Specifying the Simplices

In previous sections we learned aboutsimplices- the elements of a Delaunay complex as well as their dual Voronoi cells. We also learned than the intersections of Delaunay simplices and its duals are the critical points on distance transform defined on the sampling points. So, using the combinatorial structures we made implications about local differential properties of a continuous function. The critical points are the combining link between a continuous function and the combinatorial structure defined on the points.

Here we start with the continuous function and proceed towards the concept of critical points to make specifications about intersections of Delaunay simplices and their duals and, following, about the properties of certain simplices.

A gradient of a function defines a first-order differential equation. A solution is a curve whose tangent vectors agree with the gradient of the function. For each non-critical pointxthere is a unique solution, the flow curve that contains x. In our context the flow curve is piecewise linear which justifies the notationflow line.

Every flow line starts in a critical point and ends in a critical point or in the infinite. We adopt the notation from [Edelsbrunner, 2003] of the infinite as a critical point with infinitely large distance value.

We call it theinfinite maximum.

The distance function on a point set S measures the distance to the nearest point. The flow line follows the steepest increasing path for every point in space. How do we introduce some intuition towards the trajectory of a flow line? The distance values increase linearly with the distance to the nearest point.

So, the nearest point or nearest points push the point is space away. The direction of the gradient is then the vector starting on the nearest point and ending in the point in space - our measuring point. If more than one point is equidistant to our measuring point then the sum of vectors is the direction of the gradient.

Let us consider a two-dimensional example as illustrated in Figure 2.22. Let us consider further a closed ball with steadily increasing radius. If the closed ball centered on a pointx4(compare Figure 2.22 (a) ) in space touches a sample pointAbut does not contain any other points ofS={A, B, C}then the sample pointA pushes x4 at most and the flow is directed from the sample point throughx4 (dashed arrow).

If the closed ball centered on x4 contains two sample points (A and C) there is to decide which sample point is nearer. The nearest pushes the most until the distance to both sample points is equal (the end of the arrow throughx4in Figure 2.22 (a) ). The sum of the pushing vectors results in the new direction of the flow line (the short arrow starting on the end of the arrow thoughx4and ending inO).

If the pushing vectors are of the same length (the distance to the nearest is equal as for example in the center ofAB) and are in opposite directions then the sum is zero and the flow line stays in the

A

B

C O

x1

x2

x3

x4

(a)

A

B C

O x1

x2

x3

x4

(b)

Fig. 2.22: Flow lines on distance transform defined on pointsABCwhich build (a) an acute (b) an obtuse triangle.

meeting point. This meeting point is a critical point of the distance function. This is the case for all flow lines starting on edges opposite to an acute angle. Otherwise the flow then makes a kink (compare the flow line throughx4) and follows the line perpendicular to the line connecting the sample points -the dual Voronoi edge. This is -the case for all measuring points in space not on -the edge connecting -the points.

Increasing the radius of the ball centered onx4until it contains a third sample point, and connecting the three sample points with edges, we construct a Delaunay triangle (ABC). The flow line from our previous construction with two sample points in the ball only follows the line perpendicular to the edge connecting the sample points until the circumcenter of the triangle is reached. If the triangle is acute (Figure 2.22 (a)) then the circumcenter is contained in the triangle’s interior and the flow cannot leave. The circumcenter is equally pushed by points and in the case of an acute triangle the sum of pushing vectors is zero. In the case of an obtuse triangle (Figure 2.22 (b)) the flow does not end in the circumcenter. The sum of pushing vectors results in a vector in the direction perpendicular to the edge (AC) opposite to the obtuse angle away from the sample points.

In Figure 2.22 (b) x3 is a point on the edge AC. But there is a sphere centered onx3 containing further sample pointB. This sample pointB pushesx3the most and so the flow line onx4breaks adrift towards the circumcenter of the triangle and so the point which is equidistant to AC. The flow does not end in the center but the distance values increase along the dual Voronoi line toAC away from the sample points.

The simplices which contain the endpoint of the flow line are calledcentered and the simplices which do not contain its flow are equivocal. In 2D equivocal edges are opposite obtuse angles. The opposite sample point is too close to the sample points such that it pushes the flow away from the edge. This is the case if the circumsphere of the edge contains another sample point.

Let us extend our example to 3D. The radius of the ball centered onxis increased until it contains the fourth sample point. The convex hull of the four sample points is a Delaunay tetrahedron. If the circumsphere of three sample points of the tetrahedron does not contain any further sample point, or -as we call it now - the vertex of the tetrahedron, then there are two c-ases to consider. The triangle built of these three sample points is acute - then it again is calledcentered. If the triangle is obtuse, then the flow is contained in the affine hull of the triangle but not the triangle itself. In such a case the triangle is calledconfident.

If the circumsphere of the triangle does contain the further vertex of the tetrahedron then we again have the case where the vertex pushes the flow away from the simplex. Such a triangle isequivocal.

In [Edelsbrunner, 2003] the specifications are summed up under the following properties:

Fact 2.51(Centered Simplex [Edelsbrunner, 2003]). A Delaunay simplexσwith its dual Voronoi cell ν is centered if and only if the intersection ofσandν is not empty. The intersection is a critical point of f and its index is the dimension ofσ.

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In our 2D example in Figure 2.22 (a) the triangle and its edges are centered. The triangle contains its own circumcenter which is its dual Voronoi 0-cell. The half-lines starting in the circumcenter and going through the centers of the edges are the dual Voronoi 1-cells of the edges. Since the intersection is not empty, the edges are centered.

Fact 2.52(Equivocal Simplex [Edelsbrunner, 2003]). A Delaunay simplex σwith its dual Voronoi cell ν is equivocal if and only if the intersection ofν and the affine hull ofσis empty.

The 2D example in Figure 2.22 (b) illustrates the triangle with one equivocal edge AC. Its dual starts in the circumcenter O of the triangle and extends perpendicularly away from the center of the edgeAB. The intersection is empty even if we expand the edge to a line.

The flow starting inside a Delaunay simplex necessarily crosses the circumcenter of its greatest dimensional coface. The flow starting inside a tetrahedron of a 3D Delaunay complex ends or crosses the circumcenter of the tetrahedron. The affine expansion of a triangle divides the space into the half-space containing the tetrahedron and the half-space not containing any point of the tetrahedron. Equivalent consideration is valid in 2D. Since the flow is pushed through the equivocal triangle and necessarily intersects the circumcenter of the tetrahedron we conclude that the circumcenter is on the other side of the plane containing the triangle:

Corollary 2.53(Equivocal Separates the Simplex from its Circumcenter). Let σbe an−1-dimenional equivocal proper face of τ in n-D Delaunay complex. Then the affine expansion of σ divides the n-dimensional space into a half-space containing the τ and a half-space containing the circumcenter of τ.

We use the result of Corollary 2.53 as intuition in the following 3D examples. For example in Figure 2.23 (a,b) the triangleABDis equivocal.

Fact 2.54 (Confident Simplex [Edelsbrunner, 2003]). A Delaunay simplexσ with its dual Voronoi cell ν is confident if and only if the intersection of σandν is empty and the intersection ofν and the affine hull ofσis not empty.

The greatest-dimensional Delaunay simplex is never equivocal since its affine expansion covers the whole space and, so, its dual Voronoi. The triangle in Figure 2.22 (b) is not centered since it does not contain its own circumcenter: consequently, the triangle is confident. The lower-dimensional 2D Delaunay simplices cannot be confident. This property occurs first in higher dimensions.

In the following we are going to investigate the possible constellations and relations of different Delaunay simplices. In Figure 2.23 (a) for example we observe a lower-dimensional 2-simplex ( the triangleABCin 3D Delaunay triangulation ) whose affine expansion, denoted by the gray circumcircle, contains its circumcenter, denoted by a small opaque ball. Notice, the dual of the triangle is the half-line starting in the circumcenter of the tetrahedron ( the small opaque ball in the center of the wired ball ) and going through the circumcenter of the triangle. The dual Voronoi half-line does not intersect the triangle: consequently, the triangle is confident.

The dual Voronoi half-lines are included into Figure 2.23 (b). The bottom triangle is acute but neither the triangle nor its affine expansion intersects its dual. We conclude that the bottom triangle is equivocal.

Observation 2.55 (In a Cell with One Equivocal All Triangles can be Acute). Let τ be a Delaunay triangle and σbe its only proper equivocal face. All proper faces of τ may be acute triangles.

Not-Gabriel Simplices

In the context of Delaunay triangulation the concept of Gabriel graph originally introduced in [Gabriel and Sokal, 1969] plays an important role and is the basis for our further investigations.

Definition 2.56. Let D be a nD Delaunay complex and σ∈D be am < n dimensional simplex. Let σ be the convex hull of {p1, . . . , pm} ⊂ D points. σ is called Gabriel or is said to have the Gabriel -property if its circumball does not contain any further points D but {p1, . . . , pm} otherwise σ is called not-Gabriel

(a) (b)

Fig. 2.23: (a)ABDis the only proper equivocal face of the cell. ABC is confident. BC is equivocal but not a face of ABD. (b) EquivocalABDis the only face of tetrahedronABCS. All

proper faces ofABCDare acute triangles.

The edge AC in Figure 2.22 (b) does contain the point B. We observe that the pointB is the one point which pushes the flow on the points on the edge away from the triangle. In fact, we may observe that if a circumball of a simplex contains a further point, this point will be nearer to the circumcenter than the vertices of the simplex and consequently will push the flow away from the circumcenter before it can reach it. It wollows that the simplex can neither be centered nor confident. In Theorem 2.57 we prove that the not-Gabriel property is given if and only if the simplex is equivocal.

Theorem 2.57(Equivocal is equivalent to Not-Gabriel). Let τ be a Delaunay simplex. σ is its proper equivocal face if and only if the circumsphere of σcontains a further vertex ofτ (not-Gabriel).

Proof: Iτ is a 3-simplex (tetrahedron) andσis a 2-simplex (triangle).

If equivocal then not-Gabriel: The dual Voronoiν toσis an edge between the circumcenter ofτ and the circumcenter of the neighboring proper coface ofσ. The affine hull ofσdivides the space into a half-space containingτ and a half-space not containing any point ofτ. Ifν does not intersect the affine hull of σ, then the circumcenter ofτ is separated by the affine hull ofσ from τ. It follows τ does not contain its own circumcenter.

The intersection of the affine hull of σ and the circumsphere ofτ is the circumcircle of σ. Let the distance between the circumcentercσ ofσand the circumcentercτ ofτ bepand the circumradius ofτ be r. Letαbe the angle betweencσ,cτ and one of the vertices ofσ. Since the edge betweencσ, cτ is orthogonal to the circumcircle ofσ,α=^p_r.

Let the vertex opposite to σbevand let the angle betweenv,cτ andcσ be α⁰. Since the affine hull ofσseparatesτ fromcτ: 0≤α⁰< α. Then the squared distance betweenv andcσisr²−2rpcosα+p² which is maximized byα⁰→α, and since limα⁰→α(r²−2rpcosα+p²) =r²−p²which is the circumradius ofσ,σcontainsτ.

If not-Gabriel then equivocal: The circumsphere ofσcontains the vertex v opposite toσ. The distancedbetween the circumcentercσ ofσandv is less than the circumradiusrσ ofσ. Sincerσ≤rτ, it followsd < rτ. Since the convex hull ofσseparates the space into a half-space completely containing τ and the half-space not containing any point in τ and d < rτ, it follows that cτ is not in the same half-space asτ and the dual toσcannot intersect the affine hull of σ.

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