizations

D

B

¥

-1

0

+1C

A

A

B C

D

x

w

A B C D

2D lines in [x,w]

xX

w=

P1

Topology

CD

+¥

0B-1A

-¥

x

w

C

D+¥

0B-1A

-¥

360o generates P1 twice

P2 Space Visualizations

w

x

y A

CBD

X

Y

A

CB

D

x

y

DB C

A

w

w

x

y A

CBD

X

Y

A

CB

D

ef

g

ef

g

hh

P2 Space Visualizations

How Many Triangles?

Four

Other Space Topologies

X

Y

A

CB

D

ef

g

ef

g

hh

A

CB

D

e f g

e f g

h

j

h

j

Point on a Line

0Ax By Cw+ + =

0A

x y w B

C

=

P L = 0

L

P

L

Parametric Line

,a b a b= +P R S

1 2 3

1 2 3

R R R

S S Sa b

=

P

=P aV

P = a V

R

S

P

b

a

Implicitize Parametric Line

V

R

S

R = 1,0 V

S = 0,1 V

R

=

1,0 V

S 0,1 V

L =

Implicitize Parametric Line

VR

S

1,0 V

0,1 V

1,0 V

0,1 V

1,0 V

0,1 V

= -2

1,0 V

0,1 V

1,0 V

0,1 V

= -

Implicitize Parametric Line

V

R

S

V

V

= L

Parametric

Implicit

Quadratic Curve

Point on a Quadratic Curve

2

2

2

2 2

2

0

Ax Bxy Cxw

Dy Eyw

Fw

+ +

+ +

+ =

0A B C x

x y w B D E y

C E F w

=

P Q P = 0

P

Q

Tangent to Quadratic at P

2

2

2

, , 2 2

2

x y w Ax Bxy Cxw

Dy Eyw

Fw

= + +

+ +

+

Q

2

x

y

w

Ax By Cw

Bx Dy Ew

Cx Ey Fw

+ +

= + + + +

Q

Q

Q

PQ = L

P

Q

L

2

A B C x

C D E y

B C F w

=

Intersect Line And Quadratic

= 0P Q P

P

Q

P = a V

= 0Qa V V a

2x2 Matrix

= 0a q a

Parametric line Implicit Quadratic

Double Root Means Tangent

Q VV

Q VV

= 0

q

q

=

P

Q

Q VVq =

Reinterpret Diagram Fragment

V

V

1 1

1 2 3

2 2

1 2 3

3 3

0 1

1 0

R SR R R

R SS S S

R S

= -

L

1 2 3

1 2 3

R R R

S S S

=

V

L

R

S

1

2

3

L

L

L

=

L

1 2 2 1 1 3 3 1

2 1 1 2 2 3 3 2

3 1 1 3 3 2 2 3

0

0

0

R S R S R S R S

R S R S R S R S

R S R S R S R S

- -

= - - - -

3 2

3 1

2 1

0

0

0

L L

L L

L L

- = - -

Line Tangent To Quadric

=

V

V

L

Q

Q

= 0L L

L

Q

Q VV

Q VV

= 0

q

q

=

Klebsch Transfer Principle

Klebsch Transfer Principle

Q

L

Q

Q

= 0L L

q

q

= 0

Condition for a double root in 2D (P1)

L

Condition for tangency in 3D (P2)

Make substitution

Parametric Quadratic curves

2 22x y w

x y w

x y w

A A A

t ts s B B B x y w

C C C

=

y yx x w w

y yx x w w

A BA B A B tt s x y w

B CB C B C s

=

= x,y,w

t

t

QP

= x,y,wts2 QT

2 2

2 2

2 2

, 2

, 2

, 2

x x x

y y y

w w w

x s t A t B ts C s

y s t A t B ts C s

w s t A t B ts C s

= + +

= + +

= + +

Points and Tangents

t

Qp

tT

t

Qp

u

t

Qp

t

t

Qp

u

=t

Qp

t

t

Qp

u

-t

Qp

t

t

Qp

u

T =

Points and Tangents

t

Qp

tT

t

Qp

u

T =t

Qpt

t Qpu

- 1/2

Points and Tangents

t

Qp

t

u

Qp

u

t

Qpu

t

u

0 1=

1 1=If Bezier Control Points

Plus:

Homogeneous

Scales of Points

Implicitizing A Parametric

Quadratic Curve

=

t

t

pQP

0A B C x

x y w B D E y

C E F w

=

y yx x w w

y yx x w w

A BA B A B tt s x y w

B CB C B C s

=

p Q p = 0

p

Given

Want

Implicitizing A Parametric Quadratic

Curve p

L=

t

t

L

t

QL

t

QP

p

M

=

t

t

M

t

QM

t

QP

=

t

t

LM

t

QLM

t

a,b a,bQP

p

L

M

aL+bM LM a,b=

Implicitizing A Parametric Quadratic

Curve

=

t

t

LM

t

QLM

t

a,b a,bQP

L

M

aL+bMp

LMLM

LMLM

QP QP

QPQP

=

QLMQLM

QLM QLM

= 0

=

LM

LM

p = 0p p

QPQP

QP QP

Use resultant:

Implicitizing A Parametric Quadratic Curve

p

= 0pp

QPQP

QP QP

=

t

t

pQP

= 0QP

QP

QP

=

t

u

-

t

u

t

u

Given the parametric form: The implicit form is:

Line Tangency to Param. Quadratic Curve

p

= 0pp

QPQP

QP QP

=

t

t

pQP

QPQP

QP QP

QPQP

QPQP

=

LL

= 0QP

QP

QP

=

t

u

-

t

u

t

u

0 =

QPQP

QP QP

L

QPQP

QP QP

L

QPQP

QP QP

QPQP

=

QP

QP

QP

QP

QP

QP

Sign?

In terms of the implicitized

parametric form:

The line tangency condition is:

Sign is

positive

Singularity in a Param. Quadratic Curve

= 0

QPQP

QP QP

QPQP

t

Qp

t

u

Qp

u

t

Qpu

t

Qp

t

u

Qp

u

t

Qp

u=

=

t

u

-

t

u

t

u

QP

QP

QP

t

u

t

u

t

u

= 0QP

QP

QP

What does mean?

= 0 implies

E

FB

LL

Q

R

F/E

B/E

Q

R

S

t

Qp

t

u

Qp

u

t

Qp

u=QP

QP

QP

t

u

t

u

t

u

= x,y,w

t

t

QP 0

0 1

1

y yx x w w

y yx x w w

A BA B A B tt t

B CB C B C t

=

t u

t

u

1 0=

0 1=

1=

QP

QP

QP

B

C

A

=

Forms of Singularity Condition

Forms of Singularity Condition F/E

B/E

Q

R

S

= x,y,w

t

t

QP y yx x w w

y yx x w w

A BA B A B tt s

B CB C B C s

=

= x,y,wts2 QT

2 22

x y w

x y w

x y w

A A A

t ts s B B B

C C C

=

QP

QP

QP

B

C

A

= X

W

Y

=QT=

QT

QT

A

C

B

XY

W

A

C

B

LL

Q

x,y,w coplanar in coefficient space

Sub-Forms of Singularity F/E

B/E

Q

R

S

= x,y,wts2 QT

2 2

0 0

2 0 1

0 0

x

x

x

A

t ts s B

C

=

A

C

B

X

Y

W

A

C

B

W

YX

X

Y

2 2

1

0

1

2

x

x

x

A

B

C

t sX

ts

=

+= X

Y

2

1

0

0

2

x

x

x

A

B

C

tX

ts

=

= X

Y

2 2

1

0

1

2

x

x

x

A

B

C

t sX

ts

= -

-=

A

C

B

W

Y

X

Types of Quadratic

F/E

B/E

x

w

y

= x,y,wts2 QT

2 22

x y w

x y w

x y w

A A A

t ts s B B B

C C C

=

X

Y

X

Y

X

Y

F/E

B/E

x

w

y F/E

B/E

x

w

y

X

Y

F/E

B/E

x

w

y

Cubic Curve in P2 3 2 2 3

2 2

2 2

3

3 3

3 6 3

2 3

0

Ax Bx y Cxy Dy

Ex w Fxyw Gyw

Hxw Jyw

Kw

+ + +

+ + +

+ +

+ =

0A B E B C F E F H x x

x y w B C F C D G F G J y y

E F H F G J H J K w w

=

P = 0C

P

P

P

C

Cubic Curve Tangent at P

= 0 C

P

P

L =

P

C

L

Is Line L Tangent to Cubic

CC

L

LL

L

CC

L L = 0

CC

CC

= 0

Clebsch Translation Principle

C

L

Wierstrass Standard Form

2 3Y X cX d= + +

-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

c

d

3 2 3 2 0x cxw dw y w+ + - =

C

C

C

C

4

8

9I c= = -

C C

CC C

C

6

8

9I d= =

3 3 3 6 0x y w m xyw+ + + =

c,d parameters along lines

generate congruent curves

Different lines

generate different curves

Cubic Continuum

c

d

3Y X cX d= + +

Nodes on a Cubic

C

C

C

C

4I =

C C

CC C

C

6I =

1 = 0C

P

P

0=

3 2

4 66I I = -

Loop and Serpentine

c

d

22 3 3 2 1 2Y X X X X= - + = - +

22 3 3 2 1 2Y X X X X= - - = + -

3 2

4 6

3 3 2

6

8

3 3 2

I I

c d

= -

= - +

Parametric Cubic Curve

Cp

p

p

p

3 2 2 33 3

x y w

x y w

x y w

x y w

A A A

B B Bt t s ts s x y w

C C C

D D D

=

A B B CA B B C A B B Cy y y yx x x x w w w w

B C C D B C C D B C C Dx x x x y y y y w w w w

t tt s x y w

s s

=

= x,y,wts3 4 CT

Parametric Cubic Curve

t

Cp

t

t

u

Cp

u

u

t

Cp

u

t

tCp

u

u

Parametric vs Implicit

P I

Parametrizable Cubics

= 0 C

0,0,1

0 =

0,0,1

C

P

P3 2 2 3

2 2

2 2

3

3 3

3 6 3

2 3

Ax Bx y Cxy Dy

Ex w Fxyw Gy w

Hxw Jyw

Kw

+ + +

+ + + =

+ + +

P

H=J=K=0

Node at origin

3 2 2 3

2 2

3 3

3 2

Ax Bx y Cxy Dyw

Ex Fxy Gy

+ + +=

- + +

Parametrizable Cubics

C

P

P3 2 2 3

2 2

2 2

3

3 3

3 6 3

3 3

Ax Bx y Cxy Dy

Ex w Fxyw Gy w

Hxw Jyw

Kw

+ + +

+ + + =

+ + +

P

3 2 2 3

2 2

3 3

3 2

Ax Bx y Cxy Dyx y w x y

Ex Fxy Gy

+ + + =

- + +

3 2 2 2 2 3 3 2 2 33 6 3 , 3 6 3 , 3 3

x y w

Ex Fx y Gxy Ex y Fxy Gy Ax Bx y Cxy Dy

=

- - - - - - + + +

Parametrizable only if has Node

Tangent to Parametric Cubic Curve

t

Cp

t

t

t

Cp

u

t

t

Cp

t

t

t

Cp

u

t

T =t

Cp

t

t

t

Cpu

t

= - 1/2

Inflection point

Cpt

t

t

Cpt

t

u

Cpu

t

u

positionFirst

derivative

Second

derivative

Cpt

t

tCp

t

t

u

Cp

u

t

u

= 0 Cpt

t

Cp

t

Cp

=

u

t

t

u

t

u

=

t

u

-

t

u

t

u

Inflection point

Cp

pi

pi

pi

Cp

Cp

= G

pi

pi

pi

= 0

G is Type 111 1

Type 11

Type 12

Type 3?

Inflection points are colinear

Cp

pi

pi

pi

Cp

Cp

= 0 = Cp

pi

pi

pi

Cp

Cp

K

KK

=

Cp

Cp

K

Double Points

Cp

Cp

Cp=

pi

pi

piCp

pi

pi

pi

Cp

Cp

=G

pi

pi

pi

= G GH

Parameters at double point are roots of Hessian(G)

Implicitization of Parametric Quad,Cubic

QpQp

Qp Qp

p p

CD

CD

CD

CD

CD

CD

CD

CD

CD

CD

CD

CD

+36+36

p

p

pp

p

p

CP

CP

CP

CP

CP

CP

CP

CP

CP

CP

CP

CP

QR QR

QR QR

Quadratic

Cubic

P1 resultant P2 implicitization

Analyzing Cubic Curves

KL

M

N

C

LM

N

KK

K+=

The Other Form of G

Cp

p

p

p

A B B CA B B C A B B Cy y y yx x x x w w w w

B C C D B C C D B C C Dx x x x y y y y w w w w

t tt s

s s

=

= x,y,wts3 4 CT

3 2 2 33 3

x y w

x y w

x y w

x y w

A A A

B B Bt t s ts s

C C C

D D D

=

Cp

Cp

Cp

= G

A B B C

B C C D

=

CT

CT

CT

G=

A

B

C

D

=

det , det , det , det

x y w x y w x y w x y w

x y w x y w x y w x y w

x y w x y w x y w x y w

B B B A A A A A A A A A

A C C C B C C C C B B B D B B B

D D D D D D D D D C C C

= = - = = -

x y

w

Forms and Singularities of Curve

G is type 1

11G is type 21G is type 111

G is type 3

xy w

xy

w

x

y

w

3 2 2 33 3

x y w

x y w

x y w

x y w

A A A

B B Bt t s ts s x y w

C C C

D D D

=

Look at plane in coefficient space containing x,y,w cubics