Linear precision for parametric patches
aa r X i v : . [ m a t h . AG ] F e b LINEAR PRECISION FOR PARAMETRIC PATCHES
LUIS DAVID GARCIA-PUENTE AND FRANK SOTTILE
Abstract.
We give a precise mathematical formulation for the notions of a parametricpatch and linear precision, and establish their elementary properties. We relate linearprecision to the geometry of a particular linear projection, giving necessary (and quiterestrictive) conditions for a patch to possess linear precision. A main focus is on linearprecision for Krasauskas’ toric patches, which we show is equivalent to a certain rationalmap on CP d being a birational isomorphism. Lastly, we establish the connection betweenlinear precision for toric surface patches and maximum likelihood degree for discrete ex-ponential families in algebraic statistics, and show how iterative proportional fitting maybe used to compute toric patches. Introduction
B´ezier curves and surfaces are the fundamental units in geometric modeling. There aretwo basic shapes for surfaces—triangular B´ezier patches and rectangular tensor productpatches. Multi-sided patches are needed for some applications, and there are several controlpoint schemes for C ∞ multi-sided patches. These include the S -patches of Loop andDeRose [17], Warren’s hexagon [22], Karˇciauskas’s M -patches [12], and the multi-sidedtoric B´ezier patches of Krasauskas [14]. (Relationships between these and other patchesare discussed in [13].) Parametric patches are general control point schemes for C ∞ patcheswhose shape is a polygon or polytope. They include the patch schemes just mentioned, aswell as barycentric coordinates for polygons and polytopes [9, 21, 23].The success and widespread adoption of B´ezier and tensor-product patches is due inpart to their possessing many useful mathematical properties. Some, such as affine in-variance and the convex hull property, are built into their definitions and also hold forthe more general parametric patches. Other properties, such as de Castlejau’s algorithmfor computing B´ezier patches, come from the specific form of their Bernstein polynomialblending functions. Linear precision is the ability of a parametric patch to replicate linearfunctions. When the blending functions of a parametric patch correspond to the vertices ofa polytope, these blending functions give barycentric coordinates precisely when the patchhas linear precision. In this way, blending functions for a parametric patch having linearprecision are barycentric coordinates for general control point schemes. Mathematics Subject Classification.
Key words and phrases. tensor product B´ezier surfaces; triangular B´ezier surface patches; Barycentriccoordinates; Iterative proportional fitting.Work of Sottile supported by NSF CAREER grant DMS-0538734, by the Institute for Mathematics andits Applications with funds provided by the National Science Foundation, and by Peter Gritzmann of theTechnische Universit¨at M¨unchen.
For us, a (parametric) patch is a collection of non-negative blending functions indexedby a finite set A of points in R d , where the common domain of the blending functions isthe convex hull ∆ of A . A collection of control points in R ℓ indexed by A is used to definea map from ∆ to R ℓ . The blending functions determine the internal structure of this mapand the basic shape, ∆, of its image, while the control points determine how the imagelies in R ℓ . Choosing the control points to be the points of A gives the tautological map,and the patch has linear precision when this tautological map is the identity on ∆. Weshow that every patch has a unique reparametrization (composing the blending functionswith a homeomorphism of ∆) having linear precision (Theorem 1.11). This generalizesTheorem 8.5 of [19], which was for toric patches.The blending functions of a patch may be arbitrary non-negative C ∞ functions. A patchis rational if it has a reparametrization having polynomial blending functions. For a rationalpatch, a map given by control points corresponds to a linear projection of a projectivealgebraic variety associated to its blending functions. Its unique reparametrization havinglinear precision has rational blending functions if and only if a certain canonical mapdefined on this variety is a birational isomorphism (Theorem 2.9), which implies that thisvariety has a maximally degenerate position with respect to a canonical linear subspacegiven by the set A (Theorem 2.10).We apply this analysis to Krasauskas’s toric patches, which are rational. The shape ofa toric patch is a pair ( A , w ) where A is a set of integer points in Z d and w is a collectionof positive real numbers indexed by A . We show that iterative proportional fitting [5],a simple numerical algorithm from statistics, computes the blending functions that havelinear precision. This algorithm was suggested to us by Bernd Sturmfels. It may form abasis for algorithms to manipulate these patches.Krasauskas [15, Problem 3] asked whether any toric patches (besides the classical B´eziersimploids [6]) admit a rational reparametrization having linear precision. A toric patch ofshape ( A , w ) corresponds to a homogeneous polynomial f = f A ,w whose dehomogenizationis the sum of monomials with exponents A and coefficients w . The toric patch admits arational reparametrization having linear precision if and only if the toric differential D toric f := h x ∂∂x f : x ∂∂x f : · · · : x d ∂∂x d f i : CP d −−→ CP d defines a birational isomorphism (Corollary 3.13). This analysis of linear precision is usedin [11] to classify which toric surface patches can have linear precision.In Krasauskas’s question about linear precision, he allowed the points A indexing theblending functions of toric patches to move within their (fixed) convex hull, keeping thesame blending functions (See Example 3.16). Our analysis of linear precision for generalparametric patches will help to address that version of his question.In Section 1, we define parametric patches and show that every patch has a uniquereparametrization that has linear precision. In Section 2 we show that if a rational patchhas linear precision, then an algebraic variety we obtain from the blending functions hasan exceptional position with respect to a certain linear space. We define toric patchesin Section 3, and in Section 4 we explain how iterative proportional fitting computes theblending functions for toric patches that have linear precision. INEAR PRECISION FOR PARAMETRIC PATCHES 3 Linear precision for parametric patches
We interpret the standard definition of a mapping via control points and blending func-tions (see for example [13, § C ∞ ) where defined and real-valued unless otherwise stated. Let R > bethe set of strictly positive real numbers and R ≥ the set of non-negative real numbers.Let A be a finite set of points in R d , which we shall use as geometrically meaningfulindices. A control point scheme for parametric patches, or ( parametric ) patch for short, isa collection β = { β a | a ∈ A} of non-negative functions, called blending functions . Thecommon domain of the blending functions is the convex hull ∆ of A , which we call the domain polytope . We will always assume that ∆ is full dimensional in that it has dimension d . We assume that the blending functions do not vanish simultaneously at any point of ∆.That is, the blending functions have no base points in their domain.A set { b a | a ∈ A} ⊂ R ℓ of control points gives a map F : ∆ → R ℓ defined by(1.1) F ( x ) := P a ∈A β a ( x ) b a P a ∈A β a ( x ) . The denominator in (1.1) is positive on ∆ and so the map F is well-defined. Remark 1.2.
Positive weights, or shape parameters, { w a ∈ R > | a ∈ A} scaling theblending functions are often included in the definition of the map F . We have insteadabsorbed them into our notion of blending functions. See Remarks 1.7 and 3.3.The blending functions { β a | a ∈ A} are normalized if they form a partition of unity, X a ∈A β a ( x ) = 1 . As the denominator in (1.1) is (strictly) positive on ∆, we may divide each blending functionby this denominator to obtain normalized blending functions.Redefine β a := β a P a ∈A β a . Then β a ( x ) ≥ X a ∈A β a ( x ) = 1 , for x ∈ ∆. For normalized blending functions, the formula (1.1) becomes(1.3) F ( x ) = X a ∈A β a ( x ) b a . We deduce two fundamental properties of parametric patches. • Convex hull property.
For x ∈ ∆, F ( x ) is a convex combination of the controlpoints. Thus F (∆) lies in the convex hull of the control points. • Affine invariance.
If Λ is an affine function on R ℓ , then Λ( F ( x )) is the map withthe same blending functions, but with control points { Λ( b a ) | a ∈ A} .A patch is non-degenerate if its blending functions are linearly independent. This impliesin particular that F (∆) cannot collapse into a point unless all the control points are equal.We make our key definition. LUIS DAVID GARCIA-PUENTE AND FRANK SOTTILE
Definition 1.4.
A patch { β a | a ∈ A} has linear precision if for every affine function Λ( x )defined on R d , Λ( x ) = X a ∈A β a ( x ) Λ( a ) . That is, if the patch can replicate affine functions. This notion depends strongly on thepositions of the points in A .The tautological map τ of a given patch is the map (1.3) when the control points aretaken to be the corresponding points of A ,(1.5) τ ( x ) := X a ∈A β a ( x ) a . By the convex hull property, τ (∆) ⊂ ∆. By affine invariance, the patch has linear precisionif and only if τ is the identity function, τ ( x ) = x . We record this fact. Proposition 1.6.
A patch has linear precision if and only if its tautological map is theidentity map on the domain polytope ∆ . Remark 1.7.
This is a more restrictive notion of linear precision than is typically consid-ered. It is more common to define a patch to be a collection { β i | i = 1 , . . . , n } of blendingfunctions, and this has linear precision if we have x = P ni =1 w i β i ( x ) a i P ni =1 w i β i ( x ) , for some non-negative weights { w i | i = 1 , . . . , n } and some points A = { a i | i = 1 , . . . , n } whose convex hull is the domain polytope ∆ of the blending functions.Our more restrictive definition of patches and linear precision, where we incorporatethe weights and the points A into our definition of blending functions, allows us to deriveprecise criteria with which to study linear precision. Our intention is to employ thesecriteria to study patches in the generality in which other authors had worked. Example 1.8.
Fix an integer n > A := { in | ≤ i ≤ n } be the set of n +1equally distributed points in the unit interval. The i th Bernstein polynomial is β i := (cid:0) ni (cid:1) x i (1 − x ) n − i , which we associate to the point in ∈ A . These form a partition of unity, n X i =0 β i ( x ) = n X i =0 (cid:18) ni (cid:19) x i (1 − x ) n − i = (cid:0) x + (1 − x ) (cid:1) n = 1 , and are therefore normalized blending functions. Given control points b i ∈ R , for-mula (1.1) becomes F ( x ) = n X i =0 (cid:18) ni (cid:19) x i (1 − x ) n − i b i , which is the classical formula for a B´ezier curve of degree n in R . INEAR PRECISION FOR PARAMETRIC PATCHES 5
This patch has linear precision. First, note that in · (cid:0) ni (cid:1) = (cid:0) n − i − (cid:1) . Then τ ( x ) is n X i =0 (cid:18) ni (cid:19) x i (1 − x ) n − i · in = n X i =1 (cid:18) n − i − (cid:19) x i (1 − x ) n − i = x · n − X j =0 (cid:18) n − j (cid:19) x j (1 − x ) n − − j = x . Example 1.9.
Let A be the vertices of the hexagon, given below by its defining inequalities.11 − − − y >
01 + x − y > − x >
01 + x > y − x >
01 + y > ✁✁☛❅❅❘ ✛✲ ❅❅■✁✁✕ Let the blending function associated to a vertex be the product of the linear forms definingedges that do not contain that vertex.(1.10) β [ − − ] := (1 + x − y )(1 − y )(1 − x )(1 + y − x ) β [ − ] := (1 − y )(1 − x )(1 + y − x )(1 + y ) β [ ] := (1 − x )(1 + y − x )(1 + y )(1 + x ) β [ ] := (1 + y − x )(1 + y )(1 + x )(1 + x − y ) β [ ] := (1 + y )(1 + x )(1 + x − y )(1 − y ) β [ − ] := (1 + x )(1 + x − y )(1 − y )(1 − x )The normalized blending functions have denominator ϕ ( x, y ) := 2(3 + xy − x − y ), whichis strictly positive on ∆. These normalized blending functions have linear precision. X a ∈A β a ( x, y ) a = ϕ ( x, y ) (cid:20) xy (cid:21) . In fact, these normalized blending functions are Wachspress’s barycentric coordinates [21]for this hexagon.When A is the set of vertices of ∆ (so that A is in convex position), then normalizedblending functions of a patch with linear precision are barycentric coordinates for ∆. By LUIS DAVID GARCIA-PUENTE AND FRANK SOTTILE this we mean in the sense of [9, 21, 23]: nonnegative functions indexed by the extremepoints of ∆ that have linear precision in the sense of Definition 1.4.A reparametrization of a patch { β a | a ∈ A} by a homeomorphism ψ : ∆ → ∆ is anew patch with blending functions β a ◦ ψ for a ∈ A . A patch is proper if its tautologicalmap (1.5) is a homeomorphism. This condition is necessary for the patch to have linearprecision. If a patch is proper, then reparametrizing it by the inverse τ − of τ gives anew patch whose tautological map is τ ◦ τ − , the identity function, and τ − is the uniquehomeomorphism of ∆ with this property. We record this straightforward, yet fundamentalresult about linear precision. Theorem 1.11.
A proper patch has a unique reparametrization that has linear precision.
Theorem 1.11 suggests that it will be fruitful to discuss patches up to reparametrization.In Section 2, we show that a natural geometric object X β associated to a patch is invariantunder reparametrizations and thus represents the patch up to reparametrization.A patch is rational if it has a reparametrization whose normalized blending functionsare rational functions (quotients of polynomials). A patch has rational linear precision if itis proper and its reparametrization having linear precision has blending functions that arerational functions. Such a patch is necessarily rational. We seek criteria that determinewhen a rational patch has rational linear precision.Theorem 1.11 and our preceding discussion concerns reparametrizations of a patch havinglinear precision. We may also alter a patch by multiplying its blending functions by positivereal numbers, typically called weights, or by moving the points of A that are not vertices ofthe domain polytope ∆. This second change does not alter any map (1.1) given by controlpoints, but it will change the tautological map τ (1.5). Our analysis in the next sectionmay be helpful in addressing whether it is possible to tune a given patch (using weightsor moving the points in A ) to obtain one that has rational linear precision. Example 3.16(due to Krasauskas and Karˇciauskas) shows how one toric patch may be tuned to achievelinear precision. 2. The geometry of linear precision
We introduce an algebraic-geometric formulation of patches to clarify the discussion inSection 1 and to provide tools with which to understand and apply Theorem 1.11. Thisleads us to discuss the relevance for geometric modeling of the subtle difference betweenrational varieties and unirational varieties. Lastly, we use this geometric formulation togive a geometric characterization of when a patch has rational linear precision. We recom-mend the text [3] for additional background on algebraic geometry. We first review linearprojections, which are the geometric counterpart of control points.2.1.
Linear projections.
We consider R ℓ as a subset of the ℓ -dimensional real projectivespace RP ℓ via the embedding R ℓ ∋ z [1 , z ] ∈ RP ℓ . INEAR PRECISION FOR PARAMETRIC PATCHES 7
A point [ z , z , . . . , z ℓ ] ∈ RP ℓ lies in this copy of R ℓ if and only if z = 0. In that case, thecorresponding point of R ℓ is(2.1) (cid:16) z z , z z , . . . , z ℓ z (cid:17) . Let
A ⊂ R d be a finite set of points and write RP A for the real projective space whosecoordinates are indexed by A . Let b = { b a ∈ R ℓ | a ∈ A} be a collection of control points,which we regard as points in RP ℓ so that b = { [1 , b a ] ∈ RP ℓ | a ∈ A} .Given a point y = [ y a | a ∈ A ] ∈ RP A , if the sum(2.2) X a ∈A y a · (1 , b a ) ∈ R ℓ +1 is non-zero then it represents a point π b ( y ) in RP ℓ . This map π b is a linear projection (2.3) π b : RP A −−→ RP ℓ . We use a broken arrow −→ to indicate that π b is defined only on the complement ofthe set E b ⊂ RP A where the sum (2.2) vanishes. This linear subspace is the center ofprojection .When the control points affinely span R ℓ , the center of projection E b has codimension ℓ + 1 in RP A . The inverse image π − b ( x ) of a point x ∈ RP ℓ is the set H \ E b , where H ⊂ RP A is a linear subspace of codimension ℓ containing E b . If L is any ℓ -dimensionallinear subspace of RP A that does not meet E b , then π b maps L isomorphically to RP ℓ .Identifying L with RP ℓ gives an explicit description of the map π b . If y ∈ RP A \ E b , then π b ( y ) = L ∩ (cid:0) y, E b (cid:1) , the intersection with L of the linear span of y and the center E b . If y ∈ L , then its inverseimage under π b is ( y, E b ) \ E b .For example, Figure 1 shows the effect of a linear projection π on a cubic space curve C . The center of projection is a line, E , which meets the curve in a point, B . π ✲ E ❍❍❥ B ❍❍❥ y, E ✄✄✎ y ✛ L ✛ C ❙❙❙♦ ✻ π − ( y ) Figure 1.
A linear projection π with center E . LUIS DAVID GARCIA-PUENTE AND FRANK SOTTILE
Geometric formulation of a patch.
Consider (1.1) from the point of view of pro-jective geometry. Given a patch β = { β a ( x ) | a ∈ A} with A ⊂ R d , then(2.4) x [ β a ( x ) | a ∈ A ]defines a map β : ∆ → RP A whose a th coordinate is the blending function β a ( x ).The map β is unchanged if we multiply all blending functions by the same positivefunction ϕ ( x ) defined on ∆. Thus we obtain the same map if we use instead the normalizedblending functions. The image X β of ∆ under β is a closed subset of RP A (but not Zariskiclosed!), as ∆ is compact and the map β is continuous. It is non-degenerate (does not liein a hyperplane) exactly when the patch is non-degenerate.Suppose now that we have control points b = { b a ∈ R ℓ | a ∈ A} and consider thecomposition of the map β and the linear projection π b ∆ β −−→ X β ⊂ RP A π b −−→ RP ℓ . By our assumption on the positivity of the blending functions, formula (2.1) implies thatthe image lies in the standard copy of R ℓ , and there it is given by the formula (1.1). Thusthis image, F (∆), is the image of X β under the projection π b given by the control points.A reparametrization of the patch β by a homeomorphism ψ : ∆ → ∆ gives a differentmap with the same image in RP A e β : ∆ ψ −−→ ∆ β −−→ X β . Thus X β is an invariant of the patch modulo reparametrization.If we take A to be our set of control points where the element a ∈ A is the controlpoint associated to the blending function β a , then the resulting linear projection π A is the tautological projection ,(2.5) π A : RP A −−→ RP d . Here, the coordinate points ( e a | a ∈ A ) are indexed by elements of A and π A ( e a ) = a ∈ RP d . The tautological map τ (1.5) is the composition∆ β −−→ X β ⊂ RP A π A −−→ RP d . The patch is proper if this map is a homeomorphism onto its image ∆ ⊂ RP d , and thepatch has linear precision if the composition is the identity map, by Proposition 1.6.With these definitions, we have the following identification of the reparametrizationhaving linear precision, which is a more precise version of Theorem 1.11. Theorem 2.6.
Suppose that X β is a proper patch and π A : X β → ∆ is the tautologicalprojection restricted to X β . Then the blending functions for X β that have linear precisionare given by the coordinates of the inverse of π A . Remark 2.7.
This geometric perspective, where a patch is first an embedding of ∆ into RP A followed by a linear projection, may have been introduced into geometric modelingin [7]. It is fundamental for Krasauskas’s toric B´ezier patches [14], and was reworked in thetutorial [19]. There, an analog of Theorem 1.11 was formulated. We take this opportunityto correct an error in notation. The tautological projection is an algebraic version of the INEAR PRECISION FOR PARAMETRIC PATCHES 9 moment map of the toric variety underlying a toric patch, and not the actual moment mapfrom symplectic geometry.2.3.
Rational varieties.
We study the algebraic relaxation of our previous notions, re-placing the real numbers by the complex numbers so that we may use notions from algebraicgeometry. A rational patch has a parametrization by rational functions. Clearing theirdenominators gives a new collection { β a | a ∈ A} of blending functions which are poly-nomials. These polynomials define complex-valued functions on C d , and so the blendingfunctions give a map β C : C d \ B −→ CP A , which is defined on the complement of the set B where all the blending functions vanish.This is called the base locus of the map β C . This map β C extends the map β : ∆ → X β .We write β for β C , P A for CP A , and in general use the same notation for maps defined oncomplex algebraic varieties as for their restrictions to subsets of their real points.As with linear projections, we write β : C d −→ P A to indicate that β is only defined on C d \ B . Such a map between algebraic varieties that is given by polynomials and definedon the complement of an algebraic subset is called a rational map .Let Y β be the Zariski closure of the image of β . One important consequence of the patchbeing rational is that X β is a full-dimensional subset of the real points of the algebraicvariety Y β . In particular, X β is defined locally in RP A by the vanishing of some polynomials,and this remains true for any image F (∆) of a map F (1.1) given by our original blendingfunctions and any choice of control points. This is important in modeling, for these implicitequations are used to compute intersections of patches.Suppose that X β has rational linear precision and that β = { β a | a ∈ A} are rationalblending functions with linear precision. By Proposition 1.6, the composition∆ β −−→ X β π A −−→ ∆is the identity map. Since ∆ has full dimension in R d , it is Zariski dense in C d , and so thecomposition of rational functions(2.8) C d β −−→ Y β π A −−→ C d is also the identity map (where it is defined).We introduce some terminology to describe this situation. A rational map ϕ : Y −→ Z between complex algebraic varieties is a birational isomorphism if there is another map ψ : Z −→ Y such that ϕ ◦ ψ and ψ ◦ ϕ are the identity maps where they are defined. Inparticular, Y has a Zariski open subset U and Z a Zariski open subset V such that ϕ | U isone-to-one between U and V . The projection π : C −→ L of Figure 1 is not a birationalisomorphism. Indeed, if x ∈ C is a point in π − ( p ), then p, E = x, E and this plane meetsthe complement C \ B of the base locus in x and one other point.Thus if the composition (2.8) is the identity map, then the map π A is a birational iso-morphism from Y β −→ CP d . We deduce the algebraic-geometric version of Proposition 1.6. Theorem 2.9.
A patch β = { β a | a ∈ A} has rational linear precision if and only if thecomplexified tautological projection π A : Y β −−→ CP d is a birational isomorphism. Using well-known properties of birational projections, we use this characterization oflinear precision to deduce necessary conditions for a patch to have linear precision. A rational variety of dimension d is one that is birational to C d . Theorem 2.10.
If a patch β = { β a | a ∈ A} has rational linear precision, then (1) Y β is a rational variety, (2) almost all codimension d planes L containing the center E A of the tautologicalprojection meet Y β in at most one point outside of E A . By condition (2), Y β has an exceptionally singular position with respect to E A . Typi-cally, Y β does not meet a given codimension d +1 plane and its intersection with a givencodimension d plane consists of deg( Y β ) points, counted with multiplicity. By Condition(2), not only does Y β meet E A , but if L has codimension d and contains E A , then most ofthe deg( Y β ) points in Y β ∩ L lie in E A . We will see this in Examples 2.11 and 3.16. Proof.
Statement (1) is immediate from Theorem 2.9.Since the birational isomorphism π A : Y β −→ CP d is the restriction of a linear projec-tion with center E A , then for almost all points y ∈ Y β \ E A , we have (cid:0) y, E A (cid:1) ∩ Y β = { y } ∪ (cid:0) E A ∩ Y β (cid:1) . Since all codimension d planes L that contain E A have the form ( y, E A ) for some y E A and almost all meet Y β , statement (2) follows.Consider the geometric situation of Figure 1. Suppose now that E is the line tangent to C at the point B . Then E ∩ C is the point B , but counted with multiplicity 2. A linearprojection with center E restricts to a birational isomorphism of C with P .2.4. Unirational varieties.
The necessary condition of Theorem 2.10(1) gives an impor-tant but subtle geometric restriction on patches that have rational linear precision.The map β provides a parametrization of an open subset of Y β by an open subset of C d . Such parametrized algebraic varieties are called unirational . Unirational curves andsurfaces are also rational, but these two notions differ for varieties of dimension three andhigher. Clemens and Griffiths [2] showed that a smooth hypersurface of degree 3 in P isnot rational (these were classically known to be unirational, e.g. by Max Noether).Thus parametric patches of dimension at least 3 will in general be unirational and notrational. This does not occur for B´ezier simploids or toric patches. Example 2.11.
Consider again the Wachspress coordinates (1.10) of Example 1.9, whichhave linear precision. Let Y β ⊂ P be the image of the blending functions (1.10). We studythe base locus of the tautological projection π A of this patch. We first find equations for Y β as an algebraic subvariety of P .If we divide β [ − − ] β [ ] + 2 β [ ] β [ − ] + β [ ] β [ − ] by the product of the linear forms definingthe hexagon we get(1 − x )(1 + y − x ) + 2(1 + x )(1 − x ) + (1 + x )(1 + x − y ) = 4 − xy , INEAR PRECISION FOR PARAMETRIC PATCHES 11 which is symmetric in x and y , and so(2.12) β [ − − ] β [ ] + 2 β [ ] β [ − ] + β [ ] β [ − ] = β [ − − ] β [ ] + 2 β [ ] β [ − ] + β [ ] β [ − ] . If we let ( y a | a ∈ A ) be natural coordinates for P A , then (2.12) gives the quadraticpolynomial which vanishes on Y β , y [ − − ] y [ ] + 2 y [ ] y [ − ] + y [ ] y [ − ] − (cid:0) y [ − − ] y [ ] + 2 y [ ] y [ − ] + y [ ] y [ − ] (cid:1) . Cyclically permuting the vertices of the hexagon gives two other quadratics that vanish on Y β , but these three sum to 0. There is another quadratic polynomial vanishing on Y β , y [ − − ] y [ ] + y [ ] y [ ] + y [ ] y [ − − ] − (cid:0) y [ ] y [ − ] + y [ − ] y [ − ] + y [ − ] y [ ] (cid:1) . There is an additional cubic relation among the blending functions, which gives a cubicpolynomial that vanishes on Y β (2.13) y [ − − ] y [ ] y [ ] − y [ ] y [ − ] y [ − ] . These relations, three independent quadratics and one cubic, define Y β as an algebraicsubset of P A = P . The quadratic equations define Y β , together with the 2-dimensionallinear space cut out by the three linear equations.(2.14) 0 = y [ − − ] + y [ ] + y [ ] + y [ ] + y [ − ] + y [ − ]0 = y [ ] − y [ − − ] + y [ ] − y [ − ]0 = y [ ] − y [ − − ] + y [ ] − y [ − ]Thus Y β has degree 7 = 2 −
1. A general codimension 2 plane L containing the center E A of the tautological projection π A will meet Y β in at most 1 point outside of E A , byTheorem 2.10(2). Since L ∩ Y β has degree at least 7, the other 6 points must lie in thebase locus E A ∩ Y β , which is in fact a reducible cubic plane curve.The three linear forms (2.14) also define the tautological projection, π A , so this linearsubspace is the center of projection E A , and the base locus B is defined in E A by thecubic (2.13) defining Y β . Let us parametrize the center E A as follows. For [ a, b, c ] ∈ P , set y [ − − ] := a + b y [ ] := − a + by [ ] := a − c y [ − ] := − a − cy [ ] := a − b + c y [ − ] := − a − b + c Then we have y [ − − ] y [ ] y [ ] − y [ ] y [ − ] y [ − ] = 2 a ( a − b + bc − c ) , the product of a linear and a quadratic form. Toric Patches
Toric patches, which were introduced by Krasauskas [14], are a class of patches basedon certain special algebraic varieties, called toric varieties. For a basic reference on toricvarieties as parametrized varieties, see the book [20], particularly Chapters 4 and 13, andthe articles [4, 19], which are introductions aimed at people in geometric modeling. Wefirst define toric patches, give some examples, reinterpret Theorem 2.9 for toric patches,and then state the main open problem about linear precision for toric patches.3.1.
Toric Patches.
We regard elements a = ( a , . . . , a d ) ∈ Z d as exponent vectors formonomials in the variables x = ( x , . . . , x d ) x a := x a x a · · · x a d d . Let
A ⊂ Z d be a finite set of integer vectors whose convex hull ∆ is a polytope of dimension d in R d . This implies that A affinely spans R d . Suppose that w = { w a ∈ R > | a ∈ A} is aset of positive weights indexed by A . These data ( A , w ) define a map ϕ A ,w : ( C × ) d → P A ,(3.1) ϕ A ,w : x [ w a x a | a ∈ A ] . Let Y A ,w be the closure of the image of ( C × ) d under the map ϕ A ,w . When the weights w a are equal, this is the toric variety parametrized by the monomials in A . In general, it isthe translate of that toric variety by the element w of the positive real torus R A > , whichacts on P A by scaling the coordinates.Let X A ,w ⊂ RP A be the closure ϕ A ,w ( R d> ). This is the non-negative part of the translatedtoric variety Y A ,w . That is, if RP A≥ ⊂ P A is the set of points having non-negative (real)homogeneous coordinates, then X A ,w = Y A ,w ∩ RP A≥ . It is homeomorphic to the convexhull ∆ of the vectors in A , as a manifold with corners [10, § A is primitive if it affinely spans Z d . We may assume that A is primitive, forthere is always some primitive set A ′ ⊂ Z d with Y A ,w = Y A ′ ,w . First note that translatingevery vector in A by a fixed vector a ′ multiplies each coordinate of the map ϕ A ,w by themonomial x a ′ , which does not change ϕ A ,w as a map to projective space. Thus we mayassume that A contains the zero vector. Since A affinely spans R d , it generates a subgroup Z A of Z d which is isomorphic to Z d . Let b , . . . , b d ∈ Z A be the basis corresponding tothe basis of Z d under an isomorphism Z A ∼ −→ Z d . If we let A ′ ⊂ Z d be the image of A under this isomorphism and B := { b , . . . , b d } , then the map ϕ A ,w factors( C × ) d χ B −−→ ( C × ) d ϕ A′ ,w −−−−→ P A , where χ B is the surjective map that sends x to ( x b , . . . , x b d ). Note that ϕ A ′ ,w is injectivebut that χ B has fibers of size | Z d / Z A| . Thus ϕ A ,w is not injective unless A is primitive. Definition 3.2. A toric patch of shape ( A , w ) is any patch { β a | a ∈ A} such that theimage, X β , of ∆ under the map (2.4) given by the blending functions is equal to thenon-negative part X A ,w of the translated toric variety Y A ,w . Remark 3.3.
Our geometric perspective that the fundamental object is the image X β requires us to absorb the weights w into our definition and distinguishes patches withdifferent choices of weights. INEAR PRECISION FOR PARAMETRIC PATCHES 13
Example 3.4.
Let A := { , , . . . , n } ⊂ Z and let w ∈ R n +1 > be any set of weights. Thenthe map ϕ A ,w is x [ w , w x, w x , . . . , w n x n ] . The map x xn − x sends ∆ = [0 , n ) to [0 , ∞ ). Precomposing ϕ A ,w with this map gives atoric patch of shape ( A , w ).[0 , n ] ∋ x h w , w (cid:0) xn − x (cid:1) , w (cid:0) xn − x (cid:1) , . . . , w n (cid:0) xn − x (cid:1) n i = [ w ( n − x ) n , w x ( n − x ) n − , w x ( n − x ) n − , . . . , w n x n ] . Observe that if we choose weights w i = (cid:0) ni (cid:1) , substitute ny for x , and remove the commonfactors of n n , then we obtain the blending functions for the B´ezier curve of Example 1.8.Note that replacing x by ny also replaces i ∈ A by in . Example 3.5.
Krasauskas [14] generalized the classical B´ezier parametrization to anypolytope with integer vertices. A polytope ∆ is defined by its facet inequalities∆ = { x ∈ R d | h i ( x ) ≥ , i = 1 , . . . , N } . Here, ∆ has N facets and for each i = 1 , . . . , N , h i ( x ) = h v i , x i + c i is the linear functiondefining the i th facet, where v i ∈ Z d is the (inward oriented) primitive vector normal tothe facet and c i ∈ Z . (Compare these to the functions in Example 1.9.)Let A ⊂ ∆ ∩ Z d be any subset of the integer points of ∆ that includes its vertices. Let w = { w a | a ∈ A} ⊂ R > be a collection of positive weights. For every a ∈ A , Krasauskasdefined toric B´ezier functions (3.6) β a ( x ) := w a h ( x ) h ( a ) h ( x ) h ( a ) · · · h N ( x ) h N ( a ) . For x ∈ ∆, these parametrize a patch X β .Krasauskas defined a toric patch to be any reparametrization of such a patch. Ourdefinition of a toric patch (Definition 3.2) agrees with Krasauskas’s, when A = ∆ ∩ Z d consists of all the integer points in a polytope: Observe that the map β in (3.6) is thecomposition of a map H : ∆ → R N given by x ( h ( x ) , . . . , h N ( x )) with a rational map ϕ : C N −→ RP A given by ϕ : ( u , . . . , u N ) [ w a u h ( a )1 · · · u h N ( a ) N | a ∈ A ] . This map ϕ factors through the map ϕ A ,w . Indeed, define a map f ∆ : ( C × ) N −→ ( C × ) n u = ( u , . . . , u N ) t = ( t , . . . , t n ) where t j := N Y i =1 u h v i , e j i i Then N Y i =1 u h i ( a ) i = N Y i =1 u c i i · N Y i =1 u h v i , a i i = u c · t a . And so ϕ ( u ) = u c · ϕ A ,w ( f ∆ ( u )). This shows that X β ⊂ X A ,w , and Krasauskas [15, § Example 3.7. A standard d -simplex of degree n is the simplex in R d , n ∆ d := { x ∈ R d | x + · · · + x d ≤ n and x i ≥ } . For a point a = ( a , . . . , a d ) in n ∆ d ∩ Z d set | a | := a + · · · + a d and let w a := (cid:0) n a (cid:1) = n ! a ! ··· a d !( n −| a | )! be the multinomial coefficient. This gives a system of weights for A = n ∆ ∩ Z d .Then Krasauskas’s toric B´ezier patch for n ∆ d has blending functions β a := (cid:0) n a (cid:1) ( n − P i x i ) n − P i a i d Y i =1 x a i i . If we substitute ny i for x i and then remove the common factors of n n , we recover theclassical Bernstein polynomials [8, § Example 3.8.
The
B´ezier simploids [6] are toric patches based on products of B´eziersimplices. Suppose that d , . . . , d m and n , . . . , n m are positive integers. Set d := d + · · · + d m . Write R d as a direct sum R d ⊕ · · · ⊕ R d m and suppose that for each i = 1 , . . . , m , thescaled simplex n i ∆ d i lies in the i th summand R d i . Consider the Minkowski sum∆ := n ∆ d + n ∆ d + · · · + n m ∆ d m ⊂ R d , which is just the product of the simplices n i ∆ d i .If x i := ( x i, , . . . , x i,d i ) are coordinates on R d i , then∆ = { x = ( x , . . . , x m ) | x i,j ≥ x i, + · · · + x i,d i ≤ n i i = 1 , . . . , m } . Let A := ∆ ∩ Z d be the set of integer points of ∆. Given an integer point a = ( a , . . . , a m ) ∈A , set w a := Q mi =1 (cid:0) n i a i (cid:1) , the product of multinomial coefficients. This gives a system ofweights, and let Y A ,w be the resulting toric patch. Krasauskas’s toric patch of shape ∆ hasblending functions β a ( x ) := m Y i =1 β a i ( x ) , where β a i ( x i ) for x i ∈ R d i are the Bernstein polynomials. The resulting patch is called a B´ezier simploid in [6].We state our main result about linear precision for toric patches, which is a usefulreformulation of linear precision for toric patches.
Theorem 3.9.
Let
A ⊂ Z d be a primitive collection of exponent vectors, w ∈ R A > be asystem of weights, and define the Laurent polynomial f = f A ,w := X a ∈A w a x a . A toric patch of shape ( A , w ) has rational linear precision if and only if the rational function ψ A ,w : C d −→ C d defined by (3.10) 1 f (cid:16) x ∂∂x f, x ∂∂x f, . . . , x d ∂∂x d f (cid:17) is a birational isomorphism. INEAR PRECISION FOR PARAMETRIC PATCHES 15
Remark 3.11.
The map (3.10) has an interesting reformulation in terms of toric deriva-tives. The i th toric derivative of a Laurent polynomial f is x i ∂∂x i f . The toric differential D torus f is the vector whose components are the toric derivatives of f . Thus the map ψ A ,w is the logarithmic toric differential D torus log f .In the proof below, we show that the map ψ A ,w comes from a map C d −→ CP d ,(3.12) x h f : x ∂∂x f : x ∂∂x f : · · · : x d ∂∂x d f (cid:3) Let F A ,w be the homogenization of f with respect to a new variable x . Thendeg( F A ,w ) F A ,w = d X i =0 x i ∂∂x i F A ,w . Thus, after homogenizing and a linear change of coordinates, the map (3.12) is defined bythe formula D torus F A ,w = h x ∂∂x F A ,w : x ∂∂x F A ,w : · · · : x d ∂∂x d F A ,w (cid:3) This last form of the map (3.12) gives an appealing reformulation of Theorem 3.9 whichwe feel is the most useful for further analysis of linear precision for toric patches.
Corollary 3.13.
The toric patch of shape ( A , w ) has linear precision if and only if themap D torus F A ,w : CP d −→ CP d is a birational isomorphism.Proof of Theorem 3.9. By Theorem 2.9, a toric patch of shape ( A , w ) has linear precisionif and only if the complexified tautological projection π A : Y A ,w −−→ CP d is a birational isomorphism. Precomposing this with the defining parametrization of Y A ,w , ϕ A w : ( C × ) d ֒ → Y A ,w (which is a rational map C d −→ Y A ,w ), gives a map(3.14) C d ϕ A ,w −−→ Y A ,w π A −−→ CP d x [ w a x a | a ∈ A ] X a ∈A w a x a (1 , a ) . The initial (0th) coordinate of this map is X a ∈A w a x a which is the polynomial f A ,w . For i > i th coordinate of this composition is X a ∈A a i · w a x a = x i ∂∂x i f A ,w . If we divide the composite map (3.14) by its 0th coordinate, f A ,w , we obtain the map ψ A ,w .This map is birational if and only if π A is birational, as ϕ A ,w is birational. Example 3.15.
Let A := { , , . . . , n } ⊂ Z as in Example 3.4. Let w ∈ R n +1 > be any setof weights. Then the polynomial f = f A ,w is f ( x ) = w + w x + · · · + w n x n . Suppose that this toric patch has rational linear precision. By Theorem 3.9, the log-arithmic toric differential xf ′ ( x ) /f ( x ) is a birational isomorphism. Thus the fraction xf ′ ( x ) /f ( x ) reduces to a quotient of linear polynomials and so f ( x ) is necessarily a purepower, say a ( x + α ) n with α = 0.We conclude that the weights are w i = α n − i (cid:0) ni (cid:1) , which are not quite the weights in theB´ezier curve 1.8. If we rescale x (which does not change Y A ,w ), setting y = x/α , then theweights become | α | n (cid:0) ni (cid:1) . Removing the common factor of | α | n , reveals that this toric patchis the classical B´ezier curve of Example 1.8.We close with two problems. Problem 1.
Classify the toric patches of dimension d that have linear precision.The analysis of this section was recently used to classify toric surface patches ( d = 2)having linear precision [11]. It remains an open problem to understand how to tune a toricpatch (moving the points A ) to achieve linear precision. Problem 2.
Classify the toric patches of dimension d that may be tuned to have linearprecision.This is a strictly larger class of patches. For example, the classification of [11] shows ifa toric surface patch has linear precision, then it is triangular or quadrilateral, but it ispossible to tune a pentagonal patch to achieve linear precision. Example 3.16.
We discuss Krasauskas’s version [15] of Karˇciauskas’s pentagonal M -patch [12], which has linear precision. Let A be the integer points in the pentagonshown below, and let { w a | a ∈ A} be the indicated weights.(0 ,
2) (1 , ,
1) (1 ,
1) (2 , ,
0) (1 ,
0) (2 , A { w a | a ∈ A} These define a toric patch X A ,w and translated toric variety Y A ,w . This subvariety of P has degree 7 and is defined by 14 quadratic equations, all of the form y a y b − y c y d where a , b , c , d ∈ A with a + b = c + d . This has a monomial parametrization ϕ A ,w : ( s, t ) [ s a t a | a = ( a , a ) ∈ A ] . INEAR PRECISION FOR PARAMETRIC PATCHES 17
Its composition with the tautological projection π A gives a map g : P −→ P of degree 4,with 3 base points at ( − , − − , − ), and ( − , − π A ) by moving the non-extreme points of A within the pentagon. Let B consist of the vertices of the pentagon, together with thethree non-vertices (0 ,
2) (1 , , ) ( , ) (2 , ,
0) ( ,
0) (2 , B The composition of ϕ A ,w with this new tautological projection π B gives the rational map( s, t ) (cid:16) s (2 s + t + 3)( s + 1)(2 s + 2 t + 3) , t ( s + 2 t + 3)( t + 1)(2 s + 2 t + 3) (cid:17) . This is given by quadrics and has 3 base points ( − , − , − ), and ( − , · −
3, and is therefore a birational isomorphism. By Theorem 2.9, thetuned toric patch (with the lattice points A replaced by B ) has linear precision.Thus there is a rational parametrization β : → X A ,w whose composition with thetautological projection π B is the identity map. Indeed, Krasauskas [15] gives the followingmodification of the toric B´ezier functions and shows that the map π B ◦ β is the identity.(3.17) β [ ] := 3(2 − s ) (3 − s − t ) (2 − t ) β [ ] := 5(2 − s )(3 − s − t ) (2 − t ) s (3 − s − t ) β [ ] := 5(3 − s − t ) t (2 − s ) (3 − s − t ) (2 − t ) β [ ] := 7(3 − s − t ) t (2 − t )(3 − s − t )(2 − s ) s (3 − s − t ) β [ ] := 2(3 − s − t )(2 − t ) s (3 − s − t ) β [ ] := 2(3 − s − t ) t (2 − s ) s (3 − s − t ) β [ ] := 2(3 − s − t ) t (2 − t ) s (3 − s − t ) β [ ] := 2(3 − s − t ) t (2 − s ) s (3 − s − t )This is a modification of the toric B´ezier functions for the pentagon. If, in the defini-tion (3.6), we replace the form s by s (3 − s − t ) and the form t by (3 − s − t ) t , then weget these modified blending functions (3.17). Let us consider the geometry of linear precision for this tuned patch in the spirit ofSection 2. The projection π B is defined by the three linear forms. X := y [ ] + 2 y [ ] + y [ ] + 2 y [ ] + y [ ] Y := y [ ] + 2 y [ ] + y [ ] + 2 y [ ] + y [ ] Z := y [ ] + y [ ] + y [ ] + y [ ] + y [ ] + y [ ] + y [ ] + y [ ]Its center E B is defined by the vanishing of these three forms. A general linear subspace L of codimension 2 containing E B is defined by equations of the form X = xZ and Y = yZ . This subspace meets Y A ,w in four points. One point lies outside of E B while three pointslie in the center E B . We give the three points.3 0 − − − − − − y a of a point y ∈ RP A at the position of the point a .4. Iterative proportional fitting for toric patches
The toric patch X A ,w from geometric modeling appears naturally in algebraic statistics inthe form of a toric model, which leads to a dictionary between the subjects. We show thata toric patch has rational linear precision if and only if it has maximum likelihood degree1 as a statistical model. As a consequence, we present a new family of toric patches withrational linear precision. Finally, we propose the iterative proportional fitting algorithmfrom statistics as a tool to compute the unique reparametrization of a toric patch havinglinear precision.In algebraic statistics, the image of R d> under the map ϕ A ,w (3.1) is known as a toricmodel [18, § log-linear models , as the logarithms ofthe coordinates of ϕ A ,w are linear functions in the logarithms of the coordinates of R d> , or discrete exponential families , as the coordinates of ϕ A ,w are exponentials of the coordinatesof R d , which are themselves logarithms of the coordinates of R d> .We identify the non-negative orthant RP A≥ with the probability simplex∆ A := { y ∈ R A≥ | | y | := X a ∈A y a = 1 } , and so we may also regard X A ,w as a subvariety of ∆ A . INEAR PRECISION FOR PARAMETRIC PATCHES 19
The tautological map π A (2.5) appears in statistics. Given (normalized) data q ∈ ∆ A ,the problem of maximum likelihood estimation is to find the toric model parameters (apoint in R d> ) that best explain the data q . By Lemma 4 in [5], the maximum likelihoodestimate for the log-linear model X A ,w is the unique point p ∈ X A ,w such that π A ( p ) = π A ( q ) so that p = π − A ( π A ( q )) . Thus inverting the tautological projection is necessary for maximum likelihood estimation.Catanese, et. al. [1] defined the maximum likelihood degree of the model X A ,w to bethe degree of π − A , as an algebraic function. Equivalently, this is the number of complexsolutions to the critical equations of the likelihood function, which is the degree of thetautological map π A from the Zariski closure Y A ,w of X A ,w to CP d . By definition, a toricpatch has rational linear precision if π − A is a rational function (an algebraic function ofdegree 1). Proposition 4.1.
A toric patch X A ,w has rational linear precision if and only if X A ,w hasmaximum likelihood degree . An important family of toric models called decomposable graphical models are known tohave maximum likelihood degree 1 [16, p. 91]. Therefore, these models, which are not ingeneral B´ezier simploids, have linear precision. They typically have large d > π − A . Their algorithm requires that the data A be in a normal, homogeneousform.Observe first that the toric patch X A ,w does not change if we translate all elements of A by a fixed vector b , ( a a + b ), so we may assume that A lies in the positive orthant R d> . Scaling the exponent vectors in A by a fixed positive scalar t ∈ R > also does notchange X A ,w as x x t is a homeomorphism of R > that extends to a homeomorphism of R d> . Thus we may assume that A lies in the standard simplex ∆ d in R d :∆ d = { x ∈ R d | x ≥ | x | ≤ } . Lastly, we lift this to the probability simplex ∆ + d ∈ R d +1 ≥ ,∆ + d := { y ∈ R d | y i ≥ | y | = 1 } , by A ∋ a a + := (1 −| a | , a ) ∈ ∆ + d . Since for t ∈ R > and x ∈ R d> ,( t, tx ) a + = t −| a | ( tx ) a = tx a , we see that replacing A by its lifted version A + also does not change X A ,w . We remarkthat this is just a way to homogenize the data for our problem.Since these are affine transformations, the property of affine invariance for patches showsthat the affine map that is the composition of these transformations intertwines the originaltautological projection with the new one.We describe the algorithm of iterative proportional fitting , which is Theorem 1 in [5]. Proposition 4.2.
Suppose that
A ⊂ ∆ d and y ∈ conv ( A ) . Then the sequence { p ( n ) | n = 0 , , . . . } whose a -coordinates are defined by p (0) a := w a and, for n ≥ , p ( n +1) a := p ( n ) a · y a ( π A ( p ( n ) )) a , converges to the unique point p ∈ X A ,w such that π A ( p ) = π A ( y ) . We remark that if A is not homogenized then to compute π − A ( y ) for y ∈ conv( A ),we first put A into homogeneous form A + using an affine map ψ , and then use iterativeproportional fitting to compute π − A + ( ψ ( y )) = π − A ( y ). We also call this modification ofthe algorithm of Proposition 4.2 iterative proportional fitting. Thus iterative proportionalfitting computes the inverse image of the tautological projection, which, by Theorem 2.6gives the unique parametrization of X A ,w having linear precision. Corollary 4.3.
Iterative proportional fitting computes the unique parametrization of atoric patch having linear precision.
Little is known about the covergence of iterative proportional fitting. We think that itis an interesting question to investigate this convergence, particularly how it may relate tomaximum likelihood degree.
References
1. Fabrizio Catanese, Serkan Ho¸sten, Amit Khetan, and Bernd Sturmfels,
The maximum likelihood degree ,Amer. J. Math. (2006), no. 3, 671–697.2. C. Herbert Clemens and Phillip A. Griffiths,
The intermediate Jacobian of the cubic threefold , Ann.of Math. (2) (1972), 281–356.3. D. Cox, J. Little, and D. O’Shea, Ideals, varieties, algorithms: An introduction to computationalalgebraic geometry and commutative algebra , UTM, Springer-Verlag, New York, 1992.4. David Cox,
What is a toric variety? , Topics in algebraic geometry and geometric modeling, Contemp.Math., vol. 334, Amer. Math. Soc., Providence, RI, 2003, pp. 203–223.5. J. N. Darroch and D. Ratcliff,
Generalized iterative scaling for log-linear models , Ann. Math. Statist. (1972), 1470–1480.6. T. DeRose, R. Goldman, H. Hagen, and S. Mann, Functional composition algorithms via blossoming ,ACM Trans. on Graphics (1993), 113–135.7. Tony D. DeRose, Rational B´ezier curves and surfaces on projective domains , NURBS for curve andsurface design (Tempe, AZ, 1990), SIAM, Philadelphia, PA, 1991, pp. 35–45.8. Gerald Farin,
Curves and surfaces for computer-aided geometric design , Computer Science and Scien-tific Computing, Academic Press Inc., San Diego, CA, 1997.9. Michael S. Floater,
Mean value coordinates , Comput. Aided Geom. Design (2003), no. 1, 19–27.10. William Fulton, Introduction to toric varieties , Annals of Mathematics Studies, vol. 131, PrincetonUniversity Press, Princeton, NJ, 1993, The William H. Roever Lectures in Geometry.11. Hans-Christian Graf van Bothmer, Kristian Ranestad, and Frank Sottile,
On linear precision for toricsurface patches , 2007, in preparation.12. K¸estutis Karˇciauskas,
Rational M -patches and tensor-border patches , Topics in algebraic geometry andgeometric modeling, Contemp. Math., vol. 334, Amer. Math. Soc., Providence, RI, 2003, pp. 101–128.13. K¸estutis Karˇciauskas and Rimvydas Krasauskas, Comparison of different multisided patches usingalgebraic geometry , Curve and Surface Design: Saint-Malo 1999 (P.-J. Laurent, P. Sablonniere, andL.L. Schumaker, eds.), Vanderbilt University Press, Nashville, 2000, pp. 163–172.
INEAR PRECISION FOR PARAMETRIC PATCHES 21
14. Rimvydas Krasauskas,
Toric surface patches , Adv. Comput. Math. (2002), no. 1-2, 89–133, Ad-vances in geometrical algorithms and representations.15. , B´ezier patches on almost toric surfaces , Algebraic geometry and geometric modeling, Math.Vis., Springer, Berlin, 2006, pp. 135–150.16. Steffen L. Lauritzen,
Graphical models , Oxford Statistical Science Series, vol. 17, The Clarendon PressOxford University Press, New York, 1996, Oxford Science Publications.17. Charles T. Loop and Tony D. DeRose,
A multisided generalization of B´ezier surfaces , ACM Trans.Graph. (1989), no. 3, 204–234.18. Lior Pachter and Bernd Sturmfels (eds.), Algebraic statistics for computational biology , CambridgeUniversity Press, New York, 2005.19. Frank Sottile,
Toric ideals, real toric varieties, and the moment map , Topics in algebraic geometry andgeometric modeling, Contemp. Math., vol. 334, Amer. Math. Soc., Providence, RI, 2003, pp. 225–240.20. Bernd Sturmfels,
Gr¨obner bases and convex polytopes , American Mathematical Society, Providence,RI, 1996.21. Eugene L. Wachspress,
A rational finite element basis , Academic Press, Inc. [A subsidiary of HarcourtBrace Jovanovich, Publishers], New York, 1975, Mathematics in Science and Engineering, Vol. 114.22. Joe Warren,
Creating multisided rational B´ezier surfaces using base points , ACM Trans. Graph. (1992), no. 2, 127–139.23. , Barycentric coordinates for convex polytopes , Adv. Comput. Math. (1996), no. 2, 97–108(1997). Department of Mathematics and Statistics, Sam Houston State University, Huntsville,TX 77341, USA
E-mail address : [email protected] URL : Department of Mathematics, Texas A&M University, College Station, TX 77843, USA
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