Introduction to Gröbner Bases for Geometric Modeling Geometric & Solid Modeling 1989 Christoph M....
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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 fj for
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.
}{ 11 rr fhfhg
source: Hoffmann
},{ 1 rffI
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
0),,( 1 nxxf • 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 field k 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.
m
i
ii xa
0
source: Hoffmann
* for non-0 elements
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 ) that is not the root of any polynomial in k[x].
11
2210
m
m uauauaa
source: Hoffmann
Algebraic Concepts:Multivariate Polynomials
jnjj en
m
j
eej xxxa ,,2,1
121
• 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. 1222 zyx
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] – g be 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
fixed} |],...,[{ 1 fxxkgffI n
*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.
source: Hoffmann
}],...,[|...{ 12211 nirr xxkgfgfgfgFI
* Not necessarily unique.
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 set V(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 .
],...,[ 1 nxxkI
source: Hoffmann
fI
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. source: Hoffmann
3
2
tz
ty
tx
3232 00 xzzyyx
*see Hoffman’s Section 7.2.6 for subtleties related to this statement.
Algebraic Concepts:Algebraic Sets and Varieties (cont.)
• Given generators F = { 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.)
source: Hoffmann
• 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.
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)
0:1 zxg
source: Hoffmann
0:
0:222
2
2221
rzyf
ryxf
0:2 zxg
211 , ffII
112 , gfII
213 , gfII
0:1 zxg
0:2 zxg
112 , gfII 213 , gfII
)()()( 321 IVIVIV
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.
source: Hoffmann
02:
01:22
2
221
zyf
yxf
Bezout’s Theorem*: 2 irreducible surfaces of degree m and n intersect in a curve of degree mn. *allowing complex coordinates, points at infinity
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
rr fhfhfhg ...2211
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:
1. x
2. If then for all power products w.
3. If u and v are not yet ordered by rules 1 and 2, then order them lexicographically as strings.
0 ,21
21 ie
nee exxx n
source: Hoffmann
nxxx 211
vu vwuw
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:
source: Hoffmann
)()( glppflpp
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.
– g is in normal form NF(g, F) if it cannot be reduced further. Note: normal form need not be unique.
source: Hoffmann
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 basis G of ideal • Set of polynomials generating F.
• Rewriting algorithm using G produces unique normal forms.
– Ideal membership algorithm using G:
source: Hoffmann
Gröbner Bases:Buchberger’s Theorem & Construction
• Algorithm will consist of 2 operations:1. Consider a polynomial, and bring it into normal form
with respect to some set of generators G.2. 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
Gröbner Bases:Buchberger’s Theorem & Construction (continued)
source: Hoffmann
Gröbner Bases:Buchberger’s Theorem & Construction (continued)
Buchberger’s Theorem: foundation of algorithm
source: Hoffmann
Gröbner basis construction algorithm