Introduction to Gröbner Bases for Geometric Modeling

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Introduction to Gröbner Bases for Geometric Modeling. Geometric & Solid Modeling 1989 Christoph M. Hoffmann. Algebraic Geometry. Branch of mathematics. Express geometric facts in algebraic terms in order to interpret algebraic theorems geometrically.

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### Introduction to Gröbner Bases for Geometric Modeling

Geometric & Solid Modeling

1989

Christoph M. Hoffmann

Algebraic Geometry
• Branch of mathematics.
• Express geometric facts in algebraic terms in order to interpret algebraic theorems geometrically.
• Computations for geometric objects using symbolic manipulation.
• Surface intersection, finding singularities, and more…
• Historically, methods have been computationally intensive, so they have been used with discretion.

source: Hoffmann

Goal
• Operate on geometric object(s) by solving systems of algebraic equations.
• “Ideal”: (informal partial definition) Set of polynomials describing a geometric object symbolically.
• Considering algebraic combinations of algebraic equations (without changing solution) can facilitate solution.
• Ideal is the set of algebraic combinations (to be defined more rigorously later).
• Gröbner basis of an ideal: special set of polynomials defining the ideal.
• Many algorithmic problems can be solved easily with this basis.
• One example (focus of our lecture): abstract ideal membership problem:
• Is a given polynomial g in a given ideal I ?
• Equivalently: can g be expressed as an algebraic combination of the fjfor some polynomials hj?
• Answer this using Gröbner basis of the ideal.
• Rough geometric interpretation: g can be expressed this way when surface g = 0 contains all points that are common intersection of surfaces fj = 0.

source: Hoffmann

Overview
• Algebraic Concepts
• Fields, rings, polynomials
• Field extension
• Multivariate polynomials and ideals
• Algebraic sets and varieties
• Gröbner Bases
• Lexicographic term ordering and leading terms
• Rewriting and normal-form algorithms
• Membership test for ideals
• Buchberger’s theorem and construction of Gröbner bases
• For discussion of geometric modeling applications of Gröbner bases, see Hoffmann’s book.
• e.g. Solving simultaneous algebraic expressions to find:
• surface intersections
• singularities

source: Hoffmann

Algebraic Concepts:Fields, Rings, and Polynomials
• Consider single algebraic equation:
• Values of xi’s are from a field.(Recall from earlier in semester.)
• Elements can be added, subtracted, multiplied, divided*.
• Ground fieldk is the choice of field .
• Univariate polynomial over k is of form:
• Coefficients are numbers in k.
• k[x] = all univariate polynomials using x’s.
• It is a ring (recall from earlier in semester): addition, subtraction, multiplication, but not necessarily division.
• Can a given polynomial be factored?
• Depends on ground field
• e.g. x2+1 factors over complex numbers but not real numbers.
• Reducible: polynomial can be factored over ground field.
• Irreducible: polynomial cannot be factored over ground field.

* for non-0 elements

source: Hoffmann

Algebraic Concepts:Field Extension
• Field extension: enlarging a field by adjoining (adding) new element(s) to it.
• Algebraic Extension:
• Adjoin an element u that is a root of a polynomial (of degree m) in k[x].
• Resulting elements in extended field k(u)are of form:
• e.g. extending real numbers to complex numbers by adjoining i
• i is root of x2+1, so m=2 and extended field elements are of form a + bi
• e.g. extending rational numbers to algebraic numbers by adjoining roots of all univariate polynomials (with rational coefficients)
• Transcendental Extension:
• Adjoin an element (such as p) that is not the root of any polynomial in k[x].

source: Hoffmann

Algebraic Concepts:Multivariate Polynomials
• Multivariate polynomial over k is of form:
• Coefficients are numbers in k.
• Exponents are nonnegative integers.
• k[x1,…,xn] = all multivariate polynomials using x’s.
• It is a ring: addition, subtraction, multiplication, but not necessarily division.
• Can a given polynomial be factored?
• Depends on ground field (as in univariate case)
• Reducible: polynomial can be factored over ground field.
• Irreducible: polynomial cannot be factored over ground field.
• Absolutely Irreducible: polynomial cannot be factored over any ground field.
• e.g.

source: Hoffmann

Algebraic Concepts:Ideals
• For ground field k, let:
• kn be the n-dimensional affine space over k.
• mathematical physicist John Baez: "An affine space is a vector space that's forgotten its origin”.
• Points in kn are n-tuples (x1,…,xn), with xi’s having values in k.
• f be an irreducible multivariate polynomial in k[x1,…,xn]
• gbe a multivariate polynomial in k[x1,…,xn]
• f = 0 be the hypersurface in kn defined by f
• Since hypersurface gf = 0 includes f = 0, view f as intersection of all surfaces of form gf = 0
• is an ideal*
• g varies over k[x1,…,xn]
• Consider the ideal as the description of the surface f.
• Ideal is closed under addition and subtraction.
• Product of an element of k[x1,…,xn]with a polynomial in the ideal is in the ideal.

source: Hoffmann and others

*Ideals are defined more generally in algebra.

Algebraic Concepts:Ideals (continued)
• Let F be a finite set of polynomials f1, f2,…, fr in k[x1,…,xn]
• Algebraic combinations of the fi form an ideal generated by F (a generating set*):
• generators: { f, g }
• Goal: find generating sets, with special properties, that are useful for solving geometric problems.

* Not necessarily unique.

source: Hoffmann

Algebraic Concepts:Algebraic Sets
• Let be the ideal generated by the finite set of polynomials F = { f1, f2,…, fr }.
• Let p = (a1,…, an) be a point in kn such that g(p) = 0 for every g in I.
• Set of all such points p is the algebraic setV(I) of I.
• It is sufficient that fi(p) = 0 for every generator fi in F.
• In 3D, the algebraic surface f = 0 is the algebraic set of the ideal .

source: Hoffmann

Algebraic Concepts:Algebraic Sets (cont.)
• Intersection of two algebraic surfaces f, g in 3D is an algebraic space curve.
• The curve is the algebraic set of the ideal.
• But, not every algebraic space curve can be defined as the intersection of 2 surfaces.
• Example where 3 are needed*: twisted cubic (in parametric form):
• Can define twisted cubic using 3 surfaces: paraboloid with two cubic surfaces:
• Motivation for considering ideals with generating sets containing > 2 polynomials.

*see Hoffman’s Section 7.2.6 for subtleties related to this statement.

source: Hoffmann

Algebraic Concepts:Algebraic Sets and Varieties (cont.)
• Given generatorsF = { f1, f2,…, fr }, the algebraic set defined by F in kn has dimension n-r
• If equations fi = 0 are algebraically independent.
• Complication: some of ideal’s components may have different dimensions.

source: Hoffmann

Algebraic Concepts:Algebraic Sets and Varieties (cont.)
• Consider algebraic set V(I) for ideal I in kn.
• V(I) is reducible when V(I) is union of > 2 point sets, each defined separately by an ideal.
• Analogous to polynomial factorization:
• Multivariate polynomial f that factors describes surface consisting of several components
• Each component is an irreducible factor of f.
• V(I) is irreducible implies V(I) is a variety.

source: Hoffmann

Algebraic Concepts:Algebraic Sets and Varieties (cont.)
• Example: Intersection curve of 2 cylinders:
• Intersection lies in 2 planes:

and

• Irreducible ellipse in plane is is algebraic set in ideal generated by { f1,g1 }.
• Irreducible ellipse in plane is is algebraic set in ideal generated by { f1,g2 }.
• Ideal is reducible.
• Decomposes into and
• Algebraic set
• Varieties: V(I2) and V(I3)

source: Hoffmann

Algebraic Concepts:Algebraic Sets and Varieties (cont.)
• Example: Intersection curve of 2 cylinders:
• Intersection curve is not reducible
• These 2 component curves cannot be defined separately by polynomials.
• Rationale: Bezout’s Theorem implies intersection curve has degree 4. Furthermore:
• Union of 2 curves of degree m and n is a reducible curve of degree m + n.
• If intersection curve were reducible, then consider degree combinations for component curves (total = 4):
• 1 + 3: illegal since neither has degree 1.
• 2 + 2: illegal since neither is planar.
• Conclusion: intersection curve irreducible.
• Bezout’s Theorem also implies that twisted cubic cannot be defined algebraically as intersection of 2 surfaces:
• Twisted cubic has degree 3.
• Bezout’s Theorem would imply it is intersection of plane and cubic surface.
• But twisted cubic is not planar; hence contradiction.

Bezout’s Theorem*: 2 irreducible surfaces of degree m and n intersect in a curve of degree mn. *allowing complex coordinates, points at infinity

source: Hoffmann

Gröbner Bases:Formulating Ideal Membership Problem
• Can help to solve geometric modeling problems such as intersection of implicit surfaces (see Hoffmann Sections 7.4-7.8).
• Here we only treat the ideal membership problem for illustrative purposes:
• “Given a finite set of polynomials F = { f1, f2,…, fr }, and a polynomial g, decide whether g is in the ideal generated by F; that is, whether g can be written in the form where the hi are polynomials.”
• Strategy: rewrite g until original question can be easily answered.

source: Hoffmann

Gröbner Bases:Lexicographic Term Ordering and Leading Terms
• Need to judge if “this polynomial is simpler than that one.”
• Power Product:
• Lexicographic ordering of power products:
• x
• If then for all power products w.
• If u and v are not yet ordered by rules 1 and 2, then order them lexicographically as strings.

Example for n=2 on board...

source: Hoffmann

Gröbner Bases:Lexicographic Term Ordering and Leading Terms
• Each term in a polynomial g is a coefficient combined with a power product.
• Leading term lt(g) of g: term whose power product is largest with respect to ordering
• lcf (g) =leading coefficient of lt(g)
• lpp (g) =leading power product of lt(g)
• Definition: Polynomial f is simpler than polynomial g if:

Example 7.1 on board...

source: Hoffmann

Gröbner Bases:Rewriting and Normal-Form Algorithms
• Given polynomial g and set of polynomials F = { f1, f2,…, fr }
• Rewrite/simplify g using polynomials in F.
• gis in normal formNF(g, F) if it cannot be reduced further. Note: normal form need not be unique.

source: Hoffmann

Example 7.2 on board...

Gröbner Bases:Rewriting and Normal-Form Algorithms
• If normal form from rewriting algorithm is unique
• then g is in ideal when NF(g, F) = 0.
• This motivates search for generating sets that produce unique normal forms.

source: Hoffmann

Gröbner Bases:A Membership Test for Ideals
• Goal: Rewrite g to decide whether g is in the ideal generated by F.
• Gröbner basisG of ideal
• Set of polynomials generating F.
• Rewriting algorithm using G produces unique normal forms.
• Ideal membership algorithm usingG:

source: Hoffmann

Example 7.3 on board...

Gröbner Bases:Buchberger’s Theorem & Construction
• Algorithm will consist of 2 operations:
• Consider a polynomial, and bring it into normal form with respect to some set of generators G.
• From certain generator pairs, compute S-polynomials (see definition on next slide) and add their normal forms to the generator set.
• G starts as input set F of polynomials
• G is transformed into a Gröbner basis.
• Some Implementation Issues:
• Coefficient arithmetic must be exact.
• Rational arithmetic can be used.
• Size of generator set can be large.
• Reduced Gröbner bases can be developed.

source: Hoffmann

Example 7.4 on board...

source: Hoffmann

Buchberger’s Theorem:

foundation of algorithm

Gröbner basis construction algorithm

Example 7.5 on board...

source: Hoffmann