Positive Geometries for Barycentric Interpolation
aa r X i v : . [ c s . C G ] J a n Positive Geometries for Barycentric Interpolation
M´arton Vaitkus Budapest University of Technology and Economics
Abstract
We propose a novel theoretical framework for barycentric in-terpolation, using concepts recently developed in mathematicalphysics. Generalized barycentric coordinates are defined simi-larly to Shepard’s method, using positive geometries – subsetswhich possess a rational function naturally associated to theirboundaries. Positive geometries generalize certain properties ofsimplices and convex polytopes to a large variety of geometricobjects. Our framework unifies several previous constructions,including the definition of Wachspress coordinates over poly-topes in terms of adjoints and dual polytopes. We also discusspotential applications to interpolation in 3D line space, mean-value coordinates and splines.
The interpolation of scalar- or vector-valued data is an impor-tant task in many fields, including numerical analysis, geomet-ric modeling and computer graphics.
Barycentric coordinates can be defined over segments, triangles and simplices as wellas more complex shapes, such as polytopes [4]. These include
Wachspress coordinates [17] which are rational functions de-fined for convex polytopes. Wachspress coordinates were alsogeneralized to subsets bounded by non-linear hypersurfaces [2]and can be defined in several equivalent ways [18, 11, 9]. Wedescribe a theoretical framework that clarifies the relationshipbetween these different formulations, and provides opportuni-ties for novel generalizations.We propose a set of basic building blocks for barycentric in-terpolation methods: positive geometries . Positive geometrieshave been recently introduced in the theoretical physics liter-ature [1], but their application to interpolation problems hasnot been explored before. Besides simplices and polytopes, thecategory of positive geometries includes objects with similarcombinatorial properties, such as polycons [17] or “positive”parts of toric varieties [15] and Grassmannians [12, 6]. A posi-tive geometry carries a “canonical” differential form: a rationalfunction with the properties of a signed volume, that has itspoles (where the denominator vanishes) along the boundariesof the “positive” region. Crucially, these poles have a recursivestructure, i.e. restricting a canonical form to a boundary com-ponent (via a generalization of taking complex residues) resultsin another canonical form. Thus, positive geometries sharemany properties with polytopes – most importantly, that com-plicated objects can be constructed by adding together simplerones. To define interpolation in terms of canonical forms, weuse a variant of Shepard’s method [4, 13]: barycentric coordi-nates are ratios tending to ∞∞ along the interpolated bound-aries. For polytopes, our construction recovers the definitionof Wachspress coordinates in terms of dual volumes [9].In this preliminary work, our goal is to introduce the theoryof positive geometries to our audience, and demonstrate how they generalize earlier constructions for barycentric interpola-tion. After some motivating observations (Section 2), we give adefinition of positive geometries and their canonical forms, alsogiving some examples (Section 3). We then describe how to usethe canonical forms of positive geometries for barycentric inter-polation (Section 4). Finally, we discuss potential applicationsof this framework to interpolation in line space, and possibleextensions to non-rational barycentric coordinates and splines(Section 5). Let us consider the basic example of linear interpolation overa segment [ a, b ] ⊂ R . The standard formula f ( x ) = b − xb − a f ( a ) + x − ab − a f ( b ) , (2.1)can be written in an alternative barycentric form [8]: f ( x ) = x − a f ( a ) − x − b f ( b ) − b − a ( x − a )( x − b ) , (2.2)as a rational function with both the numerator and the denom-inator having poles at the boundaries a and b .Consider next linear interpolation over triangles( p , p , p ) ⊂ R in the plane. The usual formula forbarycentric interpolation gives f ( x ) = A f ( p ) + A f ( p ) + A f ( p ) A , (2.3)which can be reorganized into the equivalent form f ( x ) = A A f ( p ) + A A f ( p ) + A A f ( p ) AA A A , (2.4)with the notations shown in Figure 1. A A A xp p p p i − p i p i +1 x A i − ,i A i,i +1 C i Figure 1: Notations for barycentric coordinates.Wachspress proposed generalized barycentric coordinatesover convex polytopes [17]. For a planar n -gon with vertices p , . . . p n − these coordinates are defined as: f ( x ) = n X i =0 C i A i − ,i A i,i +1 P k C k A k − ,k − ··· A k +1 ,k +2 Q k A k,k +1 f ( p i ) , (2.5)1igure 2: Canonical rational function of a triangle.Figure 3: Canonical rational function of a pentagon.where the indices are cyclical modulo n – see Figure 1.In each of these cases, some terms in the numerator anddenominator approach infinity along an interpolated subset,which is reminiscent of Shepard’s method [8, 13]. The rationalfunctions in the denominators are illustrated in Figure 2 andFigure 3. The common pattern involves functions with polesalong the boundaries of some shape. Our goal is to make thisidea rigorous using the notion of a positive geometry. The central concept of our work is that of a positive geome-try – a relatively new concept originating from mathematicalphysics. In this chapter, we give an informal overview, andrefer to [1] for technical details. For the necessary backgroundin projective and algebraic geometry, see e.g. [12, 7].
Positive geometries were introduced as generalizations ofshapes with a recursive structure similar to polytopes, definedby some sort of “positivity” criterion (e.g. the interior of poly-topes, or totally positive matrices [5]). The definition was mo-tivated by recent developments in quantum physics, where thesolution of differential equations in Fourier space led to rationalfunctions with poles at the boundaries of certain regions [1].
Definition 1. A positive geometry is a pair ( X, X ≥ ) , where X is a d -dimensional complex projective algebraic variety, and X ≥ is an oriented semi-algebraic subset of its real part, witha unique differential d -form Ω( X, X ≥ ) called its canonicalform , defined by recursion on the boundary dimension: • If d = 0 , then X is an (oriented) point, and Ω( X, X ≥ ) = ± depending on the orientation. • If d > , then the boundary components of X ≥ are them-selves positive geometries, and the multivariate residue(see [1, A.3] or [7, Ch. 5]) along a component is the canon-ical form for the associated positive geometry. We stress that a positive geometry is determined by an am-bient complex manifold (most often a projective space P n ),together with a “positive” real subset, thus many different pos-itive geometries could be associated to the same ambient space.For positive geometries over projective spaces with homoge-neous coordinates x = [ x : x : . . . x d ], the canonical form canalways be written in terms of a rational function [1, C.1]:Ω( X, X ≥ ) = C ( x ) ω ( x ) , (3.6)where C is called the canonical rational function of thepositive geometry and ω ( x ) = 1 d ! x dx ∧ . . . ∧ dx d + . . . + ( − d x d dx ∧ . . . ∧ dx d − is the standard measure on projective d -space. Some elementary examples of positive geometries are the fol-lowing: • Segments , bounded by two points a = [ a : 1] and b =[ b : 1] in the projective line P parameterized using ho-mogeneous coordinates. The canonical form at the point x = [ x : y ] is Ω( P , [ a, b ]) = h b , a ih x , a i h x , b i ω ( x ) , (3.7)where h v , w i := det( v , w ) denotes the determinant of vec-tors of homogeneous coordinates. • Simplices in projective spaces. For triangles formed bythree points p i = [ x i : y i : 1] , i = 0 , , ⊂ P parameterized using homogeneous coordi-nates x = [ x : y : z ], the canonical form isΩ( P , ∆) = 12 h p , p , p i h x , p , p i h x , p , p i h x , p , p i ω ( x ) . (3.8)Both of these forms are invariant under independent scalingof the homogeneous coordinates of the vertices, as well as thepoint of evaluation, and are thus well-defined functions overprojective spaces.Many other examples of positive geometries were identified[1, Ch. 5]: • Planar regions bounded by a conic section and a line. • Regions in projective 3-space bounded by a quadric or cu-bic surface and a plane. • Positive, real parts of
Grassmannians (manifolds of k -planes in n -dimensional space, denoted G ( k, n )). • Positive, real parts of toric varieties [15].These examples – called generalized simplices – all havecanonical forms with constant numerators.The canonical forms of positive geometries are additive, i.e.the canonical form of a union is the sum of the canonical formsof its parts. The poles along boundaries meeting with oppo-site orientation will cancel, so that only poles on the exterior2oundary remain (technically, this is true for a “signed trian-gulation of the empty set” – see [1, Ch. 3]). This impliesthat the canonical forms of more complicated regions can bedetermined by “triangulating” them. It follows that convexpolytopes – which can be triangulated in the usual sense – arealso positive geometries.In analogy with generalized simplices, there are also various generalized polytopes : • Convex regions of the projective plane bounded by straightlines and conics, which are examples of polycons , as definedby Wachspress [17]. • Grassmann polytopes , in particular
Amplituhedra , whichare generalizations of cyclic polytopes into Grassmannians.Generalized polytopes have canonical forms with a non-constant numerator. In fact, for polytopes and polycons thenumerator is known as the adjoint polynomial [10, 2].
A canonical form often has a natural geometric interpretation.In particular, the canonical rational function of a convex pro-jective polytope P ⊂ P d is the signed volume of its polar dual P ∗ x ⊂ ( P d ) ∗ , as shown in [1, Ch. 7.4.1]. The polar dual is theintersection of the dual projective cone of the polytope, withthe dual hyperplane of the point x ∈ P , and its volume is arational function over P computed by the integral formulaVol( P ∗ x ) = 1 d ! Z y ∈ P x ∗ x T y ) d +1 ω ( y ) . (3.9) We claim that the canonical forms of positive geometries canbe used to define barycentric interpolation in a way similar toShepard’s method.Consider barycentric (linear) interpolation over a segment .Define the weight function for the endpoint a as the ratio ofcanonical forms for positive geometries over the projective line λ a ( x ) = Ω( P , [ a, x ∗ ])Ω( P , [ a, b ]) , (4.10)where x ∗ is the projective dual of the point x . If we choosecoordinates so that x is the origin and x ∗ is the point at infinity,we get the barycentric formula (2.2) for linear interpolation.The same construction applies to a triangle as well. For thethe vertex p i , we take the ratio of two canonical forms over theprojective plane – that of the original triangle, and the trianglebounded by the two sides meeting at p i together with the dualline of the current point x . Using the notations of Figure 4: λ ∆ i ( x ) = Ω( P , ∆( l ki , l ij , x ∗ ))Ω( P , ∆( l ij , l jk , l ki )) . (4.11)This is simply the usual formula (2.4) for barycentric interpo-lation. Along each of the sides, the forms (technically, theirresidues) restrict to linear interpolation over a segment, as ex-pected.For a convex polytope P ⊂ P we follow the same recipe –the denominator for the vertex p i will be the canonical formof P , while the numerator is the canonical form defined by thesides meeting at the vertex, with the dual line of the currentpoint: λ Pi ( x ) = Ω( P , ∆( l ( i − i , l i ( i +1) , x ∗ ))Ω( P , P ) . (4.12) xp j p k p i x ∗ l ij l ki l jk Figure 4: Triangles used for barycentric coordinates.These weight functions are the Wachpress coordinates (2.5)over P . The generalization to higher-dimensional simplices and(simplicial) polytopes is straightforward.In each case, the numerators are defined by positive geome-tries that form a signed triangulation of the domain. The addi-tivity of canonical forms under unions then implies that theirsum is the canonical form of the original polytope, and thusthese functions form a partition of unity .While these triangulations are not the most natural with re-spect to the original polytope, they correspond to a natural tri-angulation of the polar dual polytope including the origin (i.e.the current point x ). Recalling the interpretation of canonicalforms as dual volumes, the numerator is seen as the volume ofthe dual pyramid formed by the current point and the dual faceof the vertex. Thus, we connect to the earlier work of [9], whocharacterized Wachspress coordinates as ratios of polar dualvolumes. Note that a triangulation of the polar dual throughits vertices is analogous to Warren’s triangulation-based defi-nition of the adjoint polynomial for a polytope [18].Wachspress coordinates can be generalized also to regionsbounded by subsets bounded by higher-order algebraic vari-eties, such as polycons [17]. We omit the discussion of thesecases to conform to spatial limitations. Our approach to barycentric interpolation with canonical formsof positive geometries unifies earlier approaches using adjointpolynomials, dual volumes and Shepard-like interpolation. Anadvantage of this framework is that it extends to positive ge-ometries other than polytopes, such as Grassmannians andtoric varieties.We also mention that the definition of a positive geometryembeds the domain into a higher-dimensional complex mani-fold, thus our approach can be viewed as a multivariate gen-eralization of complex analytical methods (contour integrals,residues) used in univariate approximation theory [16].
As was mentioned previously, certain “positive” subsets ofGrassmannians are also examples of positive geometries. Pointsin a Grassmannian G ( k, n ) can be represented by k × n matrices,and the positive geometry known as the positive Grassmannian corresponds to totally positive matrices, which are of great in-terest for approximation theory and geometric modeling [5].The Grassmannian of 2-planes in 4-space, G (2 ,
4) – which isalso the manifold of lines within 3-space – is particularly in-teresting for many applications [12]. G (2 ,
4) is a 4-dimensional3ypersurface in P cut out by a quadratic equation in Pl¨uckercoordinates, and the positive Grassmannian is a semi-algebraicsubset with non-linear boundaries. The line-geometric ana-logue of a simplex might be related to the tetrahedral line com-plex , the boundary of which is the set of lines defined by theedges of a tetrahedron. The combinatorial structure of thisboundary is that of an octahedron – an example of a hyper-simplex – as shown in Figure 5, where each vertex represents aline along an edge of the tetrahedron, while each face representslines through either a vertex or a face [6]. p p p p l l l l l l Figure 5: Tetrahedral line complex and corresponding hyper-simplex. Shaded faces correspond to lines through vertices.Being a positive geometry, the positive Grassmannian has acanonical form, i.e. a rational function of Pl¨ucker coordinateswith poles along its boundaries [1, Ch. 5.5]. This suggests thatgeneralizations of barycentric coordinates into line space mightbe possible using our framework.
Canonical forms are defined by rational functions, so Mean-Value Coordinates (MVCs) [4] – defined by transcendentalfunctions – are apparently incompatible with the proposedframework. We could adapt the approach of [14], where MVCsare expressed as dual Shepard interpolants, after deforming theoriginal boundary to a unit circle around the current point. Adisc is not a positive geometry in the usual sense – it lacks zero-dimensional boundary components, for example. Nevertheless,we can easily find a projectively well-defined function with sin-gularities along a projective conic C given by the quadraticequation x T Qx = 0 (see [1, Ch. 10] for a lengthy discussion):Ω( P , C ) = π det( Q ) ( x T Qx ) ω ( x ) . (5.13)Observe the similarity with the transfinite form of MVCs [4,Ch. 10], when C is a circle. This kind of canonical function isnot rational and its singularities along C are not simple poles,but branch points (similar to the origin of the complex plane forfractional powers and logarithms). The authors of [1] also iden-tified such transcendental generalizations of positive geometriesas promising subjects for future research. Formulas such as the dual volume (3.9) also appear in the con-text of multivariate (box/simplex/cone) splines [3], as Laplacetransforms of indicator functions. This suggests that splinesand barycentric interpolants could both fit within an even moregeneral theoretical framework related to integral geometry.
Acknowledgements
This project has been supported by the Hungarian ScientificResearch Fund (OTKA, No.124727). The author thanks Tam´asV´arady for support and P´eter Salvi for interesting discussions.
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