EECS487:Interactive

ComputerGraphicsLecture36:Parametricsurfaces

•  Sweptsurfaces

•  Geometriccontinuity

•  Béziercurvesandpatches

Extruded/SweptSurfaces

Yu,Terzopoulos

Consideracurveinspaceasbeingsweptout

byamovingpoint:p(u) = [x(u) y(u) z(u)]T

• aswevaryuthepointmovesthroughspace

• thecurveisthepathtakenbythepoint

Similarlywecanthinkofasurface:

s(u, t) = [x(u, t) y(u, t) z(u, t)]T

• asbeingsweptoutbyaprofilecurve

alongatrajectorycurve

• thesetofpointsvisitedbythecurveduringitsmotiondefinesthesurface

GeneralSweepSurfaces

Fussell,Durand,Terzopoulos

Trajectorypathmaybeanyarbitrarycurve

Theprofilecurvemaybetransformed

asitmovesalongthepath

• scaled,rotatedwithrespecttopathorientation,…

Example:surfaces(u, t) isformedbyaprofile

curveinthexy-planep(u) = [x(u) y(u) 0 1] T

extrudedalongthez-axis:s(u, t): T(t)p(u):x(u, t) = x(u), y(u, t) = y(u), 0 ≤ u ≤ 1, z(u, t) = t, zmin ≤ t ≤ zmax

Extruded/SweptSurfaces

Differentprofilecurves,

sametrajectorycurve

Schulze

SurfacesofRevolution

Schulze,Durand

Userotationaroundan

axisinsteadoftranslation

alongapath

• or,extrusionwherethe

trajectorycurveisacircle • s(u, t): R(t)p(u)

ABananaasaGeneralizedCylinder

Whatwespecify

•  amostlycircularprofile

•  aspineforthebanana•  ascalingfunction

Periodicallyalongthespine

•  placeacrosssection•  scaleitappropriately•  connecttoprevioussection

crosssection scalingfunction

Yu,Snyder

GeneralSweepSurfaces

Thetrajectorycurveislikeaspine

•  sweepingtheprofilecurve“skins”asurfacearoundthe

trajectorycurve

•  theshapeofthespinecontrolstheshapeoftheobject

•  niceforanimation:•  don’thavetocontrolthesurface•  justreshapethespineandthesurfacefollowsalong

Yu,Chenney,Terzopoulos

GeneralSweepSurfaces

Foreverypointalongq(t),layp(u)sothatopcoincideswithq(t)

Thisgivesuslocationsalongq(t),howaboutorientation?

1.  fixedorstatic:alignsp(u)withanaxis

2.  allowssmoothlyvaryingorientationthat“follows”the

orientationofq(t):howtospecifytheorientationofq(t)?

p(u) q(t)

op

Curless

p(u)

q(t)

op

s(u, t)

DifferentialGeometryofCurves

Uses:

• defineorientationofsweptsurfaces

•  computevelocityofanimation

•  computenormalsofsurfaces

•  analyzesmoothness/continuity

Tangent:

Thevelocityofmovement,1stderivativewithrespecttotq’(t) = (x’(t), y’(t), z’(t))orq’(t) ≈ (q(t+Δt) – q(t))/Δt•  ||q’(t)||isthespeedofmovement

•  normalizedtangentt(t) = q’(t)/||q’(t)||isthedirectionofmovement

•  thenumericformofforwarddifferenceisusefulifq(t)isablackbox

Thetangentprovidesuswiththefirstofthree

orientationsforsweptsurfaces

Durand

Forsmoothmotion,wewantcontinuous1stand2nd

derivativeswithrespecttotimedq/dt

Buttodescribeshape,wecouldaskforcontinuity

withrespecttoequalsteps(arclength):dq/ds

Arclengthparameterization:

equalstepsinparameter

spacesmapstoequal

distancesalongthecurve

•  intrinsictoshapeofcurve,notdependentonanyparticularcoordinatesystem

ArcLengthParameterization

TP3,Hart [Curless]

q(t) q(s)

Ifsisthelengthofcurvefromq(0)toq(t),q(s) canbeexpressedintermsoft:q(s) = {q(t): s(t) = s},e.g.,

Unlessmovingatconstant

speed,arclength

travelledisnot

proportionaltopassingtime:

•  i.e.,equalstepsintime(t)doesnotnecessarilygiveequaldistancesinarclength(s)

ArcLengthParameterization

TP3,Hart,Curless

q(t) q(s)

t = 0, s = s(0) = 0

t = 1, s = s(1) = curvelength

s(t) = q '(τ ) dτ

0

t

∫ = x '(τ )2 + y '(τ )2 + z '(τ )2 dτ0

t

∫usually

cannotbe

evaluated

analytically

Instead:

•  pre-computeasetofvariablearclengthssiforpointsonthe

curveusingtparameterization

•  tofindthecorrespondingpoint(pk)

onthecurveforagivensk,linearly

interpolatethepointsofthe2nearestarclengthstoeitherside

siandsi+1,si ≤ sk ≤ si+1:

ArcLengthbyLinearInterpolation

pk =

si+1 − sk

si+1 − si

pi +sk − si

si+1 − si

pi+1

TP3

CurvatureandNormal

lowcurvature

highcurvatureDurand

Curvature(κ):derivativeoftangentwithrespecttoarclength(dt(s)/ds)•  howfastthecurvepullsaway

fromastraightline

•  alwaysorthogonaltotangent

•  constantforacircle•  zeroforastraightline

Normal:normalizedcurvature

•  vectorpointstothecenterofcurvature

•  the2ndof3orientationsforsweptsurfaces

TorsionandBinormal

Béchet

Torsion:deviationofthecurve

fromtheplaneformedbythe

tangentandnormalvectors

•  zeroforaplanecurve•  binormalvectorpointstothewinding

directionofthespacecurve

•  the3rdof3orientationsforsweptsurfaces

Acurveisa1Dmanifoldinaspaceofhigherdimension

• Plane(2D)curves,describedby:

•  position,tangent,curvature

• Space/skew(3D)curves,describedby:•  position,tangent,curvature,torsion

FrenetFrame

Givenacurve q(t)wecanattachasmoothlyvarying

coordinatesystemconsistingofthreebasisvectors

(reparameterizedtoarclength):

•  tangent:t(s) = q’(s(t))(normalized)

•  normal:n(s) = t’(s)/||t’(s)||•  binormal:b(s) = n×t

DuetoJeanFrédéricFrenet(1847)andJosephAlfredSerret(1851)

Aswemovealongq(t),theFrenetframe(t(s),b(s),n(s))variessmoothly

Curless

q(t)

q’(s)

q’’(s)

normalplane

formedbynandb

osculatingplane

formedbynandt

FrenetSweptSurfaces

Curless,wikipedia

Orienttheprofilecurve p(u)usingtheFrenetframeofthetrajectoryq(t)•  putp(u)inthenormalplaneofq(t)•  placeoponq(t)•  alignpx(u)withb•  alignpy(u)with−n

Ifq(t)isacircle,yougetasurfaceofrevolutionexactly!

q(t)

py(u)

px(u)

op

p(u)

Variations

Curless,Funkhouser

q(t)

p(u)

q

b t

n

Severalvariationsarepossible:

•  scalep(u)asitmoves,

possiblyscaledto||q(t)||• morphp(u)intosomeothercurve

f(u)asitmovesalongq(t)

ProblemswithSweptSurfaces

Whathappensatinflectionpoints?

•  curvaturegoestozero•  thennormalflips!

•  resultinginanon-smoothsweptsurface

Also,difficulttoavoidself-intersection

Curless,Fussell,Durand,Chenney

Free-formSurfaces

Sweptsurfacesaregreat,butwewould

liketorepresent“free-form”(asymmetric,

irregular)curvesandsurfaces

Wewouldalsoliketogivemodelbuilders

anintuitivecontrolofasmoothshape

•  specifyobjectswithafewcontrolpoints

•  resultinginvisuallypleasing(smooth)objects

Schulze

CAGD(Computer-AidedGeometricDesign):

areaofCGdealingwithfree-formshapes

1960’s:•  theneedformathematicalrepresentationsoffree-form

shapesbecameapparentintheautomotiveand

aeronauticindustries

•  PauldeCasteljau&PierreBézierindependentlydevelopedthetheoryofpolynomialcurves&surfaces

•  whichbecamethebasictoolfordescribingandrendering

free-formshapes

PolynomialSurfaces

TP3

ParametricPatches

Parametriccurvesandsurfacesgiveandrequire

fewerdegreesofcontrolthanpolygonalmeshes

•  userscontrolafewpoints

•  programsmoothlyfillsintherest

•  representationprovidesanalyticalexpressionsfornormals,tangents,etc.

Surfaceispartitionedintopatches:

• piecewiseparametricsurfaces(3Dsplines)

• eachdefinedbycontrolpointsforming

acontrolnet

Mostpopularformodelingare

Bézier,B-splines,andNURBS

• we’llstudytheseas2Dsplinesfirst,

thenwe’llusethemas3DpatchesFunkhouser,FvDFHFig.11.44

MeasuresofJoint

Smoothness

Parametriccontinuity:

• continuousbyparametert• usefulfortrajectories• 0thorder,C0�curvesegmentsmeet(joinpoint):f2(0) = f1(1)

• 1storder,C1�

1stderivatives,velocities,areequalatjoinpoint:f2’(0) = f1’(1)

• 2ndorder,C2�

2ndderivatives,accelerations,areequalatjoinpoint

y (t )

x(t )

SC1C0

C2

Join point

Foley,vanDam92

�

C1

�

C0

�

C2y(t)

x(t)

Hodgins,Marschner

C0 C1 C2

JointSmoothness

C0continuous

•  curve/surfacehasnobreaks/gaps/holes• modelis“watertight”

C1continuous

• model“lookssmooth,no

facets”(butsometimes

lookslikealumpypotato)

C2continuous

•  looksmorepolished:

smoothspecularhighlights

DurandC2almosteverywhere C1only

C 0

C1

C2

G1

G2

MeasuresofJointSmoothness

Geometriccontinuity:

• continuousbyparameters (arclength)• usefulfordefiningshapes

• 1storder,G1

1stderivatives,tangents,areinthesame

directionandofproportionalmagnitude

atjoinpoint:f2’(0) = k f1’(1), k > 0

• 2ndorder,G2

2ndderivatives,curvatures,

areproportionalatjoinpoint

� GncontinuityisusuallyaweakerconstraintthanCncontinuity

(e.g.,the“speed”alongthecurvedoesnotmatter)

ButneitherformofcontinuityisguaranteedbytheotherShirley,Marschner

G1butnotC1whentangentdirectiondoesn’tchange,

butthemagnitudechangesabruptly

y (t )

x (t )

P1P2

Q3P3

Q2Q1

TV2

TV3

�

C1�

G1

y(t)

x(t)

Rockwoodetal.,Marschner,FvD

G1notC1 C1notG1

Rockwoodetal.,Marschner

Whenthecurvep(t)goestozero,velocitychanges

direction,andstartsagain

p(t)

CubicSplines

Arepresentationofcubicsplineconsistsof:

•  fourcontrolpoints(whyfour?)

•  thesearecompletelyuserspecified

•  determineasetofblendingfunctions

Thereisnosingle“best”representationofcubicspline:

*n/awhensomeofthecontrol“points”aretangents,notpoints

Cubic Interpolate? Local? Continuity Affine? Convex*? VD*?

Hermite � � C1 � n/a n/a

Cardinal

(Catmull-Rom)

except

endpoints

� C1 � no no

Bézier endpoints � C1 � � �

natural � � C2 � n/a n/a

B-splines � � C2 � � �

Watt

BézierCurve

NamedafterPierreBézier,

acardesigneratRenault

Independentlydevelopedby

PauldeCasteljauatCitroën

Hasanintuitivegeometric“feel”,

easytocontrol•  commoninterfaceforcreatingcurvesin

drawingprograms

• usedinfontdesign(Postscript)

Foley&vanDam

BézierCurve

Usesanarbitrarynumberof

controlpoints(notjustcubic)

• thefirstandlastcontrolpointsinterpolatethecurve

• therestapproximatethecurve,controlpointiexertsthestrongestattractionatu = i/n, 1 ≤ i < n−1, 0 ≤ u ≤ 1

• tangentatthestartofthecurveisproportionaltothevectorbetweenthefirstandsecondcontrolpoints

• tangentattheendofthecurveisproportionaltothevectorbetweenthesecondlastandlastcontrolpoints

• then-thderivativeatthestart(end)ofthecurvedependsonthefirst(last)n+1controlpoints

p0

p1

p2

p3

p1-p

0

p3-p2

f’(0)=3(p1-p0)

f’(1)=3(p3-p2)

cubic

Shirley

deCasteljauAlgorithm

AgeometricevaluationschemeforBézier:

createsBéziercurveiteratively

Tocomputef(u):•  connectadjacentcontrolpointswith

straightlinesintoacontrolpolygon

•  createtheuinterpolatepoints,u ∈ [0,1],ontheselines•  ateachiteration,therearen-1suchpoints

•  connectthenewpointswithstraightlines

•  repeatuntilonlyonenewpointiscreated

u

u

u

1–u 1–u

1-u

1–u

u1–u

1–u

uu

deCasteljauQuadraticBézier

4/5

1/5 4/5

1/5 1/5

4/5

quadratic

Letu = 45

pk = p0 + 45 (p1 − p0 ) = 1

5 p0 − 45 p1

q0 = 15 ( 15 p0 + 4

5 p1)+ 45 ( 15 p1 + 4

5 p2 )Ormoregenerally:

f (u) = (1− u)((1− u)p0 + up1)+ u((1− u)p1 + up2 )whichisthequadraticBéziercurve:

f (u) = (1− u)2p0 + 2u(1− u)p1 + u2p2

AquadraticBéziercurvehas3controlpoints

deCasteljauCubicBézierGivenfourcontrolpointsp0, p1, p2, p3,use

deCasteljaualgorithmtobuildacubicBéziercurve

f(u), 0 ≤ u ≤ 1,withp0 = f(0), p3 = f(1)asshown:

1

2 3

4 1

2 3

4

�

p0

�

p1

�

p2

�

p3

q0 = p0 + u p1 − p0( )= (1− u)p0 + up1

q1 = (1− u)p1 + up2q2 = (1− u)p2 + up3 u = ½

u = ¼u = ¾

1–u

u

u

1–u

u 1–u

q0

q1

q2

u

1–u u

1–ur0 r1u 1–u

f(u)

r0 = (1− u)q0 + uq1r1 = (1− u)q1 + uq2

f(u) = (1− u)r0 + ur1f(u) = (1− u)3p0 + 3u(1− u)

2p1 + 3u2 (1− u)p2 + u

3p3

Drawoutthecurvebysweepingthroughtime

deCasteljauCubicBézier

[wikipedia]

f(u) = (1− u)3p0 + 3u(1− u)2p1 + 3u

2 (1− u)p2 + u3p3

Thenset:

f '(0) = 3(p1 − p0 )f '(1) = 3(p3 − p2 )

CubicBézierCurve

Controlpointsconsistofendpointinterpolationsand

derivatives:

Constraintmatrix

Basismatrix:

C =

1 0 0 01 1

3 0 01 2

313 0

1 1 1 1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

B = C−1 =

1 0 0 0−3 3 0 03 −6 3 0−1 3 −3 1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

f(u) = a0 + u1a1 + u2a2 + u3a3

p0 = f(0) = a0 + 01a1 + 02a2 + 03a3

p3 = f(1) = a0 + 11a1 + 12a2 + 13a33(p1 − p0 ) = f '(0) = a1 + 2*0

1a2 + 3*02a3

p1 = 13 (f '(0)+ 3p0 ) = a0 + 1

3 a1 + 02a2 + 0

3a33(p3 − p2 ) = f '(1) = a1 + 2* 1

1a2 + 3*12a3

p2 = 13 (3p3 − f '(1)) =1a0 + 2

3 a1 + 13 a2 − 0a3

Blendingfunctions:

CubicBézierCurve

Watt,Hodgins

f (u) = (1− u)3p0 + 3u(1− u)2p1 + 3u2 (1− u)p2 + u3p3

= (1− 3u + 3u2 − u3 )p0 + (3u − 6u2 + 3u3 )p1 + (3u

2 − 3u3 )p2 + u3p3

x

y

b(u)

u

uB = bi (u)i=0

n

∑

b0,3(u)

b0,3(u)

b1,3(u)

b1,3(u)

b2,3(u)

b2,3(u)

b3,3(u)

b3,3(u)

CubicBézierProperties

Properties:

•  eachbispecifiestheinfluenceof pi

•  convexhull:∑bi = 1, bi ≥ 0•  interpolatesonlyatp0andp3•  b0 =1atu = 0,b3 =1atu = 1•  b1andb2 neverreach1

•  thebasisfunctionsareeverywhere

non-zero,exceptattheendpoints

�thecontrolpointsdonotexert

localcontrol

•  thecurvesaresymmetric:reversing

thecontrolpointsyieldsthesame

curve

b0

b1 b2

b3

u

b(u)

u = 1 u = 0

p0

p2

p3

p1

Durand,Hodgins

Watt

Non-LocalControl

Everycontrolpointaffectseverypoint

onthecurve(excepttheendpoints)

Movingasinglecontrolpointaffects

thewholecurve!

Curless

VariationDiminishingProperty

Béziercurveshavethevariationdiminishingproperty:

eachisnomore“wiggly”thanitscontrolpolygon

�doesnotcrossalinemorethanitscontrolpolygon

VariousBéziercurves,ofdegrees2-6:

Shirley

BernsteinBasisPolynomials

Theblending/basisfunctionsforBéziercurvescan

ingeneralbeexpressedastheBernsteinbasis

polynomials:

Béziercurveeqn:

bk,n (u) =nk

⎛⎝⎜

⎞⎠⎟uk (1− u)n−k = n!

k!(n − k)!uk (1− u)n−k

f(u) = n!k!(n − k)!

uk (1− u)n−kpkk=0

n

∑

Shirley

Multiple-segmentcubicBéziercurvecanachieve

•  G1continuityif:q0 = f2(0) = f1(1) = p3

and(q1 − q0) = k(p3 − p2),thethreepoints(p2,p3 = q0,andq1)arecollinear

•  ifyouchangedoneofthesethree,youmustchangetheothers,butonly

needtochangethesethree,

notp1forexample�localsupport

•  C1continuityifk = 1

•  can’tguaranteeC2orhighercontinuity•  eachadditionaldegreeofcontinuityrestrictsthepositionofanadditionalcontrolpoint�cubicBézierhasnonetospare

�

p3 = q0�

p2

�

q1�

f1(u)

�

f 2(u)

JoiningBézierCurves

Shirley

BézierCurve/SurfaceProblems

TomakealongcontinuouscurvewithBézier

segmentsrequiresusingmanysegments

Maintainingcontinuityrequiresconstraintsonthe

controlpointpositions

•  theusercannotarbitrarilymovecontrolpointsand

automaticallymaintaincontinuity

•  theconstraintsmustbeexplicitlymaintained

•  itisnotintuitivetohavecontrolpointsthatarenotfree

Consider:B-spline