M.Sc Thesis | |

M.Sc Student | Segall Aviv |
---|---|

Subject | 2D Simulation and Mapping using the Cauchy-Green Complex Barycentric Coordinates |

Department | Department of Computer Science |

Supervisor | ASSOCIATE PROF. Mirela Ben-Chen |

Full Thesis text |

Conformal
maps are especially useful in geometry processing for computing shape
preserving deformations, image warping and manipulating harmonic functions. The
Cauchy-Green coordinates are complex-valued barycentric coordinates, which can
be used to parameterize a space of conformal maps from a planar domain bounded
by a simple polygon. In this work, we use the Cauchy-Green coordinates to
simulate 2D potential flow with interactive control, and to construct conformal
maps between planar domains.

The Hele-Shaw flow describes the slow flow of a
viscous liquid between two parallel plates separated by a small gap. In some
configurations such a flow generates instabilities known as Saffman-Taylor
fingers, yielding intricate visual patterns which have been an inspiration for
artists, yet are quite difficult to simulate efficiently. Formulating the equations
with our framework allows us to efficiently simulate the flow and to provide
the user with interactive control over the behavior of the fingers.
Additionally, we show that the Cauchy-Green coordinates are applicable to the
exterior of the domain, and use them for simulating two fluids with different
viscosities.

The Riemann mapping theorem guarantees that
there exists a conformal map between any two simply connected planar domains,
yet computing this map efficiently is challenging. One of the main challenges
is finding the boundary correspondence between the two domains. We use the
Cauchy-Green coordinates for parameterizing the space of conformal maps from
the source domain, and propose an alternating minimization algorithm for
constructing a boundary-approximating conformal map, which implicitly finds a
boundary correspondence. We enrich the space of solutions by generalizing the
setup to quasi-conformal maps, and allow the user to interactively control the
result using point-to-point and stroke-to-stroke constraints. Finally, we show
applications to stroke based deformation and constrained texture mapping.