The output table gives you the dimensions of the graded pieces of the module M where the degree is between a and b.
i1 : V = {{0,0},{1,0},{1,1},{0,1}};
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i2 : F = {{0,1,2},{0,2,3}};
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i3 : E = {{0,1},{0,2},{0,3},{1,2},{2,3}};
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i4 : M=splineModule(V,F,E,2); |
i5 : splineDimensionTable(0,8,M)
+---------+-+-+-+--+--+--+--+--+--+
o5 = |Degree |0|1|2|3 |4 |5 |6 |7 |8 |
+---------+-+-+-+--+--+--+--+--+--+
|Dimension|1|3|6|11|18|27|38|51|66|
+---------+-+-+-+--+--+--+--+--+--+
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The table above records the dimensions dim$S^2_d(\Delta)$ (i.e. splines on $\Delta$ of smoothness 2 and degree at most d) for $d=$0,..,8.
You may instead input the list L={V,F,E} (or L={V,F}) of the vertices, facets and codimension one faces of the complex $\Delta$.
i6 : V = {{0,0},{1,0},{1,1},{0,1}};
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i7 : F = {{0,1,2},{0,2,3}};
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i8 : L = {V,F,E};
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i9 : splineDimensionTable(0,8,L,2)
+---------+-+-+-+--+--+--+--+--+--+
o9 = |Degree |0|1|2|3 |4 |5 |6 |7 |8 |
+---------+-+-+-+--+--+--+--+--+--+
|Dimension|1|3|6|11|18|27|38|51|66|
+---------+-+-+-+--+--+--+--+--+--+
|
The following complex, known as the Morgan-Scot partition, illustrates the subtle changes in dimension of spline spaces which may occur depending on geometry.
i10 : V = {{-1,-1},{1,-1},{0,1},{10,10},{-10,10},{0,-10}};
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i11 : V'= {{-1,-1},{1,-1},{0,1},{10,10},{-10,10},{1,-10}};
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i12 : F = {{0,1,2},{2,3,4},{0,4,5},{1,3,5},{1,2,3},{0,2,4},{0,1,5}};
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i13 : M = splineModule(V,F,1); |
i14 : M' = splineModule(V',F,1); |
i15 : splineDimensionTable(0,4,M)
+---------+-+-+-+--+--+
o15 = |Degree |0|1|2|3 |4 |
+---------+-+-+-+--+--+
|Dimension|1|3|7|16|33|
+---------+-+-+-+--+--+
|
i16 : splineDimensionTable(0,4,M')
+---------+-+-+-+--+--+
o16 = |Degree |0|1|2|3 |4 |
+---------+-+-+-+--+--+
|Dimension|1|3|6|16|33|
+---------+-+-+-+--+--+
|
Notice that the dimension of the space of $C^1$ quadratic splines changes depending on the geometry of $\Delta$.
The object splineDimensionTable is a method function.