Injectivity of 2D Toric Bézier Patches
Rational Bézier functions are widely used as mapping functions in surface reparameterization, finite element analysis, image warping and morphing. The injectivity (one-to-one property) of a mapping function is typically necessary for these applications. Toric Bézier patches are generalizations of classical patches (triangular, tensor product) which are defined on the convex hull of a set of integer lattice points. We give a geometric condition on the control points that we show is equivalent to the injectivity of every 2D toric Bézier patch with those control points for all possible choices of weights. This condition refines that of Craciun, et al., which only implied injectivity on the interior of a patch.
Bézier patches; toric patches; injectivity; mapping
Mapping functions play an important role in computer graphics, computer aided geometric design (CAGD), finite element analysis (FEA) and some related areas. The injectivity of mapping functions, that is, the absence of self-intersection, is crucial in image warping and morphing , free form deformation , surface reparameterization, and so on. Many authors have investigated conditions which imply injectivity. Goodman and Unsworth  proposed a sufficient condition for the injectivity of a 2D Bézier function. For the control points of a tensor product patch, their condition involves linear inequalities. For image morphing, Choi and Lee  presented a sufficient condition for the injectivity of 2D and 3D uniform cubic B-spline functions. Their condition provides a single bound for the displacements of control points that guarantees the injectivity of the cubic B-spline function. Floater  studies a sufficient condition for injectivity of convex combination mappings over triangulations.
Fig. 1 displays rational plane cubic Bézier curves with their control polygons (bold lines). The curve in Fig. 1 has no points of self-intersection. The curve in Fig. 1 has one point of self-intersection, which may be removed by varying the weights as shown in Fig. 1. The control polygon of the first curve is in convex position, so there are no positive weights for which the resulting Bézier curve has self-intersection. For the other control polygon there are weights (e.g. Fig. 1) such that the resulting Bézier curve has a point of self-intersection. The cited works provide conditions which imply no self-intersection. Our purpose is different: We give conditions on the control points for 2D patches which are equivalent to there being no self-intersection for any choice of positive weights.
The basic units in the geometric modeling of surfaces are rational Bézier simplices and tensor product patches. Krasauskas  introduced toric Bézier patches as a natural extension of classical rational patches and their higher-dimensional generalizations, the Bézier simploids by DeRose, et al. . The theory of toric patches is based upon real toric varieties from algebraic geometry , and they provide a general framework in which to pose many questions concerning classical rational patches.
To study dynamical systems arising from chemical reaction networks, Craciun et al.  prove an injectivity theorem for certain maps. This was adapted in  to give a geometric condition on a set of control points which implies that the resulting toric Bézier patch has no self-intersection, for any choice of positive weights. That result contains a minor flaw in that it only guarantees injectivity in the interior of a patch. We correct that flaw, at least for 2D patches, showing that the condition from  plus the mild additional hypothesis that the vertices correspond to distinct control points is equivalent to injectivity for every choice of positive weights.
In Section 2, we introduce toric Bézier patches as generalizations of the classical rational patches. In Section 3 we explain our condition and sketch its equivalence to the injectivity of every 2D patch with a given set of control points, for all possible weights. More details, including examples of the geometric arguments of Lemma III.5 and Corollaries III.6 and III.7 will be added in the complete version of this paper. We conclude some remarks on how to check this condition, argue that it is in fact quite natural, and interpret it in terms of piecewise linear maps.
While our main interest is in establishing a criteria valid in 3D, and in fact in all dimensions, we currently do not know how to add hypotheses to the condition of  so that the result will be equivalent to injectivity for any choice of weights in 3D.
Ii Toric Bézier patches
Let be any finite set of integer lattice points. Its convex hull is a polygon whose vertices are lattice points. This polygon is also defined by its edge inequalities,
where are linear polynomials with integer coefficients and is relatively prime.
For each integer lattice point , Krasauskas  defined the toric Bernstein polynomial
These toric Bernstein polynomials are non-negative on , and the collection of all has no common zeroes in .
Let be with coordinates indexed by elements of .
Let be a finite set. A toric Bézier patch associated with requires an assignment () of control points and a choice of weights . The toric Bézier patch is the function
written as and are understood.
The degree of a toric Bézier patch is encoded in its domain, differing from the classical patches as developed in . These two types of patches share many properties, which is explained in [8, 9]. Two properties in particular are important for us.
One is the convex hull property, that the image of under is contained in the convex hull of the control points with if is a vertex of , and the other is the boundary property, that the restriction of to an edge of is a rational Bézier curve, defined by control points and weights corresponding to lattice points of .
The boundary property may be seen directly by considering the restriction to an edge. For the convex hull property, note that as is nonnegative, is a convex combination of the control points, and if is a vertex, then is zero unless . Since the toric Bernstein polynomials are strictly positive on the interior of (and those corresponding to an edge are strictly positive on the interior of ), we may deduce a little more.
The image of the interior of lies strictly in the interior of the convex hull of the control points , and the image of the interior of an edge lies strictly within the interior of the convex hull of .
Toric Bézier patches include the classical Bézier patches and some multi-sided patches such as Warren’s polygonal surface  which is a reparameterized toric Bézier surface.
Example II.3 (Tensor product patches)
Example II.4 (Triangular Bézier patches)
Iii Injectivity of 2D toric Bézier patches
Given a finite set and a choice of control points, we consider the injectivity of toric Bézier patches as mapping functions (2), for all choices of positive weights.
Affinely independent points determine an orientation via the ordered basis of .
A choice of control points is weakly compatible if
There are affinely independent points of such that is also affinely independent, and
For any affinely independent points of with the same orientation as , if is also affinely independent, then it has the same orientation as .
Fig. 2 shows three sets of labeled points, indicating assignments between them.
The assignment between the first two sets is weakly compatible, but neither assignment to the third set is weakly compatible.
We state Theorem 3.5 of  for , which is their main result on injectivity of toric Bézier functions (it holds in any dimension). Write for the interior of .
The map is injective for all if and only if the assignment is weakly compatible.
In , the authors incorrectly stated this result as is injective on all of , even though their proof was only valid for the interior of the convex hull. Their proof showed that has no critical points in the interior, which shows that it is an open map on .
This is the best possible result with these hypotheses: Consider a bilinear patch where two control points coincide. Specifically, let and suppose that the control points are , where , except that . This assignment of control points is weakly compatible, but collapses the edge between and to the point .
This example shows that more hypotheses are needed to ensure that is injective on , and those hypotheses should imply that faces of are not collapsed. In fact, this is the only additional hypothesis needed.
A choice of control points is compatible if it is weakly compatible, and no two vertices have the same image under .
We state our main result.
The map is injective for all if and only if the assignment is compatible.
If is a vertex of , then . Theorem III.2, together with this observation, shows that if is injective for all , then is compatible.
For the other implication, suppose that is compatible. We show that the assumption that is not injective leads to a contradiction.
We first make several observations about the relative positions of the points for which are implied by compatibility. Composing with a reflection of if necessary, we may assume that if and are both affinely independent, then they induce the same orientation on .
Let be an edge of . There is some triple of points of with affinely independent where and . Indeed, if there are no such triples, then every point of lies on every line segment between two distinct points of , which implies that the points of are collinear and the line they span contains , which contradicts the first condition for weak compatibility of Definition III.1. This argument requires that there be at least two distinct points of , which follows as the endpoints of (which are vertices of ) are mapped to different points under .
Suppose that we list the points of so that if , and , then are positively oriented. Then either are collinear or positively oriented. Since there must be at least one such triple with affinely independent, we deduce the following.
Every control point lies in the intersection of closed halfspaces
for with , and this intersection has a nonempty relative interior.
For every edge of and every , the control point does not lie in the relative interior of the convex hull of .
To see this, note that the intersection of halfspaces of Lemma III.5 is either interior or exterior to the convex hull of , and if it is exterior, then it is separated from the relative interior of the convex hull by a line. If there is an edge so that this intersection lies in the interior of the convex hull of , let be a different edge. Then the positions of the points of relative to the intersection of halfspaces for leads to a contradiction.
If is compatible, then the restriction of to any edge of is injective.
To see this, fix an edge and consider the intersection of halfspaces of Lemma III.5. This intersection is exterior to the convex hull of and so consists of an unbounded polyhedron, . Consider the orthogonal projection along an unbounded direction of . Then the map is a weakly compatible choice of control points for , and so the map restricted to the edge is injective, by Theorem III.2. But this implies that the restriction of to is injective.
Proof III.8 (Proof of Theorem iii.4)
We suppose that is compatible and that is not injective. Let be distinct points with .
First, neither nor can be a point of . To see this, suppose that and let be a neighborhood of in whose closure does not contain . Then is an open set containing , so contains an open subset of in . But then points of are mapped by to points of , and so is not injective on the interior of , which contradicts Theorem III.2, as the choice of control points is weakly compatible.
Thus and are points of some edges of . They cannot be points of the same edge , for then the restriction of to is not injective, contradicting Corollary III.7. Thus they are points of different edges, and with . We cannot have one of them be an interior point of its edge, for then the relative interiors of the convex hulls of and meet, contradicting Corollary III.6.
The only possibility left is that and are vertices of , but then and , which are different, as the choice was compatible.
By definition, to check weak compatibility for 2D patches, it suffices to check determinants for each triple of points of and the corresponding control points, giving a simple algorithm. The complexity may be reduced if we start from a triangulation of , or with careful bookkeeping. Such triangulations can be obtained from control nets for tensor product patches or Bézier triangles. We will treat the complexity of checking weak compatibility in the complete version of this extended abstract.
Mapping functions that are weakly compatible exist; for example the identity assignment of control points is weakly compatible. A designer may choose weakly compatible control points for aesthetic or other reasons. For example, if only a few control points are moved such as in image warping, morphing, or reparameterization, then the control points may be weakly compatible by design, or else only a few determinants need to be computed.
For any triangulation of , the assignment of control points induces a piecewise linear map to the image. This piecewise linear map is injective (except possibly collapsing an interior simplex) for every such triangulation if and only if the assignment of control points is weakly compatible.
In this paper, we study the injectivity of toric Bézier patch geometrically. We present a simple condition on a set of control points which implies that the resulting 2D toric Bézier patch is injective, for any choice of positive weights. For higher dimension, the best result remains Theorem III.2 by Craciun et al. in  (Theorem 3.5 in ). We plan to continue this investigation of injectivity for 3D and higher dimensions in a future publication.
The authors thanks to Tim Goodman and Keith Unsworth for providing their paper . Research of Sottile is supported in part by NSF grant DMS-1001615, and the Institut Mittag-Leffler, Djursholm, Sweden. Research of Zhu is supported by the NSFC (Grant Nos. 10801024, 11071031, and U0935004), the Fundamental Research Funds for the Central Universities (DUT10ZD112, DUT11LK34), and the National Engineering Research Center of Digital Life, Guangzhou 510006, China.
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