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Geodesic Tracks: Computing Discrete Geodesics With Track-Based Steiner Point Propagation.

Wenlong Meng, Shiqing Xin, Changhe Tu

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    Summary
    This summary is machine-generated.

    This study introduces a novel method for calculating geodesic distances on triangle meshes using evenly-spaced Steiner points. This approach offers a fast and accurate solution, outperforming conventional methods for 3D graphics models.

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    Area of Science:

    • Computer Graphics
    • Computational Geometry
    • Geometric Modeling

    Background:

    • Computing geodesic distances on triangle meshes is crucial for various applications in computer graphics and geometry processing.
    • Existing methods like window propagation have limitations in accuracy and speed.
    • The need for efficient and robust geodesic distance computation on complex meshes persists.

    Purpose of the Study:

    • To present a simple, effective, and computationally efficient method for calculating geodesic distances on triangle meshes.
    • To introduce a new data structure, geodesic tracks, for encoding discrete geodesic information.
    • To demonstrate the method's applicability to challenging mesh configurations and its extensibility.

    Main Methods:

    • Placing evenly-spaced, source-independent Steiner points on mesh edges.
    • Constructing a Steiner-point graph to partition the surface into geodesic tracks.
    • Utilizing filtering rules to optimize broadcast events and computational efficiency.
    • Developing geodesic tracks as a data structure for distance queries and path tracing.

    Main Results:

    • Achieved a mean relative error of less than 0.3% with 5 Steiner points per edge on standard 3D models.
    • Demonstrated a 10x speedup compared to conventional Steiner point methods for large models (1000K faces).
    • Showed robustness on meshes with poor triangulation and non-manifold configurations.
    • Confirmed that increased Steiner points enhance accuracy with minimal computational overhead.

    Conclusions:

    • The proposed method provides an effective balance between speed and accuracy for geodesic distance computation on triangle meshes.
    • Geodesic tracks offer a versatile data structure supporting accurate path tracing and efficient distance queries.
    • The method is robust, efficient, and extensible to various mesh types and metric properties.