Splines Jehee Lee Seoul National University. Particle Motion A curve in 3-dimensional space World coordinates.

Splines

Jehee Lee

Seoul National University

Particle Motion

• A curve in 3-dimensional space

World coordinates

)(),(),()( tztytxt p

Keyframing Particle Motion

• Find a smooth function that passes through given keyframes

World coordinates

),( 00 pt

),( 11 pt

),( 22 pt

),( 33 pt

.0),,( nit ii p)(tp

Polynomial Curve

• Mathematical function vs. discrete samples– Compact– Resolution independence

• Why polynomials ?– Simple– Efficient– Easy to manipulate– Historical reasons

dcbap

tttt

dtctbtatz

dtctbtaty

dtctbtatx

zzzz

yyyy

xxxx

23

23

23

23

)(

or

)(

)(

)(

Degree and Order

• Polynomial– Order n+1 (= number of coefficients)– Degree n

011

1)( atatatatx nn

nn

Polynomial Interpolation

• Linear interpolation with a polynomial of degree one– Input: two nodes– Output: Linear polynomial

),( 00 xt

),( 11 xt

01)( atatx

1

0

1

0

1

0

1011

0001

1

1

x

x

a

a

t

t

xata

xata

Polynomial Interpolation

• Quadratic interpolation with a polynomial of degree two

),( 00 xt

),( 11 xt

012

2)( atatatx

2

1

0

2

1

0

222

211

200

2021222

1011212

0001202

1

1

1

x

x

x

a

a

a

tt

tt

tt

xatata

xatata

xatata

),( 22 xt

Polynomial Interpolation

• Polynomial interpolation of degree n

),( 00 xt

),( 11 xt

01)( atatatx nn

nnnn

nn

nn

nn

x

x

x

a

a

a

tt

tt

tt

1

0

1

0

1

11

1

01

0

1

1

1),( nn xt

Do we really need to solve the linear system ?

Lagrange Polynomial

• Weighted sum of data points and cardinal functions

• Cardinal polynomial functions

)())(()(

)())(()()(

110

110

nkkkkkk

nkkk tttttttt

tttttttttL

nn xtLxtLxtLtx )()()()( 1100

ik

iktL ik if0

if1)(

Limitation of Polynomial Interpolation

• Oscillations at the ends– Nobody uses higher-order polynomial interpolation now

• Demo– Lagrange.htm

Spline Interpolation

• Piecewise smooth curves– Low-degree (cubic for example) polynomials– Uniform vs. non-uniform knot sequences

0x

1x

nx2x

3x

Time0t 1t 2t 3t nt

Why cubic polynomials ?

• Cubic (degree of 3) polynomial is a lowest-degree polynomial representing a space curve

• Quadratic (degree of 2) is a planar curve– Eg). Font design

• Higher-degree polynomials can introduce unwanted wiggles

Interpolation and Approximation

Continuity Conditions

• To ensure a smooth transition from one section of a piecewise parametric spline to the next, we can impose various continuity conditions at the connection points

• Parametric continuity– Matching the parametric derivatives of adjoining curve sections

at their common boundary

• Geometric continuity– Geometric smoothness independent of parametrization

– parametric continuity is sufficient, but not necessary, for

geometric smoothness

Parametric Continuity

• Zero-order parametric continuity– -continuity– Means simply that the curves meet

• First-order parametric continuity– -continuity– The first derivatives of two adjoining

curve functions are equal

• Second-order parametric continuity– -continuity– Both the first and the second

derivatives of two adjoining curve functions are equal

0C

1C

2C

Geometric Continuity

• Zero-order geometric continuity– Equivalent to -continuity

• First-order geometric continuity– -continuity– The tangent directions at the ends of two adjoining

curves are equal, but their magnitudes can be different

• Second-order geometric continuity– -continuity– Both the tangent directions and curvatures at the

ends of two adjoining curves are equal

0C

1G

2G

Basis Functions

• A linear space of cubic polynomials – Monomial basis

– The coefficients do not give tangible geometric meaning

012

23

3)( atatatatx

),,,( 0123 tttt

ia

Bezier Curve

• Bernstein basis functions

• Cubic polynomial in Bernstein bases

iinni tt

i

ntB

)1()(

33

22

12

03

333

322

311

300

)1(3)1(3)1(

)()()()()(

bbbb

bbbbp

tttttt

tBtBtBtBt

333

1232

231

330

)(

)1(3)(

)1(3)(

)1()(

ttB

tttB

tttB

ttB

Bernstein Basis Functions

Bezier Control Points

• Control points (control polygon)

• Demo– Bezier.htm

3210 ,,, bbbb

Bezier Curves in Matrix Form

3

2

1

0

23

0 0 0 1

0 0 3 3

0 3 6 3

1 3 3 1

1

)(

b

b

b

b

P

ttt

GMTt B

333

1232

231

330

)(

)1(3)(

)1(3)(

)1()(

ttB

tttB

tttB

ttB

33

22

12

03

333

322

311

300

)1(3)1(3)1(

)()()()()(

bbbb

bbbbp

tttttt

tBtBtBtBt

De Casteljau Algorithm

• Subdivision of a Bezier Curve into two curve segments

Properties of Bezier Curves

• Invariance under affine transformation– Partition of unity of Bernstein basis functions

• The curve is contained in the convex hull of the control polygon

• Variation diminishing– the curve in 2D space does not oscillate about

any straight line more often than the control point polygon

10if1)(0 ttBni

Properties of Cubic Bezier Curves

• End point interpolation

• The tangent vectors to the curve at the end points are coincident with the first and last edges of the control point polygon

)(3)1('

)(3)0('

23

01

bbp

bbp

0b

1b

2b

3b

3

0

)1(

)0(

bp

bp

Properties of Cubic Bezier Curves

Bezier Surfaces

• The Cartesian (tensor) product of Bernstein basis functions

)()(),(0 0

vBuBvu nk

n

j

n

k

njjk

bp

Bezier Surface in Matrix Form

10 0 0 1

0 0 3 3

0 3 6 3

1 3 3 1

0 0 0 1

0 0 3 3

0 3 6 3

1 3 3 1

1

)()(),(

2

3

33231303

32221202

31211101

30201000

23

0 0

v

v

v

bbbb

bbbb

bbbb

bbbb

uuu

vBuBvu

TT

nk

n

j

n

k

njjk

VMGMU

bp

Bezier Splines with Tangent Conditions

• Find a piecewise Bezier curve that passes through given keyframes and tangent vectors

• Adobe Illustrator provides a typical example of user interfaces for cubic Bezier splines

),( 00 pt

),( 11 pt

),( 22 pt

),( 33 pt

Catmull-Rom Splines

• Polynomial interpolation without tangent conditions– -continuity– Local controllability

• Demo– CatmullRom.html

),( 00 pt

),( 11 pt

),( 22 pt

),( 33 pt

2)(' 11 ii

itpp

p

1C

Natural Cubic Splines

• Is it possible to achieve higher continuity ?– -continuity can be achieved from splines of

degree n

• Motivated by loftman’s spline– Long narrow strip of wood or plastic– Shaped by lead weights (called ducks)

1nC

Natural Cubic Splines

• We have 4n unknowns – n Bezier curve segments (4 control points per each segment)

• We have (4n-2) equations– 2n equations for end point interpolation– (n-1) equations for tangential continuity– (n-1) equations for second derivative continuity

• Two more equations are required !

),( 00 xt

),( 11 xt

),( nn xt),( 22 xt

),( 33 xt),( 11 nn xt

Natural Cubic Splines

• Natural spline boundary condition

• Closed boundary condition

• High-continuity, but no local controllability

• Demo– natcubic.html

– natcubicclosed.html

0)()( 0 ntxtx

)()(and)()( 00 nn txtxtxtx

B-splines

• Is it possible to achieve both -continuity and local controllability ?– B-splines can do !– But, B-splines do not interpolate any of control points

• Uniform cubic B-spline basis functions

333

2332

2331

330

6

1)(

)1333(6

1)(

)463(6

1)(

)1(6

1)(

ttB

ttttB

tttB

ttB

2C

)()()()(

)()(

333

322

311

300

3

0

3

tBtBtBtB

tBtj

jj

bbbb

bp

B-Splines in Matrix Form

3

2

1

0

23

0 1 4 1

0 3 0 3

0 3 6 3

1 3 3 1

6

11

)(

p

p

p

p

ttt

GMTtP B

33

22

12

03

333

322

311

300

)1(3)1(3)1(

)()()()()(

bbbb

bbbbp

tttttt

tBtBtBtBt

333

2332

2331

330

6

1)(

)1333(6

1)(

)463(6

1)(

)1(6

1)(

ttB

ttttB

tttB

ttB

Uniform B-spline basis functions

• Bell-shaped basis function for each control points

• Overlapping basis functions– Control points correspond to knot points

B-spline Properties

• Convex hull• Affine invariance• Variation diminishing• -continuity• Local controllability

• Demo– Bspline.html

2C

NURBS

• Non-uniform Rational B-splines– Non-uniform knot spacing– Rational polynomial

• A polynomial divided by a polynomial• Can represent conics (circles, ellipses, and hyperbolics)• Invariant under projective transformation

• Note– Uniform B-spline is a special case of non-uniform B-spline– Non-rational B-spline is a special case of rational B-spline

n

j

njj

n

j

njjj

tB

tB

t

0

0

)(

)(

)(

bp

Cubic Spline Interpolation in a B-Spline Form

Conversion BetweenSpline Representations

• Sometimes it is desirable to be able to switch from one spline representation to another

• For example, both B-spline and Bezier curves represent polynomials, so either can be used to go from one to the other

• The conversion matrix is

22

11)(

GMT

GMTtP

111

22

2211

GMMG

GMGM

Displaying Spline Curves

• Forward-difference calculation– Generate successive values recursively by

incrementing previously calculated values– For example, consider a cubic polynomial

– We want to calculate x(t) at tk for k=0,1,2,…

dctbtattx 23)(

kk tt 1

Displaying Spline Curves

• Forward-difference calculation– Two successive values of a cubic polynomial

– The forward difference is a quadratic polynomial with respect to t

dtctbtatx

dctbtattx

kkkk

kkkk

)()()()(

)(23

1

23

)()23(3

)()()(2322

1

cbatbata

txtxtx

kk

kkk

Displaying Spline Curves

• Forward-difference calculation– The second- and third-order forward difference

– Incremental evaluation of polynomial

3

21

23

232

12

6

)()()(

266

)()()(

a

txtxtx

bata

txtxtx

kkk

k

kkk

)()()()(

)()()(

)()()(

23

22

1

12

1

1

txtxtxtx

txtxtx

txtxtx

kkkk

kkk

kkk

Matrix Equations for B-splines

• Cubic B-spline curves

333

2332

2331

330

6

1)(

)1333(6

1)(

)463(6

1)(

)1(6

1)(

ttB

ttttB

tttB

ttB

3

2

1

0

32

3

2

1

0

32

3

2

1

0

33

32

31

30

333

322

311

300

1

1331

0363

0303

0141

6

11

)()()()(

)()()()()(

p

p

p

p

p

p

p

p

p

p

p

p

ppppp

Mtttttt

tBtBtBtB

tBtBtBtBt

Control PointsGeometric Matrix

Monomial Bases

Curve Refinement

• Subdivide a curve into two segments• Figures and equations were taken from

http://graphics.idav.ucdavis.edu/education/CAGDNotes/homepage.html

Binary Subdivision

• The left segment

3

2

1

0

32

3

2

1

032

81000

04100

00210

0001

1

2221

)2

()(

p

p

p

p

M

p

p

p

p

M

ttt

ttt

tPtPL

Binary Subdivision

L

L

L

L

L

ttt

ttttP

3

2

1

0

32

3

2

1

0

132

1

81000

04100

00210

0001

1)(

p

p

p

p

M

p

p

p

p

MMM

Binary Subdivision

The Subdivision Rule for Cubic B-Splines

• The new control polygon consists of– Edge points: the midpoints of the line segments– Vertex points: the weighted average of the

corresponding vertex and its two neighbors

Recursive Subdivision

• Recursive subdivision brings the control polygon to converge to a cubic B-spline curve

• Control polygon + subdivision rule– Yet another way of defining a smooth curve

Chaikin’s Algorithm

• Corner cutting (non-stationary subdivision)• Converges to a quadratic B-spline curve• Demo: subdivision.htm

Interpolating Subdivision

• The original control points are interpolated– Vertex points: The original control points– Edge points: The weighed average of the original points

• Stationary subdivision• Demo: subdivision.htm

Subdivision Surfaces

• bi-cubic uniform B-spline patch can be subdivided into four subpatches

Catmull-Clark Surfaces

• Generalization of the bi-cubic B-spline subdivision rule for arbitrary topological meshes

Catmull-Clark Surfaces

Subdivision in Action

• A Bug’s Life

Subdivision in Action

• Geri’s Game

Summary

• Polynomial interpolation– Lagrange polynomial

• Spline interpolation– Piecewise polynomial– Knot sequence– Continuity across knots

• Natural spline ( -continuity)• Catmull-Rom spline ( -continuity)

– Basis function• Bezier• B-spline

2C1C