Geometric Modeling - Max Planck...

Geometric ModelingSummer Semester 2010

Polynomial Spline CurvesPiecewise Polynomials · Splines Bases · Properties

Geometric Modeling SoSem 2010 – Polynomial Spline Curves 2 / 116

Today...

Topics:• Mathematical Background

• Interpolation & Approximation

• Polynomial Spline CurvesPiecewise Cubic Interpolation

Splines with local control

– Hermite Splines

– Bezier Splines

– Uniform B‐Splines

– Non‐Uniform B‐Splines

Polynomial Spline CurvesPiecewise Cubic Interpolation


What we have so far...

What we have so far:• Given a basis, we can interpolate and approximate points

Curves, surfaces, higher dimensional objects

Functions (heightfields) and parametric objects

Differential properties can be prescribed as well

Problem:• We need a suitable basis

• Polynomial bases don’t work for large degree (say ≥ 10)Monomials – numerical nightmare

Orthogonal polynomials: Runge’s phenomenon still limits applicability


Piecewise Polynomials

Key Idea:• Polynomials of high degree don’t work

• Therefore: Use piecewise polynomials of low degree

• What is a good degree to use?


Choosing the Degree...

Candidates:

• d = 0 (piecewise constant): not smooth

• d = 1 (piecewise linear): not smooth enough

• d = 2 (piecewise quadratic): constant 2ndderivative, still too inflexible

• d = 3 (piecewise cubic): degree of choicefor computer graphics applications


Cubic Splines

Cubic piecewise polynomials:• We can attain C2 continuity without fixing the second derivative throughout the curve

• C2 continuity is perceptually importantThe eye can see second order shading discontinuities (esp.: reflective objects)Motion: continuous position, velocity & accelerationDiscontinuous acceleration noticeable (object/camera motion)

• One more argument for cubics:Among all C2 curves that interpolate a set of points (and obey to the same end conditions), a piecewise cubic curve has the least integral acceleration (“smoothest curve you can get” in this sense).


Piecewise Cubic Interpolation

Setup:• (n+1) control points y0...yn (to be interpolated)

• For simplicity: assume uniform spacing t0...tn = (0,1,2,...,n)

• n cubic polynomial pieces p1...pn parametrized over [0...1]

• Multidimensional case: solve problem for each axis (x,y,z)

y0

y1

yn3)(

32)(

2)(

1)(

0)( tctctccxpiiii

i +++=

p1p2

pn

y2


Conditions

y0

y1

ynp1

p2

pn

y2

)1()0(:...2

)1()0(:...2

)1(:...1)0(:...1

12

2

2

2

1

1

−

−

−

==∀

==∀

==∀==∀

ii

ii

ii

ii

pdtd

pdtd

ni

pdtd

pdtd

ni

ypni

ypni (n conditions)

(n conditions)

(n‐1 conditions)

(n‐1 conditions)

(4n degrees of freedom)

2 dimensional null space(so far)


Conditions

y0

y1

ynp1

p2

pn

y2

)1()0(:...2

)1()0(:...2

)1(:...1)0(:...1

12

2

2

2

1

1

−

−

−

==∀

==∀

==∀==∀

ii

ii

ii

ii

pdtd

pdtd

ni

pdtd

pdtd

ni

ypni

ypni

0)1(

0)0(

2

2

12

2

=

=

npdtd

pdtd

additional:(n conditions)

(n conditions)

(n‐1 conditions)

(n‐1 conditions) alternative:cyclic boundary conditions

(closed curves)


Numerical Solution

Solving the system of equations:• Band matrix, bandwidth O(1)

• Can be solved in O(n) time & space for n variables ⎟

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

************

**********

*********

***


Cubics Minimize Acceleration

Theorem:• Given n data points (yi, ti) to interpolate and fixed end conditions (either prescribed 1st derivative, or zero second derivative), a piecewise cubic interpolant minimizes

the energy:

• This means: A cubic spline curve has the least square acceleration.

• Related to elastic energy: Hooke’s elastic energy of a

straight line is given by:

• I.e.: cubic spline interpolation approximates elastic beams.

∫=n

dttffE0

2)('')(

∫=n

dttE0

2)]([2)( fκf λ


Proof: Cubics Minimize Acceleration

[ ]

∫∫∫

∫

∫

++=

+=

=

nnn

n

n

dttddttdtcdttc

dttdtc

dttaaE

0

2

00

2

0

2

0

2

)('')('')(''2)(''

)('')(''

)('')(

Cubic spline: c(t)Another C2 interpolating curve: a(t)Residual: d(t) = a(t) – c(t).

Energy functional:




[ ]

( )[ ]

( ) ( ) ∫

∫

∫∫

−−−−=

−−=

−=

==

==

n

n''cn'c

nnt

t

nn

n

dttdtccacncnanc

dttdtctctatc

dttdtctdtcdttdtc

0cond. end orderfirst identical if 0

0)( for 0

cond. end orderfirst identical if 0

0)(' for 0

00

00

0

)(')(''')0(')0(')0('')(')(')(''

)(')(''')(')(')(''

)(')(''')(')('')('')(''

Integration by parts:

∫∫∫ ++=nnn

dttddttdtcdttcaE0

2

00

2 )('')('')(''2)('')(

Energy functional:

[ ] ∫∫ −= ==b

a

btat

b

a

dtxbxatbtadttbta )()(')()()(')(





[ ]

0

)()5.0('''

)(')(''')('')(''

tion)(interpola 0

11

0

0 const. piecewise0

=

+=

=

=

+==

−

=∑

∫∫

itit

n

i

nn

tdic

dttdtcdttdtc

Middle term (cont.):

∫∫∫ ++=nnn

dttddttdtcdttcaE0

2

00

2 )('')('')(''2)('')(

Energy functional:


[ ] ∫∫ −= ==b

a

btat

b

a

dtxbxatbtadttbta )()(')()()(')(




positive:energy additional

0

2

spline cubic

0

2 )('')('')( ∫∫ +=nn

dttddttcaE

Energy functional:

Positive additional energy:Any function that differs in second derivative from c will have higher energy.

⇒ c is a minimal function in terms of E.

Polynomial Spline CurvesSpline Bases with Local Control


So what’s missing?

So we have solved our problem – what’s left to do?• Target area: interactive geometric modeling

• Shape of the entire curve depends on all control points

• Changing one control point can affect the whole curve

• Not a big issue for algorithmic curve controlFitting curves to data, optimizing curves according to some objective function, etc...

• But not acceptable for modeling by humans

• “User interface problem”: We want local control.


Notation

Function design problem:• Function f: R → Rf(t) = c1b1(t) + c2b2(t) + c3b3(t) + ...

• Coefficients c1, c2, c3, ... ∈ R

Curve design problem• Function f: R → Rn

f(t) = b1(t)p1 + b2(t)p2 + b3(t)p3 + ...

• “Control points” p1, p2, p3, ... ∈ Rn

u

f(t)t

function graph

x

t

parametric curve

y

c1 c2 c3

p1

p2

p3


New Idea

Problem: Again the basis...• We want a basis such that the coefficients / control points have intuitive meaning

• Then we can just let the user edit the control points

• The effect of control point editing has to be intuitive enough to be manually controllable


Desirable Properties

Useful requirements for a spline basis:• The curve should be well behaved: smooth basis functions

• Local control: basis functions with compact support

• Affine invariance:Applying an affine map x → Ax + b to the control points should have the same effect as transforming the curveIn particular: rotation, translationOtherwise, interactive curve editing is almost impossible

• Convex hull property:Good to have: The curve is in the convex hull of its control pointsAvoids at least too weird oscillationsComputational advantages (recursive intersection tests)


Affine Invariance

Affine Invariance:• Affine map: x → Ax + b• Part I: Linear invariance – we get this automatically

Linear approach:

Therefore:

∑∑== ⎟⎟

⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

==n

i zi

yi

xi

i

n

iii

p

p

p

tbtbt1 )(

)(

)(

1)()()( pf

( ) ( )∑∑==

=⎟⎟⎠

⎞⎜⎜⎝

⎛=

n

iii

n

iii tbtbt

11)()()( AppAfA


Affine Invariance

Affine Invariance:• Affine map: x → Ax + b• Part II: Translational invariance – need some brains

For translational invariance, the sum of the basis functions must be one everywhere (for all parameter values t that are used).

This is called “partition of unity property”.

The bi form an “affine combination” of the control points pi.

This is absolutely necessary for human modeling.

( ) bbpbp

one to sum must

1111)()()()()( ⎟⎟⎠

⎞⎜⎜⎝

⎛+=+=+ ∑∑∑∑

====

n

ii

n

ii

n

iii

n

iii tbtftbtbtb


Convex Hull Property

Convex combinations:• A convex combination of a set of points {p1,...,pn} is any point of the form:

(Remark: λi ≤ 1 is redundant)

• The set of all admissible convex combinations forms the convex hull of the point set

Easy to see (simple exercise): This convex hull is the smallest set that contains all points {p1,...,pn} and every complete straight line between two elements of the set.

1,0:..1 and 1 :with 11

≤≥=∀=∑∑==

ii

n

ii

n

iii ni λλλλ p



Accordingly:• If we have this property:

the constructed curves / surfaces will be:Affine invariant (translations, linear maps)

Be restricted to the convex hull of the control points

• Corollary: Curves with this property will have linear precision, i.e.: if all control points lie on a straight line, the curve is a straight line segment, too.

• Surfaces with planar control points will be flat, too.

0)(:..1: and 1)(:1

≥=∀Ω∈∀=Ω∈∀ ∑=

tbnittbt in

ii



Convex Hull Property:• Very useful property

Avoids at least the worst oscillations (no escape from convex hull, unlike polynomial interpolation through control points)

Linear precision property is intuitive (people expect this)

Can be used fast range checks

– Test for intersection with convex hull first, then the object.

– Recursive intersection algorithms in conjunction with subdivision rules (more on this later)


Spline Techniques

Spline bases we will look at in this lecture:• Hermite interpolation

• Bezier curves & surfaces

• Uniform B‐splines

• Non‐uniform B‐splines

• [NURBS: Non‐uniform rational B‐splines](not linear, more on this later)


Spline Techniques

Two views:• Linear algebra: polynomial function spaces

Basis changes

Derivatives and continuity conditions

• Geometry: Successive linear interpolation (“blossoming”)Construct polynomial spline curves by repeated linear interpolation of control points

More intuitive explanations of properties

Mathematical formalism: Blossoming and polar forms

This part of the lecture will deal with the linear algebra view. Blossoming gets a separate chapter...

Polynomial Spline CurvesHermite Splines


Hermite Splines

Overview:• Simple spline technique, easy to implement

• Has some shortcomings

• We will look at C1 cubic Hermite splines as an example

Key Idea:• Specify position and derivatives at the endpoints of each segment

• Come up with a rule to match them easily

• Precompute basis for this purpose


Illustration

For each segment hi(t) we know:• Positions: hi(0), hi(1)

• Derivatives: ∂t hi(0), ∂t hi(1)

3)(3

2)(2

)(1

)(0)( tctctccth

iiiii +++=

h1 h2 h3


Hermite Basis

Linear system: (one dimension, one segment)

2321

33

2210

32)(')(

tctccth

tctctccth

++=

+++=

1

0

1

0

)1(')0(')1()0(

mh

mh

ph

ph

=

=

=

=

1321

01

13210

00

32 mcccmc

pcccc

pc

=++⇒

=⇒

=+++⇒

=⇒

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

1

0

1

0

3

2

1

0

3210001011110001

m

m

p

p

c

c

c

c

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−−−−

3

2

1

0

1

0

1

0

1122123301000001

c

c

c

c

m

m

p

p


Hermite Basis

Solution:

[ ]⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−−−−

=

1

0

1

0

32

1122123301000001

,,,1)(

m

m

p

p

ttttf

32

32

321

32

)(

2)(

23)(

231)(

1

0

0

ttth

tttth

ttth

ttth

m

m

p

p

+−=

+−=

−=

+−=

Basis Functions:


Hermite Basis

Properties:• hp0 and hp1 sum to one.(affine invariant w.r.t. position)

• Curve might leave convex hull ofcontrol points

Open question:• How to specify derivatives?


Illustration

Simple rule for derivatives:• Derivatives:

• “Catmull‐Rom Spline”

fi

fi+1

1

011

11

2

:)1(:)0(

})1..1{(,2

:)1(

})..2{(,2

:)0(

−

−+

−

−=∂

−=∂

−∈−

=∂

∈−

=∂

nnnt

t

iiit

iiit

ni

ni

xxf

xxf

xxf

xxf

xi‐1

xi

xi+1


Properties

Properties of this spline construction:• Interpolates original points

• Local control

• C1 continuous

• Affine invariant

• No convex hull propertyTends to “overshoot”

This can be really nasty in practice

Polynomial Spline CurvesBezier Curves


Bezier Splines

History:• Bezier splines developed

by Paul de Casteljau at Citroën (1959)

Pierre Bézier at Renault (1962)

for designing smooth free‐form parts in automotive design applications.

• Today: The standard tool for 2D curve editing,Cubic 2D Bezier curves are used almost everywhere:

Postscript, PDF, Truetype (quadratic curves), Windows GDI...

Corel Draw, Powerpoint, Illustrator, ...

• Widely used in 3D curve & surface modeling as well


All You See is Bezier Curves...


Bernstein Basis

Bezier splines use the Bernstein basis:• Bernstein basis of degree n:

• Each basis function is a polynomial of degree n.

• The basis functions form a partition of unity

(binomial theorem)

• For t ∈ [0..1], the basis functions are positive (Bi (t) ≥ 0).

{ })()(1)(0 ,...,, nnnn BBBB =( ) inini tti

ntB −−⎟⎟

⎠

⎞⎜⎜⎝

⎛= 1:)()( ) degree( function basis

nthiB −=

( ) ( ) ∑∑==

− =−⎟⎟⎠

⎞⎜⎜⎝

⎛=−+=+−=

n

i

ni

n

i

inin tBtti

ntttt

0

)(

0)(1)1()1(1

(n)


Examples

The first three Bernstein bases:

( )

( ) ( )

( ) ( )( ) 3)3(32)3(2

2)3(1

3)3(0

2)2(2

)2(1

2)2(0

)1(1

)1(0

)0(0

:13:

13:1:

:12:1:

:1:

1:

tBttB

ttBtB

tBttBtB

tBtB

B

=−=

−=−=

=−=−=

=−=

=

n=3 (cubic)B0

B1 B2

B3

n=1 (linear)

B0 B1

n=2 (quad.)

B0B1

B2


Bernstein Basis

Bernstein basis properties:•

• Basis for polynomials of degree n.

• Each basis function has its maximum at i/n.

• Recursive computation:

• Symmetry:

{ },,...,, )()(1)(0 nnnn BBBB = ( ) inini ttin

tB −−⎟⎟⎠

⎞⎜⎜⎝

⎛= 1:)()(

n=3 (cubic)

n =10

)(niB

( ) )()(1:)( )1( 1)1()( tBttBttB ninini −−− +−=}...0{ for 0)(,1)(00 nitBtB

ni ∉==with

B0

B1 B2

B3

)1()( tBtB n inni −= −


Bezier Curves

Bezier curves Properties:

• Curves:

• Considering the interval t ∈ [0..1]

• Bezier curves are affine invariant.

• Bezier curves are contained in the convex hull of the control points.

• The influence of the control points is moving along the curve with index i. Largest influence at t = i/n.

• However: A single curve segment has no fully local control.

n=3 (cubic)

n =10

)()(0

)( tBtn

i

nii∑

=

= pfB0

B1 B2

B3


Bezier Curves: Examples

Bezier Curves:

• ∑=

=n

i

niiBt

0

)()( pf

p0

p1

p2

p3

p0

p1

p2

p3

n=3 (cubic)B0

B1 B2

B3

p0p1

p2p3

p0

p1p2

p3


Matrix Form

Matrix Notation: Bezier → Monomials

[ ]⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−=

2

1

02

121022001

1)(p

p

p

f ttt

[ ]⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−−−

−=

3

2

1

0

32

1331036300330001

1)(

p

p

p

p

f tttt

(quadratic case)

(cubic case)


Format Conversion

Conversion: Compute Bezier coefficients from monomial coefficients

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

2

1

0

.)(2

.)(1

.)(0

1

121022001

c

c

c

c

c

c

Bez

Bez

Bez

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛−

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−−−

−=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

3

2

1

0

.)(3

.)(2

.)(1

.)(0

1

1331036300330001

c

c

c

c

c

c

c

c

Bez

Bez

Bez

Bez

(quadratic case)

(cubic case)


Format Conversion

Conversion: quadratic to cubic

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−−

−

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−−−

−=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

00000000

121022001

1

1331036300330001

)2(2

)2(1

)2(0

)3(3

)3(2

)3(1

)3(0

c

c

c

c

c

c

c

Convert to monomials and back to Bezier coefficients.(Other degrees similar)

Example Application: Output of TrueType fonts in Postscript.


The Other Way Round...

Reducing the degree:• Exact solution is not always possible• Approximate solution: least‐squares(function approximation)

• System of normal equations:

( )

( )∑

=

=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛m

iii

nnnnn

n

tbctf

fxb

fxb

c

c

bbbb

bbbb

1

11

1

111

)(:)(,,~

,~

~

~

~,~~,~

~,~~,~

( )

( )∑

= ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⇔m

iin

i

i

nnnn

n

tbxb

tbxb

c

c

c

bbbb

bbbb

1

11

1

111

)(,~

)(,~

~

~

~,~~,~

~,~~,~


Cubic → Quadratic Case

Reducing the degree: Cubic → Quadratic

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

61151601

101101201

201101101

60115161

~~~

353

1409

351

1409

353

141

351

141

71

3210

2

1

0

cccc

c

c

c

Polynomial Spline CurvesBezier Splines


Bezier Splines

Local control: Bezier splines• Concatenate several curve segments

• Question: Which constraints to place upon the control points in order to get C‐1, C0, C1, C2 continuity?

p0

p1

p2

p3

p0

p1

p2

p3

(i)

(i)

(i)

(i)

(i+1)

(i+1)

(i+1)

(i+1)


Derivatives

Bernstein basis properties:• Derivatives:

( ) ( )( )

( ) ( ) ( ) ( )

( ) ( )

[ ])()(

11

111

1)(!!

!1!!

!

1)(1)(

)1()1(1

}1{}1{

}1{}1{

}1{}{}1{)(

tBtBn

tti

ntt

i

nn

ttiniin

ntit

iinn

ttintiti

ntB

dtd

ni

ni

iniini

iniini

iniinini

−−−

−−−−

−−−−

−−−−

−=

⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛ −−−⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

=

−−−

−−−

=

−−−−⎟⎟⎠

⎞⎜⎜⎝

⎛=

({k} = k if k > 0, zero otherwise)


Derivatives

Bernstein basis properties:• Derivatives:

( ) ( )( )

( ) ( )(( ) ( ) )

[ ])()(2)()1(1)}(1{1)(

1)(1}1{

1)(1)(

)2()2(1

)2(2

}2{}{}1{}1{

}1{}1{}2{

}1{}1{)(2

2

tBtBtBnn

ttininttini

ttinititii

n

ttintiti

n

dtd

tBdtd

ni

ni

ni

iniini

iniini

iniinini

−−−

−−

−−−−−

−−−−−

−−−−

+−−=

−−−−+−−−

−−−−−⎟⎟⎠

⎞⎜⎜⎝

⎛=

−−−−⎟⎟⎠

⎞⎜⎜⎝

⎛=

({k} = k if k > 0, zero otherwise)


Bezier Curve Properties

Important for continuous concatenation:• Function value at {0,1}:

• First derivative vector at {0,1}

• Second derivative vector at {0,1}

( )∑=

−−⎟⎟⎠

⎞⎜⎜⎝

⎛=

n

ii

ini tti

nt

01)( pf

0)0( pf =

npf =)1(

n=3 (cubic)B0

B1 B2

B3

n =10




• First derivative vector at {0,1}:


,)0( 0pf = npf =)1(

n=3 (cubic)B0

B1 B2

B3

[ ]ni

ini

ni tBtBntdt

dpf )()()(

0

)1()1(1

=

−−− −= ∑

( )01)0f( pp −= ndtd ( )1)1f( −−= nnndt

dpp

[ ] [ ]([ ] [ ] )nnnnnnnn

nnn

tBtBtB

tBtBtBn

pp

pp

)()()(...

...)()()(

)1(11

)1(1

)1(2

1)1(

1)1(

00)1(

0

−−−

−−

−−

−−−

+−+

+−+−= n =10






,)0( 0pf = npf =)1(

n=3 (cubic)B0

B1 B2

B3

[ ]ni

ini

ni tBtBntdt

dpf )()()(

0

)1()1(1

=

−−− −= ∑

( )01)0(' ppf −= n ( )1)1(' −−= nnn ppf

[ ] [ ]([ ] [ ] )nnnnnnnn

nnn

tBtBtB

tBtBtBn

pp

pp

)()()(...

...)()()(

)1(11

)1(1

)1(2

1)1(

1)1(

00)1(

0

−−−

−−

−−

−−−

+−+

+−+−= n =10





• Second derivative vector at {0,1}:

0)0( pf =npf =)1(

n=3 (cubic)B0

B1 B2

B3

n =10

[ ]210 2)1()0('' pppf +−−= nn[ ]212)1()1('' −− +−−= nnnnn pppf

[ ]01)0(' ppf −= n[ ]1)1(' −−= nnn ppf


Bezier Spline Continuity

Rules for Bezier spline continuity:• C0 continuity:

Each spline segment interpolates the first and last control point

Therefore: Points of neighboring segments have to coincide for C0 continuity.

p1(i)

(i)p0

(i)p2

p3(i)

p0(i+1)

p1(i+1)

p2(i+1)

p3(i+1)



Rules for Bezier spline continuity:• Additional requirement for C1 continuity:

Tangent vectors are proportional to differences p1 – p0, pn – pn‐1Therefore: These vectors must be identical for C1 continuity

p1(i)

(i)p0

(i)p2

p3(i)

p0(i+1)

p1(i+1)

p2(i+1)

p3(i+1)



Rules for Bezier spline continuity:• Additional requirement for C2 continuity:

d2/dt2 vectors are prop. to (p2 – 2p1 + p0), (pn – 2pn‐1 + pn‐2)

Tangents must be the same (C2 implies C1)

Therefore: Triangle of first / last three points must be the same

p1(i)

(i)p0

(i)p2

p3(i)

p0(i+1)

p1(i+1)

p3(i+1)

p2(i+1)


In Practice

In practice:• Everyone is using cubic Bezier curves• Higher degrees rare (CAD/CAM applications)• Typically: “points & handles” interface• Four modes:

Discontinuous (two curves)

C0 Continuous (points meet)

Tangent direction continuous (handles point into the same direction, but different length)(“G1 continuous”, more on this shortly)

C1 Continuous (handle points have symmetric vectors)

• C2 is rarely supported (too restrictive, no local control)


Bezier Curve Editing

C‐1 continuity C0 continuity

G1 continuity C1 continuity


Geometric Continuity

Parametric Continuity:• C0, C1, C2... continuity.

• Does a particle moving on this curve have a smooth trajectory (position, velocity, acceleration,...)?

• Useful for animation (object movement, camera paths)

Geometric Continuity:• Is the curve itself smooth?

• C.f.: differential geometry – parametrization independent measures

• More relevant for modeling (curve design)


Geometric Continuity

Geometric Continuity:• G0 = C0: position varies continuously

• G1: tangent directions varies continuouslyIn other words: the normalized tangent varies continuously

Equivalently: The curve can be reparametrized so that it becomes C1.

Also equivalent: A unit speed parametrization would be C1.

• G2: curvature varies continuouslyEquivalently: The curve can be reparametrized so that it becomes C2.

Also equivalent: A unit speed parametrization would be C2.


Geometric Continuity for Bezier Splines

This means:

G1 continuity

This Bezier curve is G1: It can be reparametrized to become C1. (Just increase the speed for the second segment by ratio of tangent vector lengths.)

Polynomial Spline CurvesUniform Cubic B‐Splines

Geometric Modeling SoSem 2010 – Polynomial Spline Curves

Literature

Literature:• An Introduction to Splines for use in Computer Graphics and Geometric ModelingRichard H. Bartels, John C. Beatty, Brian A. BarskyMorgan Kaufmann 1987

(now hard to get, waited several month on Amazon)

67 / 116


Overview

Uniform cubic B‐splines• This is a special case of general B‐splines

(which additionally provide arbitrary degree and general and non‐uniform parametrization)

• We look at this first to get an intuition for the basic ideas and concepts for B‐splines

• Some derivations are left out – will be shown later for the general case


Overview

Improvement over cubic Bezier splines:• C2 continuity is easily attainable

• We will use only one type of basis functionsShifted in the domain to create curves with multiple segments

This principle is conceptually easier to apply in general modeling problems (e.g. as a basis for finite elements, for PDEs or variational problems)


Key Ideas

Key Ideas:• We design one basis function b(t)

• Properties:b(t) is C2 continuous.

b(t) is piecewise polynomial, degree 3 (cubic).

b(t) has local support.

Overlaying shifted b(t + i) forms a partition of unity.

b(t) ≥ 0 for all t

• In short: We build‐in all the desirable properties into the basis. Linear combinations will inherit these.

1.0

0.8

0.6

0.4

0.2

0.0 1.0 2.0 3.0 4.0


Shifted Basis Functions

Basis function:• Consists of four polynomial parts p1...p4.

• Shifted basis b(t – i): spacing of 1.

• Each interval to be used must be overlapped by 4 different bi.

1.0

0.8

0.6

0.4

0.2

0.0 1.0 2.0 3.0 4.0

1.0

0.8

0.6

0.4

0.2

0.0 1.0 2.0 3.0 4.0 5.0 6.0

basis function b(t) shifted basis functions b(t ‐ i) for [0..6]

i=0 i=1 i=2 i=3i=‐1

p1(t)

p2(t‐1)

p4(t‐3)

p3(t‐2)


Basis Function1.0

0.8

0.6

0.4

0.2

0.0 1.0 2.0 3.0 4.0

basis function b(t)

p1(t)

p2(t‐1)

p4(t‐3)

p3(t‐2)

31 6

1)( ttp =

( )322 333161)( ttttp −++=

( )323 36461)( tttp +−=

( )324 33161)( ttttp −+−=

( )( )( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

>≤


Creating Curves

Creating uniform B‐spline curves:• Choose parameter interval [nstart‐2, nend+2,], nstart


Uniform B‐Spline Curvesone segment

two segments

three segments


Discontinuities

Continuity Control• Easier than with Bezier curves

• The parametric function is always C2, by construction

• However: We can create curves with lower geometric smoothness

This will lead to a degenerate (non‐regular) parametrization

This problem is fixed with general, non‐uniform B‐Splines

• Basic idea: Double control pointsSingle points: G2 curve

Double points: G1 curve

Triple points: G0 curve


Continuity Control

single points

double point

triple point(interpolates that point)

× 1

× 2

× 3


Illustration

1.0

0.8

0.6

0.4

0.2

0.0 1.0 2.0 3.0 4.0 5.0 6.0

1.0

0.8

0.6

0.4

0.2

0.0 1.0 2.0 3.0 4.0 5.0 6.0

× 2 × 3


End Conditions

Problem:• We need at least 4 points for one spline segment,5 for two and so on.

• Means: We need s + 3 control points for s segments (rather than s + 1), two more than in spline interpolation.

• This is inconvenient...

• Simple solution:Use double or triple end points

Triple end points will be interpolated

We will get along with s + 1 control points


Knot Sequences

Specifying a uniform, cubic B‐Spline curve:• A set of control points p1, ..., pn.

• Knot multiplicities (i1, ..., ik), ij ∈ {1,2,3}, Σij = n.For example (3,1,1,1,3,1,1,1,3)

– Creates one sharp corner in the middle of the spline

– Interpolates the end points

× 3× 3 × 3

× 1 × 1× 1 × 1

× 1 × 1


Conversion to Bezier Basis

Uniform B‐Splines can be converted to the Bezier Format:

• Cubic polynomial pieces

• Each can be represented as Bezier segment

• Just a basis change...


Basis Change

31 6

1)( ttp =

( )322 333161)( ttttp −++=

( )323 36461)( tttp +−=

( )324 33161)( ttttp −+−=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−−−

−=→

133136303030

1410

61

MnUBM

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−−−

−=→

1331036300330001

MnBezM

( ) MnUBMnBezBezUB MMM →−→→ = 1

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

−

−→

)4(3

)3(2

)2(1

)1(

)4(

)3(

)2(

)1(

~~~~

i

i

i

i

BezUB

i

i

i

i

M

bbbb

bbbb

( ) ( )

( ) 3)3(32)3(2

2)3(1

3)3(0

:13:

13:1:

tBttB

ttBtB

=−=

−=−=

Uniform B‐Spline:

Bezier‐Spline:


Where does the basis come from?

How do we construct a B‐Spline basis?• Derivation of uniform cubic B‐Splines?

• Generalizations:General degree?

Non‐uniform parametrization?


Three Approaches:

Three ways to get the basis:1. The elementary approach:

Derive a linear system of equations, solve

2. Repeated convolution:d‐fold convolution of box functions

3. de‐Boor Recursion:Repeated linear interpolation


The Elementary Approach

Cubic Uniform B‐Spline Basis:• One basis function, just shifted

• Consists of four pieces p1, p2, p3, p4.

• We just need the coefficients of the pieces

• Setting up a linear system...

1.0

0.8

0.6

0.4

0.2

0.0 1.0 2.0 3.0 4.0

basis function b(t)

p1(t)

p2(t‐1)

p4(t‐3)

p3(t‐2)

pieces pi(t)



Linear system:1.0

0.8

0.6

0.4

0.2

0.0 1.0 2.0 3.0 4.0

basis function b(t)

p1(t)

p2(t‐1)

p4(t‐3)

p3(t‐2)

pieces pi(t)

121

21

21

21

0)1(0)1(0)1()0()1()0()1()0()1()0()1()0()1()0()1()0()1()0()1()0()1(

0)0(0)0(0)0(

3210

444

434343

323232

212121

111

=⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

=′′=′=

′′=′′′=′=

′′=′′′=′=

′′=′′′=′==′′=′=

pppp

ppppppppppppppppppppp

ppp


Normalization:• Completely determines the pi.

• Turns out to hold everywhere in [0,1]

• Not yet clear why

• But: if it is possible, ourconditions are sufficient

• So we have to show that it is possible

Positivity:• Not enforced; we get this accidentally (simplicity)

• Same argument: if it’s possible, the conditions are sufficient


121

21

21

21

0)1(0)1(0)1()0()1()0()1()0()1()0()1()0()1()0()1()0()1()0()1()0()1(

0)0(0)0(0)0(

3210

444

434343

323232

212121

111

=⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

=′′=′=

′′=′′′=′=

′′=′′′=′=

′′=′′′=′==′′=′=

pppp

ppppppppppppppppppppp

ppp


Properties

Minimal Support:• We have 16 conditions (cubic case)

15 for smoothness, one for normalizationEasy to see: linear independent

• Need 4 polynomial segments to get sufficiently many degrees of freedom

• Consequence: Any C2 function with 3 or less polynomial segments, and the zero function everywhere else, must be the zero function.

15 linear independent constraints, homogeneous.Zero vector is the only solution.

• Therefore: We have minimal support.


Repeated Convolution

Convolution:• Weighted average of functions

• Definition:

Example:

∫∞

∞−

−=⊗ dxtxgxftgtf )()()()(t

gf

⊗ =


Repeated Convolution

A Different Derivation:• We start with 0th degree basis functions

• Increase smoothness by convolution

Degree‐zero B‐Spline:

General‐degree B‐Spline:

⎩⎨⎧ ∈

=otherwise ,0

)1...0[ if ,1)()0(

ttb

∫∞

∞−

−=⊗= dxtxbxbtbtbtb iii )()()()()( )0()()0()()(


Illustration

1

1 20‐1

b0

1

1 20‐1

b0

1

1 20‐1

b0

b0

1

1 20‐1

1

1 20‐1

1

1 20‐1

b0

b0


Illustration

1

1 20‐1

b0

1

1 20‐1

b0

Result:• Piecewise linear B‐spline basis function

• Each convolution with b0 increases the continuity by 1.


b1

b1

Illustration

1

1 20‐1

b1

1

1 20‐1

1

1 20‐1

b0

1

0 1‐1‐2

1

0 1‐1‐2

1

0 1‐1‐2

b0

b0


Smoothness

Convolution with a box filter increases smoothness:Function f that is Ck‐1 at t = t0, and Ck everywhere else:

1..1for 0)(),()(:

)()1()(:1

0

−==Δ−=Δ

−+==⊗=

+−

+

⊗ ∫

kitfDtfdtdtf

dtdD

tFtFdttfbff

ii

i

i

ii

t

t

( )

( )

( ))()1()(

)()1(

)()1()()(

1

2

2

2

2

1

1

tftfDtfD

tftfdtd

tFtFdtd

dtdtf

dtd

dtdtf

dtd

jj

j

j

j

j

j

j

j

j

−+Δ=Δ

−+=

⎟⎠

⎞⎜⎝

⎛ −+=⎟⎠

⎞⎜⎝

⎛=

−⊗

−

−

−

−

⊗−

−

⊗


1

b2 b2 b2b2

1 20‐1

Partition of Unity

Proof by Induction:

10‐1

b0b0

1

1

1 20‐1

b2

1 20‐1

b2b0 b0b0b0

degree zero: OK degree d: given

degree d+1: followsdegree d...

1


Other Properties

Positivity:• By definition

Continuity:• Smoothness increasing property: k‐fold convolution is Ck.

Piecewise polynomial:• Easy to see: Polynomial in each interval

• Degree k for k‐fold convolution.


Consequences

Consequences:• The constructed functions are identical to the explicitly constructed ones (limited degrees of freedom)

• This means, the explicitly constructed basis has the same properties (partition of unity, positivity)


Repeated Linear Interpolation

Another way to increase smoothness:

t t

t

t ∈ [0..2]

× ×

+



Another way to increase smoothness:

t ∈ [0..3]

× ×

+

t t

t t


Plot of the Three Pieces



General Principle:

t ∈ [0..d+1]

× ×

+

t

t t


De Boor Recursion

The uniform B‐spline basis of degree d is given as:

⎩⎨⎧


Connection

Three constructions:• It can be shown that this construction is equivalent to the previous two.

• Rough idea:Convolution with a box filter increases degree by one (multiplication with linear weight in de Boor formula)

Antiderivative of box filter: evaluate at t, t+1(corresponds to overlaying Ni, Ni+1)

Need to show that the coefficients match (induction)

Polynomial Spline CurvesNon‐Uniform B‐Splines


Generalization

De Boor formula can be generalized:• Arbitrary parameter sequences (“knot sequence”)(t0, t1, ..., tn) with t0 ≤ t1 ≤ ... ≤ tn

Not necessary to have uniform spacing

Will give us more flexibility in curve design.

• Specifying one parameter value multiple times is permitted, e.g. (0, 0, 0, 1, 2, 3, 4, 4, 4, 5, 6, 7, 8, 8, 8)

This has a similar effect as multiple nodes in simple uniform B‐Splines

Will avoid irregular parametrization (means: will create a basis that itself is less smooth, not just the traced out curve)


General De Boor Recursion

Generalized Formula:

⎩⎨⎧


Uniform Case:

For comparison – the uniform case:

⎩⎨⎧


B‐Splines

Constructing a Spline Curve:• Choose a degree d.

• Choose a knot sequence (t0, t1, ..., tn) with t0 ≤ t1 ≤ ... ≤ tn(n ≥ d – 1)

• Choose control points p0, p1, ..., pm (m = n – d + 1)

• Form the spline curve:

De Boor Algorithm:• This evaluation can be expanded explicitly...

∑=

=n

ii

di tNt

0)()( pf


( ) ( ) ( ) ( )

11

0

1

11

11111

0

1

11

111

1

01

1

11

11

11

1

1

11

1

11

1

0

1

11

1

0

11

0

1

11

1

0

)(

:)(

)()(

)()(

)()(

)()(

i

n

i

di

idi

idiiii

n

i

di

idi

idiii

n

ii

di

idi

dii

di

idi

i

n

ii

di

idi

din

ii

di

idi

i

n

ii

di

idi

din

ii

di

idi

i

n

ii

di

tN

tttttt

tNtt

tttt

tNtttt

tNtt

tt

tNtttt

tNtt

tt

tNtttt

tNtt

tt

tNtf

p

ppp

pp

pp

pp

pp

p

∑

∑

∑

∑∑

∑∑

∑

+

=

−

−−+

−+−−+

=

−

−−+

−+−−

+

=−

−

−−+

+−−

−−+

−

+

=−

−

−−+

+−

=

−

−−+

−

=

−+

+

+

=

−

−−+

−

=

=

−−+−

=−

−+−=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−−

+⎥⎦

⎤⎢⎣

⎡−

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

+⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

+⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=

=

De Boor Algorithm


( ) ( )

∑

∑

∑

+

=

−−

−−−+

−+−−

−−−+

−+

=

−−

+

=

−

=

−

−+

−−

==

=

dn

i

dii

ji

ijdi

jdiji

ijdi

iji

jn

i

ji

jdi

n

ii

di

tN

tttt

tttttN

tNtf

0

)(0

)1(1

11

1)1(

11

1)(

0

)(1

1

0

)1(1

)(

: with)(

)()(

p

pppp

p

De Boor Algorithm

This means:• For t ∈ [ti‐1,...,ti), we obtain the function value f (t).• We can write this as an algorithm...


De Boor Algorithm

De Boor Algorithm:• We want to evaluate f (t) for a t ∈ [ti‐1,...,ti)• We compute:

( ) ( )

( ) ( )

)(Output

),(1)(

)()()(

: increasingFor

1

11

1)(11

)(1)(

11

11

11

11

1

)0(

t

tttttt

ttttt

ttt

ttt

j

di

ijdi

iji

ji

ji

ji

ji

ji

ijdi

jdiji

ijdi

iji

ii

−

−−−+

−−−

−

−−

−−−+

−+−−

−−−+

−

−−

=−+=

−

−+

−−

=

=

p

pp

ppp

pp

ααα


Example

( ) ( )

( ) ( )

)(Output

),(1)(

)()()(

: increasingFor

1

11

1)(11

)(1)(

11

11

11

11

1

)0(

t

tttttt

ttttt

ttt

ttt

j

di

ijdi

iji

ji

ji

ji

ji

ji

ijdi

jdiji

ijdi

iji

ii

−

−−−+

−−−

−

−−

−−−+

−+−−

−−−+

−

−−

=−+=

−

−+

−−

=

=

p

pp

ppp

pp

ααα

p1(0) p4

(0)

p3(0)

p2(0)

t1=0

t2=1t3=2

t4=3

61

41

21

t = 1.5


Data Flow

Data Flow:

)1()2()1()0(

)2(1

)1(1

)0(1

)1(1

)0(1

)0(

−−

−−−−

+−+−

−

di

diii

diii

didi

di

ppppppp

ppp


De Boor Algorithm

Some nice properties:•

• The algorithm forms only convex combinations of the original data points.

Affine invariance follows directly

Numerically stable – no cancelation or error amplification problems

Much better than transforming to monomial basis

But slower: O(d2) instead O(d) multiplications with Horner scheme in monomial basis

( ) ) somefor ()(1)()( 111 ααα ttt jijiji −−− −+= ppp


Benefits of Non‐Uniform B‐Splines

Improvements over the uniform case:• We can choose the parameter intervals freely

Typically: Use distance between control points as knot distance

Allows better adaptation to distance between control points

Curves tend to “overshoot” less (in particular: problem forB‐Spline interpolation; no convex hull property there)

Achieve more uniform speed for applications in animation

• No irregularitiesReduced smoothness, start/end conditions build into the computed bases

Advantages for rendering (no stopping at sharp corners)


Examples

The effect of parameter spacing:

uniform

first interval larger


Examples

Multiplying Knots


A Closer Look at B‐Splines

Need some more tools & properties:• Proof various properties

• Operations: Knot insertion, degree elevation, etc.

• Convert to alternative bases (e.g. Bezier, monomials)

Problem:• Indexing nightmare

• We need a better formalism to understandwhat’s really going on: blossoming & polars

• We will look at that tool first, then go into the details again

Geometric Modeling�Summer Semester 2010Today...Polynomial Spline Curves�Piecewise Cubic InterpolationWhat we have so far...Piecewise PolynomialsChoosing the Degree...Cubic SplinesPiecewise Cubic InterpolationConditionsConditionsNumerical SolutionCubics Minimize AccelerationProof: Cubics Minimize AccelerationProof: Cubics Minimize AccelerationProof: Cubics Minimize AccelerationProof: Cubics Minimize AccelerationPolynomial Spline Curves�Spline Bases with Local ControlSo what’s missing?NotationNew IdeaDesirable PropertiesAffine InvarianceAffine InvarianceConvex Hull PropertyConvex Hull PropertyConvex Hull PropertySpline TechniquesSpline TechniquesPolynomial Spline Curves�Hermite SplinesHermite SplinesIllustrationHermite BasisHermite BasisHermite BasisIllustrationPropertiesPolynomial Spline Curves�Bezier CurvesBezier SplinesAll You See is Bezier Curves...Bernstein BasisExamplesBernstein BasisBezier CurvesBezier Curves: ExamplesMatrix FormFormat ConversionFormat ConversionThe Other Way Round...Cubic Quadratic CasePolynomial Spline Curves�Bezier SplinesBezier SplinesDerivativesDerivativesBezier Curve PropertiesBezier Curve PropertiesBezier Curve PropertiesBezier Curve PropertiesBezier Spline ContinuityBezier Spline ContinuityBezier Spline ContinuityIn PracticeBezier Curve EditingGeometric ContinuityGeometric ContinuityGeometric Continuity for Bezier SplinesPolynomial Spline Curves�Uniform Cubic B-SplinesLiteratureOverviewOverviewKey IdeasShifted Basis FunctionsBasis FunctionCreating CurvesUniform B-Spline CurvesDiscontinuitiesContinuity ControlIllustrationEnd ConditionsKnot SequencesConversion to Bezier BasisBasis ChangeWhere does the basis come from?Three Approaches:The Elementary ApproachThe Elementary ApproachThe Elementary ApproachPropertiesRepeated ConvolutionRepeated ConvolutionIllustrationIllustrationIllustrationSmoothnessPartition of UnityOther PropertiesConsequencesRepeated Linear InterpolationRepeated Linear InterpolationPlot of the Three PiecesRepeated Linear InterpolationDe Boor RecursionConnectionPolynomial Spline Curves�Non-Uniform B-SplinesGeneralizationGeneral De Boor RecursionUniform Case:B-SplinesDe Boor AlgorithmDe Boor AlgorithmDe Boor AlgorithmExampleData FlowDe Boor AlgorithmBenefits of Non-Uniform B-SplinesExamplesExamplesA Closer Look at B-Splines

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