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Spherical orbifold tutte embeddings

Published: 20 July 2017 Publication History

Abstract

This work presents an algorithm for injectively parameterizing surfaces into spherical target domains called spherical orbifolds. Spherical orbifolds are cone surfaces that are generated from symmetry groups of the sphere. The surface is mapped the spherical orbifold via an extension of Tutte's embedding. This embedding is proven to be bijective under mild additional assumptions, which hold in all experiments performed.
This work also completes the adaptation of Tutte's embedding to orbifolds of the three classic geometries - Euclidean, hyperbolic and spherical - where the first two were recently addressed.
The spherical orbifold embeddings approximate conformal maps and require relatively low computational times. The constant positive curvature of the spherical orbifolds, along with the flexibility of their cone angles, enables producing embeddings with lower isometric distortion compared to their Euclidean counterparts, a fact that makes spherical orbifolds a natural candidate for surface parameterization.

Supplementary Material

MP4 File (papers-0159.mp4)

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    cover image ACM Transactions on Graphics
    ACM Transactions on Graphics  Volume 36, Issue 4
    August 2017
    2155 pages
    ISSN:0730-0301
    EISSN:1557-7368
    DOI:10.1145/3072959
    Issue’s Table of Contents
    Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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    Publication History

    Published: 20 July 2017
    Published in TOG Volume 36, Issue 4

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    Author Tags

    1. orbifolds
    2. spherical parameterization
    3. tutte's embedding

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    • Israel PBC and ISF
    • Simons Foundation
    • European Research Council (ERC)

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    • (2024)Interactive Design and Optics-Based Visualization of Arbitrary Non-Euclidean Kaleidoscopic OrbifoldsIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2023.332692730:1(1292-1301)Online publication date: 1-Jan-2024
    • (2024)Fast ellipsoidal conformal and quasi-conformal parameterization of genus-0 closed surfacesJournal of Computational and Applied Mathematics10.1016/j.cam.2024.115888447(115888)Online publication date: Sep-2024
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