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u -bases and Bernstein polynomials

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1. u-bases for polynomial systems in one variable

Ning Song,Ron Goldman

CAGD, in press

2. Division algorithms for Bernstein polynomials

Laurent Buse, Ron Goldman

CAGD, in press

Rice University, CS, Houston,USA

History:

Sederberg, Cox, Chen, Zheng

Sederberg, Implicitization using moving

curves and surfaces.Siggraph 95

Cox, The moving line ideal basis of planar rational curves.

CAGD,1998

Zheng, A direct approach to computing the u-basis of planar rational

curves, J.Symbolic Compute,2001

U-bases for polynomial systems in one variableUSTC:

Chen and Wang, The u-basis of a planar rational curve-properties and computation. Graphical Models, 2003

Chen and Sederberg, A new implicit representation of a planar rational curve with high order singularity. CAGD,2002

Wang and Chen. Implicitization and parametrization of quadratic surfaces with one simple base point, ISSAC'2008.

Chen and Wang, Computing singular points of plane rational curves, Journal of Symbolic Computation, 2008

f(t)=(f1(t),…, fk(t))

Vector polynomials p(t)=(p(t),…,pk(t)) such that

p.f=p1f1+…+pkfk≡0

syzygy modele space k-1 generators (Cox, 98)

LV(p) denotes the leading coefficient and deg(p)=n

syzygy moduleA set of k-1 vector polynomials u1,…,uk-1 is called a u-basis of the polynomial f1,…,fk or a u-basis of the syzygy module syz(f1,…,fk),if

u1,…,uk-1 are a basis of syz(f1,…,fk). That is any l∈syz(f1,…,fk),

can be expressed uniquely by

l=h1u1+…+hk-1uk-1

where hi are polynomials in t;

2. LV(u1),…,LV(uk-1) are linearly independent.

DefinitionTheorem 1 u1,…,uk-1 are u-basis for f=(f1,…,fk) with deg(u1) ≤deg(u2) ≤…≤deg(uk-1) and GCD(f1,…,fk)=1.

We have:

Every l∈syz(f1,…,fk) can be expressed by (1) with deg(hiui) ≤deg(l)

If v1,…vk-1 is a set of vector polynomials in syz(f1,…,fk) that are linearly independent with deg(v1) ≤deg(v2) ≤…≤deg(vk), then

deg(ui) ≤deg(vi) for i=1,2,…,k-1.

If v1,…,vk-1 is another u-basis for f1,…,fk with deg(v1) ≤deg(v2) ≤

…≤deg(vk), then deg(ui)=deg(vi) for i=1,2,…,k-1.

Properties of u-bases4. ui(t0) ≠0 for every parameter value t0. deg(u1) ≤deg(u2) ≤…≤deg(uk-1) and GCD(f1,…,fk)=1.

5. u1(t0),…,uk(t0) are linearly independent for every parameter

value t0.

6. The out product of u1,…,uk-1 is equal to c(f1,…,fk) for some non-zero constant c.

7. f(t0)=cS for some constant c and for S=(s1,…,sk) if and only if

u1(t0).S=…=uk-1(t0).S=0.

8.

Let u1,…,uk be k-1 polynomials in syz(f1…,fk) with deg(u1) ≤…≤deg(uk-1) and GCD(f1,…,fk)=1. Then u1,…,uk-1 form a u-basis for syz(f1,…,fk) if and only if one of the conditions holds:

1. Every element l∈syz(f1,…,fk) can be expressed as in (1) and u1,…,uk-1 are of the lowest degree.

2. Every element l ∈syz(f1,…,fk) can be expressed as in (1) with

deg(hiui) ≤deg(l) for i=1,…,k-1.

3. Every element l ∈syz(f1,…,fk) can be expressed as in (1) and deg(u1)+…+deg(uk-1)=deg(f1,…,fk).

4. The outer product of u1,…,uk-1 is equal to c(f1,…,fk) where c is a constant not equal to zero, and deg(u1)+…+deg(uk-1)=deg(f1,…,fk).

Equivalent definitionAlgorithm: deg(u1) ≤…≤deg(uk-1) and GCD(f1,…,fk)=1. Then u1,…,uk-1 form a u-basis for syz(f1,…,fk) if and only if one of the conditions holds:

Input: f=(f1(t),…,fk(t)) with GCD=1

Output: k-1 polynomial for u-bases

Step 1: Create the set of vector polynomials {ua,b}:

ua,b=(0,…,0, -fb(t),…, fa(t),…,0);

Sort {ua,b} by a,b and rename the set {ui} i=1,2,…,s.

Step 2: Set mi=LV(ui) and ni=deg(ui), i=1,2,…,s.

Step 3: Sort ni so that n1≥n2≥…≥ns and re-index ui,mi.

Step 4: Find real numbers a1,a2,…as such that

a1m1+a2m2+…+asms=0;

and at least two of the terms aimi are not zero.

An algorithm to calculate u-basesStep 5: Choose the lowest integer j such that aj≠0 , update uj by

setting

If uj≡0, discard uj and set s=s-1; otherwise set mj=LV(uj)

and nj=deg(uj).

Step 6: If s=k-1, then output the k-1 non-zero polynomials u1,…,uk-1

and stop; otherwise, go to step 3.

With common factors update uj by

Example update uj by :

Comparison update uj by

Four questions: update uj by

1. Point on curve: Given a point P0=(x0,y0,z0,w0) in homogeneous coordinates, determine if the point P lies on the curve P(t)

2. Inversion: Given a point P0=(x0,y0,z0,w0) in homogeneous coordinates on the curve P(t), find the corresponding parameter value to such that P(t0)=cP0 for some constant c.

3. Implicitization: Given a rational space curve P(t), find a low degree implicit representation for this curve.

4. Intersection: Given two rational space curves P(t), ,find

the intersection points of P(t) and

Application to non-planar curves- Example: update uj by
- (Cubic curve).
- P0=(3,0,2,1) t=1
- P1=(0,0,0,1)
- (Quartic curve)
- P0=(2,0,-1,1)
- P1=(0,1,1,1)

Implicitization: update uj by

u-bases for P(t):

p=(p1(t),p2(t),p3(t),p4(t)); q=(q1(t),q2(t),q3(t),q4(t)); r=(r1(t),r2(t),r3(t),r4(t));

p=p.(x,y,z,w);q=q.(x,y,z,w),r=r.(x,y,z,w);

S1(x,y,z,w)=Res(p,q;t); S2(x,y,z,w)=Res(q,r;t);

S3(x,y,z,w)=Res(p,r;t)

Example:

Future work: update uj by

How to eleminate these extraneous points

How to generate the lowest degree representation using u-bases?

Authors: update uj by

Laurent Buse, INRIA Sophia Antipolis , France

Ron Goldman, Rice University, USA

Problem:

operator of Bernstein polynomials.

Reason:

advantage of Bernstein basis

Division algorithms for Bernstein polynomialsThree algorithms: update uj by

1. an algorithm for dividing two univariate polynomials

2. an algorithm for calculating the GCD of an arbitrary collection

of univariate polynomials

3. an algorithm for computing u-basis for the syzygy module of an

arbitrary collection of univariate polynomials.

then extended to three or more Bernstein polynomials

ContentsBernstein polynomial: update uj by

1. Multiplication by powers of

2. Multiplication by powers of

Bernstein bases basicsmonomial polynomials update uj by ↔ Bernstein bases

Conversion between monomial and Bernstein bases Division algorithm for two Bernstein update uj by polynomials

Given two univariate polynomials f(t) ,g(t) such that degt(t)≤degt(g)

How to solve q(t) and r(t) such that g=q.f+r.

Example: update uj by

g(t)=[2,3/4,1/2,1/4,0]=t4+t, f(t)=[1,0,0,0]=t3.

We will denote the homogenization of any polynomial P(t) ∈R[t] by

A u-basis algorithm for an arbirary collection of Bernstein polynomialsWe denote by the homogenization of fi(t) of degree

We set

Example: ∈R[t] by

f1(t)=[1]=1,f2(t)=[1.0]=t, f3(t)=[1,0,0]=t2

Remove common powers of 1-t and get

U-basis for f1,f2,f3

GCD algorithm for three or more Bernstein ∈R[t] by polynomials

Thanks ! ∈R[t] by

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