Computational Conformal Geometry Library - CCGL
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Introduction

Getting CCGL
    Download
    Installation

Development Guide
    Architecture
    Performance

Gallery
    Snapshots
    Movies
Mesh Data Structure
    Viewer
Topology
    Cut Graph
    Slicer
    Double Cover
    Homology Basis
Conformal Maps
    Harmonic Map
    Spherical Harmonic Map
    LSCM
Holomorphic 1-Form
    Integration
    Harmonic 1-Form
    Holomorphic 1-Form
    Slit Map
Curvature Flow
    Euclidean Ricci Flow
    Poly Annulus Ricci Flow
    Hyperbolic Ricci Flow
    Yamabe Flow
Documentation
    Reference Book
Acknowledgements
Holomorphic 1-Form

heads
heads
heads
heads
eight.u_0.m
eight.u_1.m
eight.u_2.m
eight.u_3.m
heads
heads
heads
heads
eight.uv_0.m
eight.uv_1.m
eight.uv_2.m
eight.uv_3.m

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

  1. 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
  2. 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
  3. 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
  4. Compute a fundamental domain

    cutgraph.exe eight.m eight.cut.m
    slice.exe eight.cut.m eight.open.m
  5. 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