CMSC427 Parametric curves: Hermite, Catmull-Rom, Bezier · 2017-12-05 · Parametric curves: Hermite, Catmull-Rom, Bezier. Modeling •Creating 3D objects •How to construct complicated

CMSC427Parametriccurves:Hermite,Catmull-Rom,Bezier

Modeling

• Creating3Dobjects• Howtoconstructcomplicatedsurfaces?• Goal• Specifyobjectswithfewcontrolpoints• Resultingobjectshouldbevisuallypleasing(smooth)

• Startwithcurves,thengeneralizetosurfaces

2

Usefulness of curves

• Surfaceofrevolution

3


• Extruded/sweptsurfaces

4


• Animation• Providea“track”forobjects• Useascamerapath

5


• Generalizetosurfacepatchesusing“gridsofcurves”,nextclass

6

How to represent curves

• Specifyeverypointalongcurve?• Hardtogetprecise,smoothresults• Toomuchdata,toohardtoworkwith

• Idea:specifycurvesusingsmallnumbersofcontrolpoints• Mathematics:usepolynomials torepresent curves

Control point

7

Interpolating polynomial curves

http://en.wikipedia.org/wiki/Polynomial_interpolation

• Curvegoesthroughallcontrolpoints• Seemsmostintuitive• Surprisingly,notusuallythebestchoice• Hardtopredictbehavior• Overshoots,wiggles• Hardtoget“nice-looking”curves

Control point

Interpolating curve8

Approximating polynomial curves• Curveis“influenced”bycontrolpoints

• Varioustypes&techniquesbasedonpolynomialfunctions• Bézier curves,B-splines,NURBS

• FocusonBézier curves

Control point

9

Mathematical definition

• Avectorvaluedfunctionofonevariablex(t)• Givent,computea3Dpointx=(x,y,z)• Mayinterpretasthreefunctions x(t), y(t), z(t)• “Movingapointalongthecurve”

x

y

z

x(0.0) x(0.5) x(1.0)

x(t)

10

Tangent vector

• Derivative• Avectorthatpointsinthedirectionofmovement• Lengthofx’(t) correspondstospeed

x’(0.0) x’(0.5) x’(1.0)

x(t)

x

y

z

11

Piecewise polynomial curves• Modelcomplexshapesbysequence• Usepolylinetostorecontrolpoints

12

Continuity• Howpiecewisecurvesjoin• Ck continuity– kthderivativesmatch• Gk continuity– kthderivativesareproportional

13

• Cubiccurve(here2D)

𝑥 𝑡 = 𝑎𝑡3 + 𝑏𝑡6 + 𝑐𝑡 + 𝑑𝑦 𝑡 = 𝑒𝑡3 + 𝑓𝑡6 + 𝑔𝑡 + ℎ

• InterpolatesendpointsP0andP1• MatchestangentatendpointsT0andT1• (alsodP0anddP1inthesenotes).

Hermite curves

P0=(x0,y0)

P1 = (x1,y1)T1=<dx1,dy1>

T0=<dx0,dy0>

• Derivativeofx(t)

𝑥′ 𝑡 = 3𝑎𝑡6 + 2𝑏𝑡 + 𝑐• Sett=0and1forendpoints• Fourconstraints

𝑥 0 = 𝑑𝑥′(0) = 𝑐

𝑥 1 = 𝑎 + 𝑏 + 𝑐 + 𝑑𝑥′ 1 = 3𝑎 + 2𝑏 + 𝑐

Computing coefficients a, b, c and d

• Solvefora,b,candd𝑑 = 𝑥0

𝑐 = 𝑑𝑥0

𝑏 = −3𝑥0 + 3𝑥1 − 2𝑑𝑥0 − 𝑑𝑥1

𝑎 = 2𝑥0 − 2𝑥1 + 𝑑𝑥0 + 𝑑𝑥1

Solve for a, b, c and d

• Constraints𝑥 0 = 𝑑𝑥′(0) = 𝑐

𝑥 1 = 𝑎 + 𝑏 + 𝑐 + 𝑑𝑥′ 1 = 3𝑎 + 2𝑏 + 𝑐

• Give

0 0 0 11 1 1 10 0 1 03 2 1 0

𝑎𝑏𝑐𝑑=

𝑥0𝑥1𝑑𝑥0𝑑𝑥1

Matrix version

• Sincewehave𝑀𝐴 = 𝐺• Wecansolvewith𝐴 = 𝑀\]𝐺• AndgetHermite basismatrix𝑀\]

𝑎𝑏𝑐𝑑=

2 −2 1 1−3 3 −2 −10 0 1 01 0 0 0


Solve matrix version: basis matrix

• Toincludex,yandz,rewritewithvectorsP0,P1andtangentsT0andT1

𝒂𝒃𝒄𝒅

=

2 −2 1 1−3 3 −2 −10 0 1 01 0 0 0

𝑷𝟎𝑷𝟏𝑻𝟎𝑻𝟏

• Coefficientsa,b,canddarenowvectors

Vector version

• Rewritepolynomialasdotproduct

𝑃 𝑡 = 𝑡3 𝑡6 𝑡 1

𝒂𝒃𝒄𝒅

•

= 𝑡3 𝑡6 𝑡 1

2 −2 1 1−3 3 −2 −10 0 1 01 0 0 0

𝑷𝟎𝑷𝟏𝑻𝟎𝑻𝟏

Full polynomial version

• Insteadofpolynomialint,lookatcurveasweightedsumofP0,P1,T0andT1• 𝑥 𝑡 = 2𝑥0 − 2𝑥1 + 𝑑𝑥0 + 𝑑𝑥1 𝑡3

•+ −3𝑥0 + 3𝑥1 − 2𝑑𝑥0 − 𝑑𝑥1 𝑡6

•+ 𝑑𝑥0 𝑡

•+𝑥0

Blending functions

• Insteadofpolynomialint,lookatcurveasweightedsumofP0,P1,T0andT1

• 𝑥 𝑡 =• 2𝑡3 − 3𝑡6 + 1 𝑥0• + −2𝑡3 + 3𝑡6 𝑥1• + 𝑡3 − 2𝑡6 + 𝑡 𝑑𝑥0• + 𝑡3 − 𝑡6 𝑑𝑥1

Blending functions

•

ℎ00(𝑡) = 2𝑡3 − 3𝑡6 + 1ℎ01(𝑡) = −2𝑡3 + 3𝑡6

ℎ10(𝑡) = 𝑡3 − 2𝑡6 + 𝑡ℎ11(𝑡) = 𝑡3 − 𝑡6

Blending functions

• HaveP(-1),P0,P1andP2asinput• ComputetangentwithHmatrix


=

1 0 0 00 1 0 0−1 1 0 00 0 −1 1

𝑥o𝑥]𝑥\]𝑥6

Computing Hermite tangents

• Unifynotation𝑎𝑏𝑐𝑑=

2 −2 1 1−3 3 −2 −10 0 1 01 0 0 0

1 0 0 00 1 0 01 0 −1 00 −1 0 1


• Finalmatrix

•𝑎𝑏𝑐𝑑=

3 −3 −1 1−5 4 2 −11 0 −1 01 0 0 0


Combine with Hermite basis

• Hermite – problemwithC1continuity

Catmull-Rom curves

P0

P1

P2

P3

P4

P5left right

• Catmull-Rom– maketangentsymmetric• Definebytwoadjacentpoints• HereT3=P4-P2

Catmull-Rom curves

P0

P1

P2

P3

P4

P5

• NeedtochangeHmatrix• ½traditionalforC-Rcurves

𝑥0𝑥1𝑑𝑥0′𝑑𝑥1′

=

1 0 0 00 1 0 0

1/2 0 −1/2 0−1/2 0 0 1/2


Catmull-Rom curves

• Whichgives

𝑎𝑏𝑐𝑑=

2 −2 1 1−3 3 −2 −10 0 1 01 0 0 0

1 0 0 00 1 0 0

1/2 0 −1/2 0−1/2 0 0 1/2


• Or𝑎𝑏𝑐𝑑=

2 −2 −0.5 0.5−3.5 3 1 −0.50.5 0 0.5 01 0 0 0


Catmull-Rom curves

Bézier curves

http://en.wikipedia.org/wiki/B%C3%A9zier_curve

• Aparticularlyintuitivewaytodefinecontrolpointsforpolynomialcurves• DevelopedforCAD(computeraideddesign)andmanufacturing• Beforegames,beforemovies,CADwasthebigapplicationforCG

• PierreBézier (1962),designofautobodiesforPeugeot,http://en.wikipedia.org/wiki/Pierre_B%C3%A9zier

• PauldeCasteljau (1959),forCitroen

30

Bézier curves

• Higherorderextensionoflinearinterpolation• Controlpointsp0, p1, ...

p0

p1

p0

p1p2

p0

p1

p2

p3

Linear Quadratic Cubic31

Bézier curves

• Intuitivecontrolovercurvegivencontrolpoints• Endpointsareinterpolated,intermediatepointsareapproximated• ConvexHullproperty• Variation-diminishingproperty

32

Cubic Bézier curve

• Cubicpolynomials,mostcommoncase• Definedby4controlpoints• Twointerpolatedendpoints• Twomidpoints controlthetangentattheendpoints

p1

p0 p2

p3

x(t)•

Control polyline

33

Bézier Curve formulation

• Threealternatives,analogoustolinearcase1. Weightedaverageofcontrolpoints2. Cubicpolynomialfunctionoft3. Matrixform• Algorithmicconstruction• deCasteljaualgorithm

34

de Casteljau Algorithm

http://en.wikipedia.org/wiki/De_Casteljau's_algorithm

• Arecursive seriesoflinearinterpolations• Worksforanyorder,notonlycubic

• Notterriblyefficienttoevaluate• Otherformsmorecommonlyused

• Whystudyit?• Intuitionaboutthegeometry• Usefulforsubdivision(latertoday)

35


p0

p1

p2

p3

• Giventhecontrolpoints• Avalueoft• Heret≈0.25

36


p0

q0

p1

p2

p3

q2

q1

q0 (t) = Lerp t,p0 ,p1( )q1(t) = Lerp t,p1,p2( )q2 (t) = Lerp t,p2 ,p3( )

37


q0

q2

q1

r1

r0

r0 (t) = Lerp t,q0 (t),q1(t)( )r1(t) = Lerp t,q1(t),q2 (t)( )

38


r1x

r0 •

x(t) = Lerp t,r0 (t),r1(t)( )

39

x•

p0

p1

p2

p3

de Casteljau algorithm

•Applets• http://www2.mat.dtu.dk/people/J.Gravesen/cagd/decast.html• http://www.caffeineowl.com/graphics/2d/vectorial/bezierintro.html

40


http://en.wikipedia.org/wiki/De_Casteljau's_algorithm

Linear Quadratic

Cubic Quartic

41

x = Lerp t,r0 ,r1( )r0 = Lerp t,q0,q1( )r1 = Lerp t,q1,q2( )

q0 = Lerp t,p0 ,p1( )q1 = Lerp t,p1,p2( )q2 = Lerp t,p2 ,p3( )

p0p1p2p3

p1q0

r0 p2x q1

r1 p3q2

p4

Recursive linear interpolation

42



p0p1p2p3

p1q0

r0 p2x q1

r1 p3q2

p4


43



p0p1p2p3

p1q0

r0 p2x q1

r1 p3q2

p4


44



p0p1p2p3

p1q0

r0 p2x q1

r1 p3q2

p4


45

Expand the LERPs

q0 (t) = Lerp t,p0 ,p1( )= 1- t( )p0 + tp1

q1(t) = Lerp t,p1,p2( )= 1- t( )p1 + tp2

q2 (t) = Lerp t,p2 ,p3( )= 1- t( )p2 + tp3

r0 (t) = Lerp t,q0 (t),q1(t)( )= 1- t( ) 1- t( )p0 + tp1( )+ t 1- t( )p1 + tp2( )r1(t) = Lerp t,q1(t),q2 (t)( )= 1- t( ) 1- t( )p1 + tp2( )+ t 1- t( )p2 + tp3( )

x(t) = Lerp t,r0 (t),r1(t)( )= 1- t( ) 1- t( ) 1- t( )p0 + tp1( )+ t 1- t( )p1 + tp2( )( ) +t 1- t( ) 1- t( )p1 + tp2( )+ t 1- t( )p2 + tp3( )( )

46

Expand the LERPs

q0 (t) = Lerp t,p0 ,p1( )= 1- t( )p0 + tp1

q1(t) = Lerp t,p1,p2( )= 1- t( )p1 + tp2

q2 (t) = Lerp t,p2 ,p3( )= 1- t( )p2 + tp3



47

Expand the LERPs

q0 (t) = Lerp t,p0 ,p1( )= 1- t( )p0 + tp1

q1(t) = Lerp t,p1,p2( )= 1- t( )p1 + tp2

q2 (t) = Lerp t,p2 ,p3( )= 1- t( )p2 + tp3



48

Expand the LERPs

q0 (t) = Lerp t,p0 ,p1( )= 1- t( )p0 + tp1

q1(t) = Lerp t,p1,p2( )= 1- t( )p1 + tp2

q2 (t) = Lerp t,p2 ,p3( )= 1- t( )p2 + tp3



49

Expand the LERPs

q0 (t) = Lerp t,p0 ,p1( )= 1- t( )p0 + tp1

q1(t) = Lerp t,p1,p2( )= 1- t( )p1 + tp2

q2 (t) = Lerp t,p2 ,p3( )= 1- t( )p2 + tp3



50

!

x(t) = 1- t( ) 1- t( ) 1- t( )p0 + tp1( )+ t 1- t( )p1 + tp2( )( )+t 1- t( ) 1- t( )p1 + tp2( )+ t 1- t( )p2 + tp3( )( )

x(t) = 1- t( )3 p0 + 3 1- t( )2 tp1 + 3 1- t( )t 2p2 + t 3p3

x(t) = -t 3 + 3t 2 - 3t +1( )B0 (t )" #$$$ %$$$

p0 + 3t 3 - 6t 2 + 3t( )B1 (t )" #$$ %$$

p1+ -3t 3 + 3t 2( )

B2 (t )& '$ ($

p2 + t 3( )B3 (t ))p3

Weighted average of control points• Regroup

51

!


x(t) = 1- t( )3 p0 + 3 1- t( )2 tp1 + 3 1- t( )t 2p2 + t 3p3

x(t) = -t 3 + 3t 2 - 3t +1( )B0 (t )" #$$$ %$$$

p0 + 3t 3 - 6t 2 + 3t( )B1 (t )" #$$ %$$

p1+ -3t 3 + 3t 2( )

B2 (t )& '$ ($

p2 + t 3( )B3 (t ))p3


52

!


x(t) = 1- t( )3 p0 + 3 1- t( )2 tp1 + 3 1- t( )t 2p2 + t 3p3

x(t) = -t 3 + 3t 2 - 3t +1( )B0 (t )" #$$$ %$$$

p0 + 3t 3 - 6t 2 + 3t( )B1 (t )" #$$ %$$

p1+ -3t 3 + 3t 2( )

B2 (t )& '$ ($

p2 + t 3( )B3 (t ))p3


Bernsteinpolynomials53

• Partitionofunity,ateacht alwaysaddto1• Endpointinterpolation,B0 andB3 goto 1

x(t) = B0 t( )p0 + B1 t( )p1 + B2 t( )p2 + B3 t( )p3

The cubic Bernstein polynomials :

B0 t( )= -t 3 + 3t 2 - 3t +1

B1 t( )= 3t 3 - 6t 2 + 3t

B2 t( )= -3t 3 + 3t 2

B3 t( )= t 3

Bi (t) = 1å

Cubic Bernstein polynomials

http://en.wikipedia.org/wiki/Bernstein_polynomial

54

General Bernstein polynomials

B01 t( )= -t +1 B0

2 t( )= t 2 - 2t +1 B03 t( )= -t 3 + 3t 2 - 3t +1

B11 t( )= t B1

2 t( )= -2t 2 + 2t B13 t( )= 3t 3 - 6t 2 + 3t

B22 t( )= t 2 B2

3 t( )= -3t 3 + 3t 2

B33 t( )= t 3

Bin t( )=

ni

æèçöø÷

1- t( )n- i t( )ini

æèçöø÷=

n!i! n - i( )!

Bin t( )å = 1

55


B01 t( )= -t +1 B0

2 t( )= t 2 - 2t +1 B03 t( )= -t 3 + 3t 2 - 3t +1

B11 t( )= t B1

2 t( )= -2t 2 + 2t B13 t( )= 3t 3 - 6t 2 + 3t

B22 t( )= t 2 B2

3 t( )= -3t 3 + 3t 2

B33 t( )= t 3

Bin t( )=

ni

æèçöø÷

1- t( )n- i t( )ini

æèçöø÷=

n!i! n - i( )!

Bin t( )å = 1

56


B01 t( )= -t +1 B0

2 t( )= t 2 - 2t +1 B03 t( )= -t 3 + 3t 2 - 3t +1

B11 t( )= t B1

2 t( )= -2t 2 + 2t B13 t( )= 3t 3 - 6t 2 + 3t

B22 t( )= t 2 B2

3 t( )= -3t 3 + 3t 2

B33 t( )= t 3

Bin t( )=

ni

æèçöø÷

1- t( )n- i t( )ini

æèçöø÷=

n!i! n - i( )!

Bin t( )å = 1

57


B01 t( )= -t +1 B0

2 t( )= t 2 - 2t +1 B03 t( )= -t 3 + 3t 2 - 3t +1

B11 t( )= t B1

2 t( )= -2t 2 + 2t B13 t( )= 3t 3 - 6t 2 + 3t

B22 t( )= t 2 B2

3 t( )= -3t 3 + 3t 2

B33 t( )= t 3

Bin t( )=

ni

æèçöø÷

1- t( )n- i t( )ini

æèçöø÷=

n!i! n - i( )!

Bin t( )å = 1

Partitionofunity,endpointinterpolation

Ordern:

58

General Bézier curves

• nth-orderBernsteinpolynomialsformnth-orderBéziercurves• Béziercurvesareweightedsumofcontrolpointsusingnth-orderBernsteinpolynomials

Bin t( )=

ni

æèçöø÷1- t( )n- i t( )i

x t( )= Bin t( )pi

i=0

n

å

Bernsteinpolynomialsof ordern:

Béziercurve of ordern:

59

Bézier curve properties

• Convexhullproperty• Variationdiminishingproperty• Affineinvariance

60

Convex hull, convex combination

• Convexhullofasetofpoints• Smallestpolyhedralvolumesuchthat(i)allpointsareinit(ii)lineconnectinganytwopointsinthevolumeliescompletelyinsideit(oronitsboundary)

• Convexcombinationofthepoints• Weightedaverageofthepoints,whereweightsallbetween0 and

1,sumupto1• Anyconvexcombinationalwayslieswithintheconvexhull

p0

p1

p2

Convex hullp3

61

Convex hull property

• Bézier curveisaconvexcombinationofthecontrolpoints• Bernsteinpolynomialsaddto1ateachvalueoft

• Curveisalwaysinsidetheconvexhullofcontrolpoints• Makescurvepredictable• Allowsefficientculling,intersectiontesting,adaptivetessellation

p0

p1

p2

p3

62

Variation diminishing property

• Ifthecurveisinaplane,thismeansnostraightlineintersectsaBézier curvemoretimesthanitintersectsthecurve'scontrolpolyline• “Curveisnotmorewigglythancontrolpolyline”

Yellowline:7intersectionswithcontrolpolyline3intersectionswithcurve 63

Affine invariance

• TwowaystotransformBézier curves1. Transformthecontrolpoints,thencomputeresulting

pointoncurve2. Computepointoncurve,thentransformit

• Eitherway,getthesametransformpoint!• Curveisdefinedviaaffinecombinationofpoints(convexcombinationisspecialcaseofanaffinecombination)• Invariantunderaffinetransformations• Convexhullpropertyalwaysremains

64

• Goodforfastevaluation,precomputeconstantcoefficients(a,b,c,d)• Notmuchgeometricintuition

Start with Bernstein form:

x(t) = -t 3 + 3t 2 - 3t +1( )p0 + 3t 3 - 6t 2 + 3t( )p1 + -3t 3 + 3t 2( )p2 + t 3( )p3

Regroup into coefficients of t :x(t) = -p0 + 3p1 - 3p2 + p3( )t 3 + 3p0 - 6p1 + 3p2( )t 2 + -3p0 + 3p1( )t + p0( )1

x(t) = at 3 + bt 2 + ct + d

a = -p0 + 3p1 - 3p2 + p3( )b = 3p0 - 6p1 + 3p2( )c = -3p0 + 3p1( )d = p0( )

Cubic polynomial form

65





x(t) = at 3 + bt 2 + ct + d

a = -p0 + 3p1 - 3p2 + p3( )b = 3p0 - 6p1 + 3p2( )c = -3p0 + 3p1( )d = p0( )


66





x(t) = at 3 + bt 2 + ct + d

a = -p0 + 3p1 - 3p2 + p3( )b = 3p0 - 6p1 + 3p2( )c = -3p0 + 3p1( )d = p0( )


67





x(t) = at 3 + bt 2 + ct + d

a = -p0 + 3p1 - 3p2 + p3( )b = 3p0 - 6p1 + 3p2( )c = -3p0 + 3p1( )d = p0( )


68

• CanconstructothercubiccurvesbyjustusingdifferentbasismatrixB

• Hermite,Catmull-Rom,B-Spline,…

!

x(t) = "a"b "c déë ùû

t 3

t 2

t1

é

ë

êêêê

ù

û

úúúú

"a = -p0 + 3p1 - 3p2 + p3( )"b = 3p0 - 6p1 + 3p2( )"c = -3p0 + 3p1( )d = p0( )

x(t) = p0 p1 p2 p3[ ]

GBez

# $%%% &%%%

-1 3 -3 13 -6 3 0-3 3 0 01 0 0 0

é

ë

êêêê

ù

û

úúúú

BBez

# $%%% &%%%

t 3

t 2

t1

é

ë

êêêê

ù

û

úúúú

T'

Cubic matrix form

69

Cubic matrix form

xx (t) = p0x p1x p2x p3x[ ]

-1 3 -3 13 -6 3 0-3 3 0 01 0 0 0

é

ë

êêêê

ù

û

úúúú

t 3

t 2

t1

é

ë

êêêê

ù

û

úúúú

xy (t) = p0y p1y p2y p3yéë ùû

-1 3 -3 13 -6 3 0-3 3 0 01 0 0 0

é

ë

êêêê

ù

û

úúúú

t 3

t 2

t1

é

ë

êêêê

ù

û

úúúú

xz (t) = p0z p1z p2z p3zéë ùû

-1 3 -3 13 -6 3 0-3 3 0 01 0 0 0

é

ë

êêêê

ù

û

úúúú

t 3

t 2

t1

é

ë

êêêê

ù

û

úúúú

• 3parallelequations,inx,yandz:

70

• Bundleintoasinglematrix

• Efficientevaluation• PrecomputeC• Takeadvantageofexisting4x4matrixhardwaresupport

x(t) =p0x p1x p2x p3x

p0y p1y p2y p3y

p0z p1z p2z p3z

é

ë

êêê

ù

û

úúú

-1 3 -3 13 -6 3 0-3 3 0 01 0 0 0

é

ë

êêêê

ù

û

úúúú

t 3

t 2

t1

é

ë

êêêê

ù

û

úúúú

x(t) = GBezBBezTx(t) = C T

Matrix form

71

Drawing Bézier curves

• Generallynolow-levelsupportfordrawingsmoothcurves• I.e.,GPUdrawsonlystraightlinesegments

• Needtobreakcurvesintolinesegmentsorindividualpixels• Approximatingcurvesasseriesoflinesegmentscalledtessellation• Tessellationalgorithms• Uniformsampling• Adaptivesampling• Recursivesubdivision

72

• ApproximatecurvewithN-1 straightsegments• N choseninadvance• Evaluate

• Connectthepointswithlines• Toofewpoints?• Badapproximation• “Curve”isfaceted

• Toomanypoints?• Slowtodrawtoomanylinesegments• Segmentsmaydrawontopofeachother

x0

x2x3

Uniform sampling

!

xi = x ti( ) where ti =iN

for i = 0, 1,", N

xi =#a i

3

N 3 +#b i2

N 2 +#c iN+ d

x4

x1

x(t)

73

• Useonlyasmanylinesegmentsasyouneed• Fewersegmentswherecurveismostlyflat• Moresegmentswherecurvebends• Segmentsneversmallerthanapixel

• Variousschemesforsampling,checkingresults,decidingwhethertosamplemore

Adaptive Sampling

x(t)

74

Recursive Subdivision• Anycubic(ork-thorder)curvesegmentcanbeexpressedasacubic(ork-thorder)Béziercurve

“Anypieceofacubic(ork-thorder)curveisitselfacubic(ork-thorder)curve”

• Therefore,anyBéziercurvecanbesubdividedintosmallerBéziercurves

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• deCasteljau constructionpointsarethecontrolpointsoftwoBéziersub-segments(p0,q0,r0,x) and(x,r1,q2,p3)

xp0

p1

p2

p3

de Casteljau subdivision

q0r0

r1

q2

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Adaptive subdivision algorithm

1. UsedeCasteljauconstructiontosplitBéziersegmentinmiddle(t=0.5)

2. Foreachhalf• If“flatenough”:drawlinesegment• Else:recursefrom1.foreachhalf

• Curveisflatenoughifhullisflatenough• Testhowfarawaymidpointsarefromstraightsegmentconnectingstartandend• Ifaboutapixel,thenhullisflatenough

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Today

Curves• Introduction• Polynomialcurves• Béziercurves• DrawingBéziercurves• Piecewisecurves

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More control points

• CubicBéziercurvelimitedto4controlpoints• Cubiccurvecanonlyhaveoneinflection• Needmorecontrolpointsformorecomplexcurves

• k-1 orderBéziercurvewithk controlpoints

• Hardtocontrolandhardtoworkwith• Intermediatepointsdon’thaveobviouseffectonshape• Changinganycontrolpointchangesthewholecurve

• Wantlocalsupport• Eachcontrolpointonlyinfluencesnearbyportionofcurve

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• Sequenceofsimple(low-order)curves,end-to-end• Piecewisepolynomialcurve,orsplines

http://en.wikipedia.org/wiki/Spline_(mathematics)

• Sequenceoflinesegments• Piecewiselinear curve(linearorfirst-orderspline)

• Sequenceofcubiccurvesegments• Piecewisecubic curve,herepiecewiseBézier(cubicspline)

Piecewise curves (splines)

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Piecewise cubic Bézier curve

!

• Given 3N +1 points p0 ,p1,",p3N

• Define N Bézier segments:x0 (t) = B0 (t)p0 + B1(t)p1 + B2 (t)p2 + B3(t)p3

x1(t) = B0 (t)p3 + B1(t)p4 + B2 (t)p5 + B3(t)p6

#

xN -1(t) = B0 (t)p3N -3 + B1(t)p3N -2 + B2 (t)p3N -1 + B3(t)p3N

x0(t) x1(t)

x2(t)x3(t)

p0

p1p2

p3

p4p5

p6

p7 p8

p9

p10 p11

p12

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• Globalparameteru,0<=u<=3N


!

x(u) =

x0 ( 13 u), 0 £ u £ 3

x1( 13 u -1), 3 £ u £ 6

" "

xN -1( 13 u - (N -1)), 3N - 3 £ u £ 3N

ì

íïï

îïï

x(u) = xi 13 u - i( ), where i = 1

3 uêë úû

x0(t) x1(t)

x2(t)x3(t)

x(3.5)

x(8.75)

u=0u=12

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Continuity

• Wantsmoothcurves• C0 continuity

• Nogaps• Segmentsmatchattheendpoints

• C1 continuity:firstderivativeiswelldefined• Nocorners• Tangents/normalsareC0 continuous(nojumps)

• C2 continuity:secondderivativeiswelldefined• Tangents/normalsareC1 continuous• Importantforhighqualityreflectionsonsurfaces

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• C0 continuousbyconstruction• C1 continuousatsegmentendpointsp3i if p3i - p3i-1 = p3i+1 - p3i• C2 ishardertoget


p0

p0

p1

p2

P3

P3

p2

p1

p4

p5

p6

p6

p5

p4

C1 continuousC0 continuous84

Piecewise cubic Bézier curves

• Usedoftenin2Ddrawingprograms• Inconveniences• Musthave4or7or10or13or…(1plusamultipleof3)controlpoints• Somepointsinterpolate(endpoints),othersapproximate(handles)• Needtoimposeconstraintsoncontrolpointstoobtain

C1 continuity• C2 continuitymoredifficult

• Solutions• Userinterfaceusing“Bézierhandles”• GeneralizationtoB-splines,nexttime

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Bézier handles

• Segmentendpoints(interpolating)presentedascurvecontrolpoints• Midpoints(approximatingpoints)presentedas“handles”• CanhaveoptiontoenforceC1 continuity

[www.blender.org] Adobe Illustrator 86

Documents

CMSC427 Parametric curves: Hermite, Catmull-Rom, Bezier · 2017-12-05 · Parametric curves: Hermite, Catmull-Rom, Bezier. Modeling •Creating 3D objects •How to construct complicated