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eight.u_0.m
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eight.u_1.m
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eight.u_2.m
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eight.u_3.m
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eight.uv_0.m
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eight.uv_1.m
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eight.uv_2.m
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eight.uv_3.m
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Algorithm Description
Holomorphic 1-form computes the holomorphic 1-form basis on a high genus closed surface. According to Hodge theory, each
cohomologous class has a unique harmonic form, and corresponds to a unique holomorphic 1-form. The algorithm is linear.
Input and output
The inputs are the basis of harmonic 1-forms on the mesh, which are the output of harmonic 1-form algorithm.
- Harmonic 1-form basis, e.g. eight.w_0.m eight.w_1.m eight.w_2.m eight.w_3.m.
The output are the basis of holomorphic 1-forms,
- Holomorphic 1-form basis recorded as edge traits eight.w_0.m eight.w_1.m eight.w_2.m eight.w_3.m.
Convergence, Stability and Efficiency
The computational process is stable and converges fast.
Example
The input genus two mesh has 7k faces. The computational time is about 2 seconds on a PC with 3.0 GB of RAM, 3.60 GHz CPU.
The data set and the source can be downloaded.
Command
- Compute the homology group basis:
homology.exe eight.m eight.open
This generates eight.open_0.m eight.open_1.m eight.open_2.m eight.open_3.m
- Compute the harmonic 1-forms:
harmonicform.exe eight.m eight.open_0.m eight_w_0.m eight.u_0.m
harmonicform.exe eight.m eight.open_0.m eight_w_1.m eight.u_1.m
harmonicform.exe eight.m eight.open_0.m eight_w_2.m eight.u_2.m
harmonicform.exe eight.m eight.open_0.m eight_w_3.m eight.u_3.m
- Compute the holomorphic 1-forms:
holomorphicform.exe eight.w_0.m eight_w_1.m eight.w_2.m eight.w_3.m
The outputs are
eight.duv_0.m eight.duv_1.m eight.duv_2.m eight.duv_3.m
- Compute a fundamental domain
cutgraph.exe eight.m eight.cut.m
slice.exe eight.cut.m eight.open.m
- Integarte holomorphic 1-form on the fundamental domain
integrate.exe eight.duv_0.m eight.open.m eight.uv_0.m
integrate.exe eight.duv_1.m eight.open.m eight.uv_1.m
integrate.exe eight.duv_2.m eight.open.m eight.uv_2.m
integrate.exe eight.duv_3.m eight.open.m eight.uv_3.m
