Simplicial meshes are now standard methods to approximate geometric objects. These meshes are used in algorithms for several tasks, including numerically solving partial differential equations, finite element approximation of functions and computational dynamical systems. The quality of approximation of these algorithms depends on thegoodness of the mesh. The notion of goodness of a triangulation is defined using some geometric properties of the simplices involved. We discuss three measures to capture goodness: the first is the thickness of a simplex, which is the ratio of the minimal height to the maximal edge length of the simplex. The thickness of the triangulation is then the smallest thickness of any of its simplices. The second measure is the aspect ratio of a simplex, which is the ratio of the minimal height to the diameter of its circumsphere. Again, the aspect ratio of a triangulation is the smallest aspect ratio of its simplices. For an introduction on the connections between simplex quality in a mesh and finite element methods, we refer the reader to the survey of Shewchuk [She02].
Delaunay triangulations are one of the most popular simplicial meshes. Delaunay triangulations are unique for a given (non-degenerate) point set and they satisfy many useful structural properties. Over the last few decades fast algorithms have been developed to construct Delaunay triangulations and to update them under insertions and deletions to the point set. They have been generalized to a larger class of weighted Delaunay triangulations [Aur87]. Weighted point sets and point set perturbation techniques have been used to get good Delaunay meshes in and both in theory and practice. For an introduction on these topics, we refer the reader to the recent book on Delaunay mesh generation by Cheng, Dey, and Shewchuk [CDS13].
In the context of Delaunay triangulations, recently Boissonnat, Dyer and Ghosh [BDG13] have introduced a new quality measure called protection: intuitively, this measures how far is a Delaunay triangulation from being degenerate. More specifically, consider the circumball of a -simplex in the Delaunay triangulation of a point set in general position in . Since the point set is in general position, there are precisely points incident to this ball, and none in the interior. Protection is then defined as the maximal amount by which each circumball of a -simplex in the Delaunay triangulation can be inflated, so that it does not contain any other point of the lattice in its interior. In actual terms, protection is not a new measure but just a parameterization of the general position condition for point sets in Euclidean space. When it is clear from the context, we refer to the protection of a point set as the protection of its Delaunay triangulation. For any degenerate lattice, like the regular grid in , the protection value is . On the other hand, the permutahedral lattice, which is one of the very few lattices in general position, is known to have a high value of protection [CKW17]. Boissonnat et al. [BDG13] showed that good protection implies stability of Delaunay triangulations with respect to metric distortions and perturbations of the point set. They also proved that a good value of protection guarantees good-quality simplices [BDG13] in the Delaunay triangulation. This measure has also successfully been used to study Delaunay triangulations on manifolds, discrete Riemannian Voronoi diagrams, manifold reconstructions from point sample and anisotropic meshing [BDG17, BDGO17, BSTY15, BRW17, BWY15].
Algorithmic techniques (like weighted point sets, perturbation of point sets, and refinement method) for getting good Delaunay triangulations work well both in theory and practice in and [CDE00, ELM00, Li03]. But these techniques do not scale well to higher dimensions. There are perturbation algorithms (see [BDG14]) for points in , that give quality measures such as thickness and aspect ratio of the order , which is exponentially small in . The same thing is true for protection in , see [BDG14, BDG15]. This leads one to search for more structured points set in Euclidean space whose Delaunay triangulations would have better quality guarantees. A natural class of candidates are lattices in .
In this paper we concern ourselves with a family of lattices, which we obtain by a distortion of the integer grid in along the principal diagonal direction . We call this family of lattices as the diagonally distorted lattices. Essentially, stretching or squeezing the grid linearly along this direction gives this family of lattices. This family was first studied in [EK12] by Edelsbrunner and Kerber, with an ulterior motive to do topological analysis of high-dimensional image data. Later, this lattice was used to study covering and packing problems of Euclidean balls in different contexts [EIH18, EK11, IHKU14].
All simplices in the Delaunay triangulation of a diagonally distorted lattice are congruent. Naturally, the thickness, aspect ratio and protection of each simplex is the same and defines the parameters for the lattice. Recently, the qualities of a class of triangulations known as the Coxeter triangulations was studied in [CKW17]. This class of triangulations includes the Delaunay triangulation of the permutahedral lattice.
Outline of the paper
In Section 2 we discuss the preliminaries, including protection, the permutahedral lattice and the family of diagonally distorted lattices, and we summarize the main results of this paper. Section 3 expands on the details of our results, where we study the protection and other quality measures of the diagonally distorted lattices.
2 Background and Contributions
2.1 General notations
In this paper we work with the standard -norm in , and the distance between any two points in will be denoted by . For any point and any set , we denote the distance between and as . Given any point and a radius , the ball is open, and the ball is closed.
For , the convex hull and affine hull of will be denoted by and , respectively. For a set , we denote by the affine dimension of .
A simplex denotes the set of points . The combinatorial dimension of is , and geometric dimension of is .
A simplex is called a sub-simplex or a face (and proper face) of a simplex if (if ). For any vertex in , denotes the sub-simplex with vertex set , and denotes the distance .
We denote the circumradius and longest edge length of by and , respectively. The quality measure thickness of a simplex with combinatorial dimension is defined as
and the aspect ratio is defined as
A lattice is a countable subset of of the form , where
are linearly independent vectors in. The vectors are called representative vectors of . We will interchangeably call points in as vectors to simplify notation. Determinant of , by abuse of notation, is the absolute value of the determinant of the matrix whose columns are the vectors . Also, will denote the
-th smallest singular value of the matrix with columns. Observe that .
For a given lattice , let denote the length of smallest vector in . The following result is a direct application of Minkowski’s theorem [CSB87].
For any lattice , we have .
2.2 Voronoi diagram, Delaunay complexes and protection
Let be a subset of . For any point , Voronoi cell of is defined as the region
and for a simplex , the Voronoi cell of is defined as . The Voronoi diagram of , denoted by , is the decomposition of into Voronoi cells of simplices with vertices from . The Delaunay complex of , , is the nerve of , that is, iff . For a point , the star of , denoted by , is the set of simplices in such that .
Observe that for a lattice , (and ) can be obtained by the periodic copies of the Voronoi cell (and star) of the origin (and ). For the rest of the section, we denote by the star of the origin , when is clear from the context.
First we formally state the notion of protection as defined in [BDG13]. Consider a finite point set in .
Definition 2 (-protection of a simplex).
A simplex is -protected if such that, for all and , we have .
Definition 3 (-protection of a triangulation).
A triangulation of is said to be -protected for a non-negative real , if
For a lattice in , observe that if all the -simplices in are -protected then is -protected.
Adapting a result of Delaunay from [Del34] one can show that if all the -simplices in are -protected for some , then is a triangulation of . A lattice is degenerate if is -protected. Observe that if a lattice is degenerate then contains simplices with combinatorial dimension greater than .
Let be a lattice in such that is -protected. Then
2.3 Permutahedral Lattice
Before we explore the Permutahedral lattice, we look at a closely related lattice, the lattice: this is a -dimensional lattice consisting of the set of points which satisfy
. This lattice resides in the hyperplane. Let us call this hyperplane . One can observe that . For more details see [BA09].
The lattice, also known as the Permutahedral lattice [CSB87] is the dual lattice to . This means that it consists of points such that . Note that , both lie in , and contain the origin. The vertices of the Voronoi cell of the origin consist of all permutations of coordinates of the point
For this reason, this polytope is also called the permutahedron and lends the name permutahedral lattice to . The permutahedron has precisely vertices, each with the same norm. The representative vectors of lattice are of the form
for . This means that any point of can be expressed in the form , where for all .
It was shown in [BA09, CKR17] that is in general position, and that consists of congruent -simplices. This means that it offers non-zero protection for its simplices. Elementary calculations show that the Delaunay radius is
Let denote the quantity
Recently in [CKW17], it was shown that
Protection for is .
The normalized protection of the lattice is the ratio of the protection to the Delaunay radius, that is, .
The power protection333For more details on power protection refer to [BDGO17]. of is .
2.4 Diagonal distortion and Freudenthal triangulation
For a point set , a diagonal distortion is a perturbation along the diagonal direction . Formally, the diagonal distortion of a vector denoted by is defined as:
where , and is the distortion parameter. This distortion was introduced by Edelsbrunner and Kerber in [EK12], to build and study a family of lattices.
Here denotes the hyperplane passing through the origin, which is normal to the vector , that is, it is simply the hyperplane . is times the height of from . Also for
, the linear transformationis the identity map, while for , it projects points on to . For , the transformation moves each point closer to , where the distance moved is proportional to the height of the point from , as evident from Equation (3).
In [EK12], the authors built a family of lattices by setting . Each gives a lattice, which we call a distorted grid. It is thus natural to talk about distorted cubes of the distorted grid, which are images of a cube of under the transformation .
Let denote the -cube . A monotone chain on is a sequence of a subset of its vertices such that their coordinates are in strictly increasing lexicographic order. More precisely, a sequence of vertices is a monotone chain if for each pair with , it holds that . Each monotone chain can be interpreted as a simplex, which is the convex hull of its vertices. It turns out that the collection of all simplices defined by monotone chains triangulates the cube . This triangulation is more commonly known as the Freudenthal triangulation [Fre42] of the -cube (also known as the Kuhn subdivision [Kuh60]). This contains precisely -simplices [EK12]. See Figure 1 for a three-dimensional example.
In [EK12], the authors show that for any , the Delaunay triangulation of a distorted cube of is combinatorially equivalent to the Freudenthal triangulation of the unit -cube. For , the Delaunay triangulation of the distorted grid is degenerate, but for each , it remains non-degenerate and combinatorially stays the same [EK12].
2.5 Summary of Contributions
Distorted grid and the permutahedral lattice
Our first result is an interesting relation between the distorted grids and the permutahedral lattices. Since the distorted grid resides in the hyperplane , it is a -dimensional point set. Also, we know that is a -dimensional lattice, residing in . We show that
is the lattice.
is isometric to the lattice for .
In the light of the above result, we add a complementary observation:
is isometric to the lattice for .
Protection for distorted grids
We calculate the protection values for distorted grids, when the distortion parameter lies in the range . Let denote the Delaunay radius for the parameter . Then,
The normalized protection values for the diagonally distorted lattice are
Thickness and Aspect ratio
We further calculate the thickness and aspect ratio of the distorted grid for .
Let denote the thickness, and denote the aspect ratio of the distorted grid at parameter . Then,
One can see that the protection increases monotonically in the range and decreases monotonically to 0 in the range . The maximum protection value is attained at . Similarly, the thickness and aspect ratio are also maximized for .
In Figure 2 we plot the quality measures of the distorted grid for a few dimensions.
3 Properties of Diagonally Distorted lattices
3.1 Proof of Lemma 8
We prove the claim in two steps, first by showing that , and then showing that , which implies the result.
: consider any point . Let , which means that , . Recall the definition of lattice: it consists of all points such that is an integer for each . Also, by definition. Now consider the dot product ,
which is an integer since both . This holds for all points , so is a point of . Since each point satisfies the membership criteria for , it holds that .
: for any point to be a member of , it must have a special form: there must exist a point such that is the projection of onto . Specifically, , must hold for each . We show that has this special form.
The representative vectors of lattice [CSB87, Chap4.] are of the form
for . So can be uniquely written as the linear combination , where each is an integer. Consider the point of , such that for (note that ). We show that , which proves the claim.
Expanding , we get
Simplifying, we see that which simplifies to
Similarly, it is easy to see that . Hence, is of the form
Proof of Lemma 10
The proof idea is to find a bijection between the basis vectors of the two lattices and for , such that the bijection preserves norms of the vectors and the dot products between them.
Specifically, we choose a basis for as , where is the standard basis for . For , the standard basis in is
The bijection takes to for each . It is easy to calculate that the norm of each vector is . Moreover, it can be verified that for all . As a result, there is an bijection between the two lattices, that takes the point of where , to the point of . This bijection is an isometry because of the above conditions.
3.2 Protection of distorted grids
In this sub-section we prove Theorem 11 by studying the protection properties of the family of lattices . Throughout this sub-section, we will assume that is a value in this range, if it is not stated explicitly.
Let denote the protection of . For , the Delaunay triangulation of is degenerate and hence offers 0 protection, so . We calculate for by first calculating the protection of a specific simplex in the Delaunay triangulation and then showing that all simplices have the same protection.
Let denote the standard basis of . Let denote the simplex where is the vector sum . That means, , and more generally . It is straightforward to see that the simplex is a part of the Freudenthal triangulation of the -cube . We inspect the distorted transformation of , which we denote by ; this is a Delaunay simplex of the distorted grid . We calculate the protection for to determine the value for . To do so, we first find the circumcentre and circumradius of the simplex.
The vertices of the simplex can be written as , where , . In particular,
We denote the circumcentre of as and the radius of the circumsphere by . Setting , we see that . It follows that is a Delaunay simplex of . The circumcentre of is , which is a Voronoi vertex of . Then, the radius of the circumsphere is [EK11]. On the other hand, for , the simplex is . Then the circumcentre is and . For intermediate values of , from [EK11] we have the relation that
which can be simplified as
We now calculate . Note that is equidistant from each . Since , we have that for all , . This simplifies to . So we have a set of equations,
where is a column vector and the rows of the matrix on the left hand side contain the vertices of the simplex. Solving the system of equations, it follows that
This can be written as
can be explicitly written as the barycentric coordinates of as , where and . Note that . So the simplex is well-centered, that is, the circumcenter lies in the interior of the simplex.
Candidates for protection
To calculate the protection for , we find the vertices of which after distortion realize the protection value for . Since is a -simplex, it has facets. Consider such a facet . In the Freudenthal triangulation, the facet has two -simplices as co-faces, one being . Both these simplices are formed by adding a vertex to . Let be the vertex of that when added to forms , and let be the vertex of that forms the other -simplex with . We say that is opposite to .
Each vertex of has an opposite vertex in . We call the set of such opposite vertices as . Since there are facets, so . Note that since the combinatorial structure of the triangulation does not change with a change in , . First we calculate the protection offered by the distortions of the points of , and then show that these are precisely the points of the distorted grid which define the protection for .
Given a vertex of , we denote by the opposite vertex in . Using the monotone chain property of Freudenthal triangulation, it follows that
Note that each is distinct. We see that , for , and , so that
We next define as the protection offered by the point . After calculations, we see that and for all . Also,
This agrees with the observation that for , the distorted lattice is isometric with the lattice (Corollary 9) and hence has points defining the protection (see [CKW17]). So, we can define the offered protection as
The explicit values are