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}}; |
i2 : F = {{0,1,2},{0,2,3}}; |
i3 : E = {{0,1},{0,2},{0,3},{1,2},{2,3}}; |
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| +---------+-+-+-+--+--+--+--+--+--+ |
The table above records the dimensions dimS2d(Δ) (i.e. splines on Δ 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 Δ.
i6 : V = {{0,0},{1,0},{1,1},{0,1}}; |
i7 : F = {{0,1,2},{0,2,3}}; |
i8 : L = {V,F,E}; |
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}}; |
i11 : V'= {{-1,-1},{1,-1},{0,1},{10,10},{-10,10},{1,-10}}; |
i12 : F = {{0,1,2},{2,3,4},{0,4,5},{1,3,5},{1,2,3},{0,2,4},{0,1,5}}; |
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 C1 quadratic splines changes depending on the geometry of Δ.