EECS487:Interactive

ComputerGraphicsLecture37:•  B-splinescurves•  RationalBézierandNURBS

CubicSplinesArepresentationofcubicsplineconsistsof:

•  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 � � �

NaturalCubicSplineAnaturalcubicspline’scontrolpoints:

• positionofstartpoint• 1stderivativeofstartpoint

• 2ndderivativeofstartpoint

• positionofendpoint

•  constraintandbasismatrices:

•  subsequentsegmentsassumethepositionand1stand2nd

derivativesoftheendpointoftheprecedingsegment

C =

1 0 0 00 1 0 00 0 2 01 1 1 1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

, B = C−1 =

1 0 0 00 1 0 00 0 1

2 0−1 −1 − 1

2 1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

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

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

p1 = f '(0) = a1 + 2*01a2 + 3*0

2 a3p2 = f ''(0) = 2*11a2 + 6*0

2 a3p3 = f (1) = a0 + 11a1 + 12 a2 + 13a3

NaturalCubicSpline

Givenncontrolpoints,anaturalcubicsplinehasn−1 segments

Forsegmenti:

Set:

O’Brien

fi (0) = pi−1, i = 1,…,nfi (1) = pi , i = 1,…,n

fi' (0) = fi−1

' (1), i = 1,…,n −1fi"(0) = fi−1

" (1), i = 1,…,n −1

f1"(0) = fn

"(1) = 0

NaturalCubicSplines

Eachcurvesegment(otherthanthefirst)receives

threeoutofitsfourcontrolpointsfromthe

precedingsegment,thisgivesthecurveC2continuity

Howeverthepolynomialcoefficientsare

dependentonallncontrolpoints• controlisnotlocal:anychangeinanysegment

maychangethewholecurve

• curvetendstobeill-conditioned:asmallchange

atthebeginningcanleadtolargesubsequentchanges

B-splines

TP3,Watt

Givenn(≥ d + 1)controlpoints,aB-splinecurvehasn−dsegments

•  disthedegreeofeachB-splinesegment

•  thesegmentsarenumbereddton−1,foreaseofnotation•  numberofcontrolpointsisindependentofthedegree

•  unlikeaBézierspline,whereaddingacontrol

pointnecessarilyincreasesdegreebyone,

•  andunlikemulti-segmentedBéziercurvewheremultiplecontrol

pointssupportinganewsegmentmustbeaddedatthesametime

•  segmentdegree(d)isalsocurvedegree

B-splinesofdegreedaresaidtohaveorderk(= d + 1)

AdvantagesofB-splinesMainadvantagesofB-splines:•  numberofcontrolpointsnotlimitedbydegree(d)•  automaticCd�1continuity

•  localcontrol

TocreatealargemodelwithC2continuityand

localcontrol,youprettymuchwanttousecubic

B-splines

Asidefromthefirstsegment,eachB-spline

segmentsharesthefirstdcontrolpointswithits

precedingsegment•  soundslikenaturalspline…

howcanB-splineshavelocalcontrol?

LocalControl

UnlikenaturalsplinesandBéziercurves,B-splines’

controlpointsarenotderivatives

Insteadeachsegmentisaweighted-sumofdbasisfunctions(only),givingthecontrolpointslocal

control

HenceBasisspline

WhyisBcalledtheBasisMatrix?

PolynomialsasaVectorSpace

Polynomialsf (u) = a0 + a1u + a2u2 + … + anun

•  canbeadded:justaddthecoefficients

•  canbemultipliedbyascalar:multiplythecoefficients

•  areclosedunderadditionandmultiplicationbyscalar

•  i.e.,theresultisstillapolynomial

�It’savectorspace!

Avectorspaceisdefinedbyasetofbasis

•  linearlyindependentvectors•  linearcombinationofthebasisvectorsspansthespace

•  herevector=polynomial

Durand

CanonicalPowerBasis1, u, u2, u3, … , un

•  areindependent•  anypolynomialisalinearcombinationofthese,

a0 + a1u + a2u2 + … + anun

•  oftencalledthecanonicalbasisfunctions

JustaswithEuclideanspace,

thereareinfinitenumberofpossiblebasis

Forcubic,thebasisfunctionscouldbe,forexample:

•  1, 1+u, 1+u+u2, 1+u–u2+u3

•  u3, u3–u2, u3+u, u3+1

Durand

BasisMatrixandBasisFunctions

Abasismatrix(B)transformsthecanonicalbasis(u)toanotherbasis:

Thebi(u)’sarethebasisfunctionsoftheotherbasis(we’veknownthemastheblendingfunctions)

f (u) = ua = uBp = (1− u)p0 + up1 = bi (u)pii=0

n

∑

uB = bi (u)i=0

n

∑

B-splines

Givenncontrolpoints,therearen−dsegments

• wecallthesegmentsfi(u),d ≤ i < n•  eachsegmenthasaunitrange,0 ≤ u ≤1

• wecalltheentireB-splinecurvewithncontrolpointsf(t)

Theparametersti’swheretwosegmentsjoinare

calledknots

•  thestartandendpoints(tdandtn)

arealsocalledknots

•  therange[td, tn]isthedomain

ofaB-splinecurve

•  theparameteruofsegmentiisscaledtoti ≤ u < ti+1

y(t) P9

P3

P1

P8

P4

P7

P6

P5

P2

P0

Q9

Q8

Q7

Q6

Q5

Q4

Q3

t3

t5

t6

t7

t4

t8

t9

t10

Knot

Control pointx(t)

Foley,vanDam90

f3(u)f4(u)

f5(u)

f6(u)

f7(u)

f8(u)

f9(u)

acubicB-spline

UniformB-splines

TheknotsofauniformB-splinesarespacedat

equalintervals

y(t) P9

P3

P1

P8

P4

P7

P6

P5

P2

P0

Q9

Q8

Q7

Q6

Q5

Q4

Q3

t3

t5

t6

t7

t4

t8

t9

t10

Knot

Control pointx(t)

f3(u)f4(u)

f5(u)

f6(u)

f7(u)

f8(u)

f9(u)

acubic

B-spline

Foley,vanDam90

Arbitrarycurveshaveanuncountablenumberofparameters

Real-numberfunctionvalueexpandedintoaninfinitesetofbasisfunctions:

Approximatebytruncatingsetatsomereasonablepoint,e.g.,3:

WhatDegreeisSufficient?

O’Brien

f (u) = bi (u)pii=0

∞

∑

f (u) = bi (u)pii=0

3

∑

Inthelinearcase,thebasisfunctionsare

b0(u) = (1−u)andb1(u) = u

(a.k.a.tent/trianglebasis,thei’thfunctionsareshiftedversionsofthe0’th)

LinearB-splineSegment

0 1 2

1

James,Hanrahan

b(u) =

0 u < −11+ u −1< u < 01− u 0 < u <10 u >1

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

LinearB-splineCurve

ConsiderusinglinearB-splines(d=1,k=2)todraw

apiecewiselinearcurve(apolyline)

Todrawthecurve,weperformlinearinterpolation

ofasetofcontrolpointsp0, …, pn–1

Forsegmenti,wewritetheinterpolatinglinear

curveasfi(u) = (1−u)pi–1+ upi,where u =t − titi+1 − ti

∈ 0,1[ ]

p0 pn–1 pi–2

pi–1 pi+1 pi+2

pi

t = i–1

t = i

t = i+1

t = i+2

t = i+3

fi(u)

[Craig]

LinearB-splines

Theinfluenceofcontrolpointpionthewholecurve

isthusthe“tent/triangle”function:

ThehardestpartofworkingwithB-splinesis

keepingtrackofthetediousnotations!

[Craig]

ti ti+1 ti+2

bi(t)1

bi (t) =

t − titi+1 − ti

, ti ≤ t < ti+1,

ti+2 − tti+2 − ti+1

, ti+1 ≤ t < ti+2 ,

0, everywhereelse

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

t − titi+1 − ti

ti+2 − tti+2 − ti+1

pi−1pi

pi+1

LinearB-splines

Linearlyinterpolatingthesetofcontrolpointstodraw

thecurve:

fi (u) = (1− u)pi−1 + upi =ti+1 − tti+1 − ti

pi−1 +t − titi+1 − ti

pi , u =t − titi+1 − ti

, ti ≤ t < ti+1

fi+1(u) = (1− u)pi + upi+1 =ti+2 − tti+2 − ti+1

pi +t − ti+1ti+2 − ti+1

pi+1, u =t − ti+1ti+2 − ti+1

, ti+1 ≤ t < ti+2

[Craig]

ti–1 ti ti+1 ti+2 ti+3

f(t) upi+1upi

(1�u) pi(1�u) pi−1

pipi−1p0

pi+1pi−2 pn−1

pi+2

fi(u)

fi+1(u)

LinearB-splines

Wecanrewritethesegmentfunctionsas:

Andforthewholecurve:

wherebi(t)’sarethebasisfunctions(inthiscase,linear)

fi (t) = bi−1(t)pi−1 + bi (t)pi , ti ≤ t < ti+1fi+1(t) = bi (t)pi + bi+1(t)pi+1, ti+1 ≤ t < ti+2

[Craig]

f (t) = fi (t) = bi (t)pii=1

n−1

∑i=1

n−1

∑

fi (u) = (1− u)pi−1 + upi =ti+1 − tti+1 − ti

pi−1 +t − titi+1 − ti

pi , u =t − titi+1 − ti

, ti ≤ t < ti+1

fi+1(u) = (1− u)pi + upi+1 =ti+2 − tti+2 − ti+1

pi +t − ti+1ti+2 − ti+1

pi+1, u =t − ti+1ti+2 − ti+1

, ti+1 ≤ t < ti+2

QuadraticB-splinesQuadraticB-splines(d=2,k=3)aredrawnbytwo

interpolationsteps,similarbutdifferenttoquadratic

Bézier

WhereasdeCasteljaualgorithmperformstheiterative

interpolationsforBéziercurves,deBooralgorithm

doessoforB-splines[TP3,Buss]

4/5

1/5 4/5

1/5 1/5

4/5

Bézierquadratic:

parameter“along”edge

B-splinesquadratic:

parameter“around”

controlpoint

pi–2

pi–1

pi

qi–1

qi

f(t)

deBoorAlgorithmDeBooralgorithmisaniterativeinterpolation

algorithmthatgeneralizesdeCasteljau’salgorithm

ToevaluateaB-splinecurvef(t) atparametervaluet:1.  determinethe[ti, ti+1)inwhichtbelongs;

d ≤ i < n,thedomainofthecurveis[td, tn]2.  tocomputef(t)ofdegreed,firstinterpolatebetween

controlpointsp’s3.  then,inabottomupfashion,continuetoperformrroundsofpairwiselinearinterpolations,untilr = d,using:

[TP3,Buss]

f j ,dr (t) =

t j+k−r − tt j+k−r − t j

f j−1r−1(t)+

t − t jt j+k−r − t j

f jr−1(t),

t j ≤ t < t j+k−r ,1≤ r ≤ d,

j = i − d + r, i − d + r +1, …,i

QuadraticB-splines

UsingthedeBooralgorithmwefirstcomputeqi–1

andqi(note:overtwoknotintervals):

[TP3,Buss]

B-splinesquadratic

pi–2

pi–1

pi

qi–1

qi

f(t)

qi−1 = fi−1,21 (t) = ti+1 − t

ti+1 − ti−1pi−2 +

t − ti−1ti+1 − ti−1

pi−1, ti−1 ≤ t < ti+1

qi = fi,21 (t) = ti+2 − t

ti+2 − tipi−1 +

t − titi+2 − ti

pi , ti ≤ t < ti+2

QuadraticB-splines

Thenwelinearlyinterpolatebetweenqi–1andqiin

asecondround(r = 2)ofinterpolation:

[TP3,Buss]

B-splinesquadratic

pi–2

pi–1

pi

qi–1

qi

f(t)

fi,22 (t) = ti+1 − t

ti+1 − tiqi−1 +

t − titi+1 − ti

qi , ti ≤ t < ti+1

f (t) = ti+1 − tti+1 − ti

ti+1 − tti+1 − ti−1

pi−2

+ ti+1 − tti+1 − ti

t − ti−1ti+1 − ti−1

+ t − titi+1 − ti

ti+2 − tti+2 − ti

⎛⎝⎜

⎞⎠⎟pi−1

+ t − titi+1 − ti

t − titi+2 − ti

pi

QuadraticB-splines

Thecontrolpointpiinfluencesfi,2(t),fi+1,2(t),andfi+2,2(t),fromwhichwecanassembleitsblending

function:

[TP3,Buss]

!

bi (t) =

ti+1 − tti+1 − ti

ti+1 − tti+1 − ti−1

, ti ≤ t < ti+1,

ti+1 − tti+1 − ti

t − ti−1ti+1 − ti−1

+ t − titi+1 − ti

ti+2 − tti+2 − ti

, ti+1 ≤ t < ti+2 ,

t − titi+1 − ti

t − titi+2 − ti

, ti+2 ≤ t < ti+3,

0, everywhere!else

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

fi(t)fi+1(t)

fi+2(t) InterpolationandBasisFunctions

Bézier B-splineinterpolation deCasteljau deBoor

basisfunctionsBernstein

polynomials

Cox-deBoor

recurrence

Cox-deBoorRecurrencedeBooralgorithmconstructsbasisfunctions“bottom-

up”,whereasCox-deBoorrecurrencegeneratesthe

basisfunctions“top-down”

Letbi,k(t)beak-thorderbasisfunctionforweighting

controlpointpi,•  ifthedenominatoris0(non-uniformknots),

thebasisfunctionisdefinedtobe0•  Cox-deBoorrecurrenceessentiallytakesalinearinterpolationoflinearinterpolationsoflinearinterpolations,similartothedeCasteljaualgorithm

bi,1(t) =1, ti ≤ t < ti+1 (both ≤ forlastsegment),0, otherwise

⎧⎨⎪

⎩⎪

bi,k (t) =t − ti

ti+k−1 − tibi,k−1(t)+

ti+k − tti+k − ti+1

bi+1,k−1(t)

CubicB-splines

For4thorder(cubic)B-splines,the

recursivedefinitionstartsatbi,4(t):

base:

linear:

quadratic:

cubic:

astepfunctionof1interval

apiecewiselinear“tent”

functionspanning2intervals

apiecewisequadratic

functionspanning3intervals

apiecewisecubic

functionspanning4intervals

Illustratedwithuniformlyspacedknots

bi,4(t)

bi,1(t)

bi,3(t)

bi,2(t)

bi,1(t) =1, ti ≤ t < ti+10, otherwise

⎧⎨⎪

⎩⎪

bi,2 (t) =t − titi+1 − ti

bi,1(t)+ti+2 − tti+2 − ti+1

bi+1,1(t)

bi,3(t) =t − titi+2 − ti

bi,2 (t)+ti+3 − tti+3 − ti+1

bi+1,2 (t)

bi,4 (t) =t − titi+3 − ti

bi,3(t)+ti+4 − tti+4 − ti+1

bi+1,3(t)

UniformCubicBasisFunctionConstructedfromtheCox-deBoorrecurrence• takingadvantageoffixedintervalbetweenknots

• consideringonlyintervalsforwhichthebasisfunction

isnon-zero

�

bi(t)

ti ti+1 ti+2 ti+3 ti+4

[Buss,Shirley,Gleicher]

!

Let!ti = i,!specializing!for!i = 0:

b0,4 (t) = !

t 3

6, 0 ≤ t <1,

−3t 3 +12t 2 −12t + 46

, 1≤ t < 2,

3t 3 − 24t 2 + 60t − 446

, 2 ≤ t < 3,

−t 3 +12t 2 − 48t + 646

, 3≤ t < 4,

0, everywhere!else

⎧

⎨

⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

bi,1(t) =1, ti ≤ t < ti+10, otherwise

⎧⎨⎪

⎩⎪b1,2 (t) = t − ti( )bi,1(t)+ ti+2 − t( )bi+1,1(t)bi,3(t) =

t − ti2

bi,2 (t)+ti+3 − t2

bi+1,2 (t)

bi,4 (t) =t − ti3

bi,3(t)+ti+4 − t3

bi+1,3(t)

y(t)

P3

P1

P8

P4

P7

P6

P5P

2

P0

Q8

Q7

Q6

Q5

Q4

Q3

Knot

Control pointx(t)

P’4

P’’4

P’’4Curve

P’4Curve

CurveP4

LocalControlPropertyForuniform,multi-segmentB-splinecurves,theknotvaluesareequally

spacedandeachbasisfunctionisacopyandtranslateofeachother

WedefinetheentiresetofcurvesegmentsasoneB-splinecurveint:

Thecurveisalinearcombinationofall

thebasisfunctionsofthesegments:

f(t) = bi (t)pi

i=0

n−1

∑ , t ∈[3,n]

domainofcurve

�

b0(t)

�

b1(t)

�

b2(t)

�

b3(t)

�

b4 (t)

�

b5(t)

�

b6(t)

�

b7(t)

�

b8(t)

•  eachsegmentisinfluencedbyfour

(non-zero)basisfunctions

•  eachcontrolpointisscaledbyasinglebasisfunction

•  eachbasisfunctionisnon-zerooverfoursuccessiveintervalsint�  eachcontrolpointinfluencesfour

segments(only)�localcontrol

f(t)

BBs—2

t

—66

—56

—46

—36

—26

—16

BBs—1

BBs—3BBs0

0 1

�

bi−3

�

bi

�

bi−2

�

bi−1

t

b(t)

�

b4 (t)

t=i t=i+1 t=i+2 t=i+3 t=i+4

Foley,vanDam90

Watt00

f3(u)f8(u)f4(u)

f5(u)f6(u)

f7(u)

ConvexHullProperty

Thebasisfunctionis≥ 0andsumsto

unityintherangetitoti+4 �  allthecontrolpointsformaconvexhull

�  thewholecurveiswithintheconvexhull

Betweenknotvalues,thefourbasis

functionsarenon-zeroandsumtounity

Ateachknotvalue,onebasisfunction

“switchesoff”andanother“switcheson”,and

threebasisfunctionsarenon-zeroandsumtounity

domainofcurve

�

b0(t)

�

b1(t)

�

b2(t)

�

b3(t)

�

b4 (t)

�

b5(t)

�

b6(t)

�

b7(t)

�

b8(t)

f(t)

BBs—2

t

—66

—56

—46

—36

—26

—16

BBs—1

BBs—3BBs0

0 1

�

bi−3

�

bi

�

bi−2

�

bi−1

u

b(u)

bi−k (t)k=0

3

∑ = 1, ti ≤ t < ti+1

bi− k(t ) ≥ 0, 3 ≤ i < n, 0 ≤ k ≤ 3

�

bi(t)

ti ti+1 ti+2 ti+3 ti+4

UniformCubicB-spline

SegmentBasisFunctions

BasisfunctionsforasingleB-splinesegment���• shiftedpiecesofasinglebasisfunctiontou∈[0,1]range

Specializingfori = 0:

!

bi,4 (u) =u3

6, u = t, !0 ≤ t <1

bi−1,4 (u) =−3u3 + 3u2 + 3u +1

6, u = t −1, !1≤ t < 2

bi−2,4 (u) =3u3 − 6u2 + 4

6, u = t − 2, !2 ≤ t < 3

bi−3,4 (u) =(1− u)3

6, u = t − 3, !3≤ t < 4

f(t)

BBs—2

t

—66

—56

—46

—36

—26

—16

BBs—1

BBs—3BBs0

0 1

�

bi−3

�

bi

�

bi−2

�

bi−1

u

b(u)

[Shirley,Gleicher]

Controlpointsforonesegmentfi(u)arepi-3,pi-2,

pi-1,pi,3 ≤ i < n,recall:thecontrolpointscantakeonarbitraryvalues(geometricconstraints)

Asegmentisdescribedas:

ThecubicB-splinesegment

basismatrixis:

y(t) P9

P3

P1

P8

P4

P7

P6

P5

P2

P0

Q9

Q8

Q7

Q6

Q5

Q4

Q3

t3

t5

t6

t7

t4

t8

t9

t10

Knot

Control pointx(t)

Foley,vanDam90

f3(u)f4(u)

f5(u)

f6(u)

f7(u)

f8(u)

f9(u)

UniformCubicB-splineSegment

fi (u) = bi−3+ j (u)pi−3+ jj=0

3

∑ , u ∈[0,1]

= (1− u)3

6pi−3 +

3u3 − 6u2 + 46

pi−2 +

−3u3 + 3u2 + 3u +16

pi−1 +u3

6pi

B = 16

−1 3 −3 13 −6 3 0−3 0 3 01 4 1 0

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

BézierisNotB-spline

Relationshiptothecontrolpointsisdifferent

Bézier

B-spline

Durand

Interpolation

AB-splinecurvedoesn’thavetointerpolateany

ofitscontrolpoints

Interpolationwith

MultipleControlPoints

AB-splinecurvecanbemadetointerpolateoneor

moreofitscontrolpointsbyaddingmultiplecontrol

pointsofthesamevalue,atthelossofcontinuity

Examples:

Watt00

C2G2

C2G1 C2G0

curvebecomesastraight

lineoneithersideofthe

controlpoints

Multiplecontrolpointsreducescontinuity:the

intersectionbetweenthetwoconvexhullsshrinks

fromaregiontoalinetoapoint,andcausesthe

adjacentsegmentstobecomelinear

P0

P2

P4

P1

P3

Q3

Q4

(a)

P0

P3

P1

= P

2P

4

Q3

Q4

(b)

P0

P4

P1

= P

2 = P

3

Q3

Q4

(c)

Q3 Convex hull

Q4 Convex hull

Foley,vanDam90

Interpolationwith

MultipleControlPoints

curvebecomesa

straightlineon

eithersideofthe

controlpoints

Non-uniformB-splinesinterpolatewithoutcausing

adjacentsegmentstobecomelinearbyusing

multipleknotsinsteadofmultiplecontrolpoints

Theintervalbetweentiandti+1maybenon-uniform;

whenti = ti+1,curvesegmentfiisasinglepoint

Non-uniformB-splines

y(t) P9

P3

P1

P8

P4

P7

P6

P5

P2

P0

Q9

Q8

Q7

Q6

Q5

Q4

Q3

t3

t5

t6

t7

t4

t8

t9

t10

Knot

Control pointx(t)

Foley,vanDam90

f3(u)f4(u)

f5(u)

f6(u)

f7(u)

f8(u)

f9(u)

InterpolationwithMultipleKnotsUniformB-splines:

UniformB-splines,

multiplecontrolpoints:

Non-uniformB-splines,

multipleknots:

Watt00

curvebecomesa

straightlineon

eithersideofthe

controlpoints

curvedoesn’t

becomeastraight

line(though

continuityisstilllost)

t = 0,1, 2,3, 4, 4, 4, 5, 6, 7,8[ ]f4 (u)andf5(u)(Q4 andQ5 infigure)shrinksto0

Non-Uniform

B-splinesBasisFunctions

Becausetheintervalsbetweenknotsarenot

uniform,thereisnosinglesetofbasisfunctions

Instead,thebasisfunctionsdependontheintervals

betweenknotvaluesandaredefinedrecursivelyin

termsoflower-orderbasisfunctions(usingtheCox-

deBoorrecurrence)

ABéziercurveisreallyanon-uniformB-splineswith

no(interior)knotbetweencontrolpoints• B-splinescanberenderedasaBéziercurve,byinsertingmultipleknotsatthecontrolpoints,withnointeriorknot!

Polynomialcurvescannotrepresentconicsections/

quadricsexactly–formodelingmachineparts,e.g.

Whynot?Aconicsectionin2Disthe

perspectiveprojection

ofaparabolain3Donto

theplanez = 1,withthe

COPattheorigino

Polynomialcurvesareaffineinvariant,

butnotperspectiveinvariant

Instead,usearationalcurve,

i.e.,aratioofpolynomials:

RationalCurves

Merrell,Funkhouser,andrews.edu

f (u) = p1(u)p2 (u)

�

f(u)

�

fp (u)

RationalCubicBézierAswithhomogeneouscoordinate,arationalcurveisa

nonrationalcurvethathasbeenperspectiveprojected

CubicBézier:• addanextraweightcoordinate:wipi = (wixi, wiyi, wizi, wi)�(wiisthehomogeneouscoordinate)

•  rationalduetodivisionbyfinalweightcoordinate:

(=perspectivedivide)

• projectedtoz = 1:fp (u) =

wibi (u)pii=0

3

∑

wibi (u)i=0

3

∑

Ramamoorthi,Watt00

Ifthewi’sareall

equal,werecover

thenonrational

curve

AdvantagesofRationalCurves

Bothaffineandperspectiveinvariant

Canrepresentconicsasrationalquadratics

Weights(wi’s)provideextracontrol:

valuesaffect“tension”nearcontrolpoints

•  thewi’scannotallbesimultaneouslyzero

•  ifallthewi’sare≥ 0,thecurveisstillcontainedintheconvexhullofthecontrolpolygon

[Farin]

movingcontrolpoint changingweight

RoleoftheWeights(wi’s)

Forexample:largerw1meansthatthepre-image,

nonrationalcurvenearp1is“furtherup”inz,andtheprojectedimageis“pulled”towardsp1

[Farin,Watt]

�

f(u)

�

fp (u)

Non-UniformRationalB-Splines

f (u) =wibi,k (u)pi

i=0

3

∑

wibi,k (u)i=0

3

∑

wi = scalar weight for each control point

pi = control points

wipi = (wixi ,wiyi ,wizi ,wi )bi,k (u) = the B-splines basis functions

k = B-splines order

with:

AdvantagesofNURBS

Mostgeneral,canrepresent:• B-splines• BézierandrationalBézier•  conicsections

Properties:•  localcontrol•  convexhull(ifwi ≥ 0)•  variationdiminishing(ifwi ≥ 0)•  invariantunderbothaffineandprojectivetransformations

Standardtoolforrepresenting

freeformcurvesinCAGDapplications

cubicNURBcontrolpointscubicBéziercontrolpoints

[Farin]

HowtoChooseaSpline

Hermitecurvesaregoodforsinglesegments

whenyouknowtheparametricderivativeor

wanteasycontrolofit

Béziercurvesaregoodforsinglesegmentsor

patcheswhereausercontrolsthepoints

B-splinesaregoodforlargecontinuouscurves

andsurfaces

NURBSarethemostgeneral,andaregood

whenthatgeneralityisuseful,orwhenconic

sectionsmustbeaccuratelyrepresented(CAD)

Chenney