curve surface

8/13/2019 Curve Surface

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Representation of Curves &Surfaces

Prof. Lizhuang Ma

Shanghai Jiao Tong University


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Contents

Specialized Modeling Techniques

Polygon Meshes

Parametric Cubic Curves Parametric Bi-Cubic Surfaces

Quadric Surfaces

Specialized Modeling Techniques


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The Teapot


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Representing Polygon Meshes

Explicit representation

By a list of vertex coordinates

Pointers to a vertex list

Pointers to an edge list

)),,(),...,,,(),,,(( 222111 nnn zyxzyxzyxP


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Pointers to A Vertex List

),,,( 4321 VVVVV

)),,(),...,,,(( 444111 zyxzyx

)3,2,4(

)4,2,1(

2

1

P

P


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Pointers to An Edge List

),,,( 4321 VVVVV

)),,(),...,,,(( 444111 zyxzyx

),,(

),,(

),,,(

),,,(

),,,(

),,,(),,,(

4322

5411

1145

21244

2433

2322

1211

EEEP

EEEP

PVVE

PPVVE

PVVE

PVVEPVVE


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Parametric Cubic Curves

The cubic polynomials that define a curve

segment are of the form:

10)(

)(

)(

,

23

23

23

tdtctbtatz

dtctbtaty

dtctbtatx

zzzz

yyyy

xxxx

TtztytxtQ )]()()([)(


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The curve segment can be rewrite as

Where

zzzz

yyyy

xxxx

T

dcba

dcba

dcba

C

tttT 123

TCtztytxtQ T )]()()([)(


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Tangent Vector

Ttzdt

dty

dt

dtx

dt

dtQtQ

dt

d)]()()([)()( '

T

ttCTCdt

d

]123[ 2

T

zzzyyyxxx ctbtactbtactbta ]232323[ 222


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Continuity

Between Curve Segments G0 geometric continuity

Two curve segments join together

G1 geometric continuity

The directions (but not necessarily the magnitudes)

of the two segments tangent vectors are equal at a

join point


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Continuity

Between Curve Segments C1 continuous

The tangent vectors of the two cubic curve segments

are equal (both directions and magnitudes) at thesegments joint point

Cncontinuous

The direction and magnitude of through

the nth derivative are equal at the joint point

)]([/ tQdtd nn


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Continuity

Between Curve Segments


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Continuity

Between Curve Segments


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Three Types of


Hermite Curves Defined by two endpoints and two endpoint tangent

vectors Bzier Curves

Defined by two endpoints and two control pointswhich control the endpoint tangent vectors

Splines Defined by four control points


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Representation:

Rewrite the coefficient matrix as

where Mis a 4x4 basis matrix, Gis called the

geometry matrix So

1)(

)(

)(

)(2

3

44342414

43332313

42322212

41312111

4321t

t

t

mmmm

mmmm

mmmm

mmmm

GGGG

tz

ty

tx

tQ

TCtQ )(

MGC

4 endpoints or tangent vectors


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Where is called the blending functions

BGTMGtQ )(

TMB


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Hermite Curves

Given the endpointsP1andP4and tangent vectors atthemR1andR4

What is Hermite basis matrix MH Hermite geometry vector GH Hermite blending functions BH

By definition

GH=[P1 P4 R1 R4]


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Hermite Curves

Since

THH

T

HH

T

HH

T

HH

MGPQMGRQ

MGPQ

MGPQ

0123)1(0100)0(

1111)1(

1000)0(

4

'

1

'

4

1

0011

1110

2010

3010

4141 HHH MGRRPPG


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Hermite Curves

So

And

0011

0121

0032

1032

0011

1110

2010

3010 1

HM

TH tttttttttB 23232323 232132

HHHH BGTMGtQ )(


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Given the endpointsP1andP4and two control

pointsP2andP3which determine the endpoints

tangent vectors, such that

What is Bzier basis matrixMB Bzier geometry vector GB Bzier blending functionsBB

Bzier Curves

)(3)1(

)(3)0(

34

'

4

12'

1

PPQR

PPQR


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Bzier Curves

By definition

Then

So

HBB MGPPPP

3011

3000

0300

0301

4321

4321 PPPPGB

4141 RRPPGH

TMMGTMGtQHHBBHH

)()(

TMGTMMG BBHHBB )(


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Bzier Curves

And

HBBHHBB MGMMM

0001

0033

0363

1331

4

3

3

2

2

2

1

3 )1(3)1(3)1()( PtPttPttPttQ

T

ttttttBB 3223

)1(3)1(3)1( (Bernstein polynomials)


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Convex Hull


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The polynomial coefficients for natural cubic

splines are dependent on all n control points

Has one more degree of continuity than is inherent inthe Hermite and Bzier forms

Moving any one control point affects the entire curve

The computation time needed to invert the matrix can

interfere with rapid interactive reshaping of a curve

Spline


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B-Spline


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Cubic B-Spline

Has m+1control points

Has m-2cubic polynomial curve segments

Uniform The knots are spaced at equal intervals of the

parameter t Non-rational Not rational cubic polynomial curves

Uniform NonRational B-Splines

3,,..., 10 mPPP m

mQQQ ,...,, 43


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Curve segment Qi is defined by points

B-Spline geometry matrix

If Then


iiii PPPP ,,, 223

miPPPPG iiiiBSi 3,123

T

iiii ttttttT 1)()()(

23

1,)( iiiBsBsii tttTMGtQ


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So B-Spline basis matrix

B-Spline blending functions


TBs tttttttB 323233 1333463)1(6

1

0001

1333

4063

1331

6

1BsM


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The knot-value sequence is a nondecreasing

sequence

Allow multiple knot and the number of identical

parameter is the multiplicity Ex. (0,0,0,0,1,1,2,3,4,4,5,5,5,5)

So


)()()()()( 4,4,114,224,33 tBPtBPtBPtBPtQ iiiiiiiii


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Where isj th-order blending function for

weighting control pointPi

)()()(

)()()(

)()()(

,0

,1)(

3,1

14

43,

3

4,

2,1

13

32,

2

3,

1,1

12

21,

1

2,

1

1,

tBtt

tttB

tt

tttB

tB

tt

tttB

tt

tttB

tBtt

tttB

tt

tttB

otherwise

ttttB

i

ii

ii

ii

ii

i

ii

ii

ii

ii

i

ii

ii

ii

ii

ii

i

)(, tB ji


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Knot Multiplicity & Continuity

Since is within the convex hull of and

If is within the convex hull of ,

and and the convex hull of and ,so it

will lie on If will lie on

If will lie on both and , and the

curve becomes broken

)( itQ 23 , ii PP 1iP

)(,1 iii tQtt 23, ii PP

1iP 12 , ii PP 1iP

12 ii PP

)(,21 iiii tQttt 1iP

)(,321 iiiii tQtttt 1iP iP


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Multiplicity 1 : C2 continuity

Multiplicity 2 : C1 continuity

Multiplicity 3 : C0 continuity Multiplicity 4 : no continuity

Knot Multiplicity & Continuity


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NURBS:

NonUniform Rational B-Splines

Rational

and are defined as the ratio of two

cubic polynomials

Rational cubic polynomial curve segments areratios of polynomials

Can be Bzier, Hermite, or B-Splines

),(),( tytx

)(

)()(

tW

tXtx

)(tz

)(

)()(

tW

tYty

)(

)()(

tW

tZtz


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Parametric Bi-Cubic Surfaces

Parametric cubic curves are

So, parametric bi-cubic surfaces are

If we allow the points in Gto vary in 3D along

some path, then

Since are cubics:

, where

SMtGtGtGtGtsQ )()()()(,( 4321

TMGtQ )(

SMGtQ )(

)(tGi

TMGtGi )( 4321 iiiii ggggG


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Parametric Bi-Cubic Surfaces

So

SM

gggg

gggg

gggg

gggg

MTtsQ TT

44342414

43332313

42322212

41313211

),(

1,0, tsSMGMT TT


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Hermite Surfaces

SMGMTtsQ HHT

H

T ),( TMGtRtRtPtP HH ()()( 4141


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Bzier Surfaces

SMGMTtsQ BsBsT

Bs

T ),(


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Normals to Surfaces

Ss

MGMTtsQs

TT

),(

TTT ssMGMT 0123 2

SMGMTt

tsQs

TT

)(),(

SMGMss TT 0123 2

),(),( tsQt

tsQs

normal vector


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Quadric Surfaces

Implicit surface equation

An alternative representation

with

0 PQPT

0222222),,( 222 kjzhygxfxzeyzdxyczbyaxzyxf

1

z

yx

P

kjhg

jcef

hebdgfda

Q


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Advantages:

Computing the surface normal

Testing whether a point is on the surface Computingzgivenxandy

Calculating intersections of one surface with another

Quadric Surfaces


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Fractal Models


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Grammar-Based Models

L-grammars

A -> AA

B -> A[B]AA[B]

curve surface

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