Dimensions on the Morgan-Scott Split

The Morgan-Scott Split is one of the most studied triangulations in multivariate spline theory since it exhibits a large number of varied phenomena. On this page we give the dimension of various configurations of the split, to illustrate the complexity of the split, and also to demonstrate the power of the MDS code. The dimensions can be obtained by running the code with a sufficiently high amount of memory. For example, the entries in the row r=13, d=27 were computed using 400 Mbytes of memory (and taking about 12 hours of CPU time on a Sparc 2 station).

As illustrated in the nearby figure, the Morgan split is obtained by taking a triangle with vertices V ₀, V ₁, V ₂. inscribing another triangle with vertices V ₃, V ₄, V ₅ . and adding edges to make a triangulation with seven triangles. It can also be obtained by projecting an octahedron into the plane (where the original triangle forms the image of the eight face of the octahedron).

The MDS code has several versions of the Morgan-Scott split on the list of built-in triangulations. In these illustrations we consider eight different configurations. Emphasis is on parallel pairs of edges emanating from the same vertex since they tend to cause dimensions to become non-generic. Note that for the effects to occur the edges must be exactly parallel which is possible to obtain because the coordinates of vertices are all integer.

For each configuration the following information is given:

• A description of its distinguishing features.
• The coordinates of the vertices.
• A link to a binary file that can be read by the MDS code. It will most likely show up empty on your browser, but you can download like any other file and then read it into your downloaded copy of the MDS code.
• A link to the description of the split generated by the MDS code.
• A link to a graphical image of the particular split. Pairs of parallel edges emanating from the same vertex are drawn in red.

Example 1: A generic configuration.

This configuration contains no recognized geometric artifacts. In all cases the dimension equals the generic value.

• Vertices: (80,488), (520,488), (300,113), (344,338), (254,332), (285,408).
• binary file.
• MDS code description.
• image

Example 2: One parallel pair of edges.

This configuration contains just one pair of parallel edges.

• Vertices: (80,488), (520,488), (300,113), (344,338), (300,363), (285,425).
• binary file.
• MDS code description.
• image

Example 3: Two parallel pairs of edges.

This configuration contains pairs of parallel edges. However, these do not form a singular vertex.

• Vertices: (80,488), (520,488), (300,113), (312,280), (254,332), (324,447).
• binary file.
• MDS code description.
• image

Example 4: A singular Vertex

This configuration contains two pairs of parallel edges which intersect to from a singular vertex.

• Vertices: (80,488), (520,488), (300,114), (344,338), (256,263), (355,394).
• binary file.
• MDS code description.
• image

Example 5: Three pairs of parallel edges.

This configuration contains three pairs of parallel edges. It has been used by Wang to construct a finite element.

• Vertices: (80,490), (521,490), (297,112), (330,274), (205,382), (363,436).
• binary file.
• MDS code description.
• image

Example 6: The symmetric Split.

This configuration is symmetric in the sense defined by Diener and discussed here.

• Vertices: (80,488), (520,488), (300,113), (344,338), (256,338), (300,413).
• binary file.
• MDS code description.
• image

Example 7: A partially symmetric split.

This configuration has vertices V ₄ and V ₅ symmetrically placed with respect to the line from V ₀ V ₃. Thus is exhibits part of Diener's symmetry.

• Vertices: (75,500), (525,500), (325,100), (355,340), (265,340), (303,452).
• binary file.
• MDS code description.
• image

Example 8: Diener's Example.

This strange looking configuration was given by Diener as an example of a configuration that satisfies an algebraic condition that came up in his analysis. It is reproduced as Diener gave, it with two of the edges of the inner triangle defined by the unit vectors in the plane.

• Vertices: (580,76), (20,126), (132,526), (356,176), (412,176), (356,126).
• binary file.
• MDS code description.
• image

Dimensions

The following table gives dimension of various spline spaces on the above described configurations. For each case, the true dimension is computed and compared to Schumaker's lower bound (for that particular configuration). When the dimension and the lower bound agree, their common value is listed. If they disagree, the dimension is listed, followed by a slash, and the discrepancy. Thus, for example, the dimension on the symmetric split when r=1 and d=2 is 7 and the lower bound is 6.

m and n are the dimension of the linear system that must be analyzed. They are included to illustrate the capacity of the MDS code.

For clarity, r and d are listed on a yellow background, and the matrix dimensions on a purple background. A cyan background indicates that the dimension of the spline space actually equals the dimension of the space of polynomials of degree d. Thus in that case all splines are in fact polynomials. A green background indicates that there are nontrivial splines, and the true dimension exceeds Schumaker's lower bound.

At present, except in the special case of the symmetric split and d=2r, we have no theory that explains the entries on a green background (except for configuration 8 in the case that d=2r, see Dwight Diener, Instability in the dimension of spaces of bivariate piecewise polynomials of degree 2r and smoothness order r. SIMA J. Numer. Anal., 27 (1990), pp. 543-551)

The dimension on the Morgan-split is completely understood (in the sense that we know a minimal determining set) in the case that d>3r+1. However, for the range of r considered here, discrepancies only occur for d< 2r+1. It is an open question whether discrepancies occur for larger of values of d, either on the Morgan-Scott split for values of r not considered here, or on configurations other than the Morgan-Scott split.

For r>11 the dimensions were computed twice, once modulo p = 2³¹-1=2,147,483,647, and once modulo p = 2³¹-17=2,147,483,629. The computed dimensions were identical, increasing one' confidence in the results of residual arithmetic.

The figures for r=13 were computed using 500 Mbytes of memory and took about 3 days of CPU time.

[15-Apr-1999]