Additive geometric kernel computation algorithms in 3D

Date

2024-7

Author

Asiler, Merve

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The geometric kernel defines points inside or on the boundary of a shape, ensuring shape visibility. In this thesis, we present two different approaches to compute the kernel in 3D. First, we introduce a novel approach to approximate the geometric kernel. Our algorithm uses scattered rays to identify sample points on the kernel surface, leveraging them to identify surface vertices. Computing the convex hull of these points yields an approximate kernel representation that remains inside or on the kernel surface. The parametric structure of our solution allows for different levels of accuracy, enabling the user to tailor the approximation to their specific needs, as well as computing the kernel itself. Second, we present the KerGen algorithm (Kernel Generation) to compute the kernel. KerGen employs efficient plane-plane and line-plane intersections, alongside point classifications based on positions relative to planes. This approach allows for the incremental addition of kernel vertices and edges in a simple and systematic way. The output is a polygon mesh that represents the surface of the kernel, not an approximation. Extensive comparisons with CGAL and Polyhedron Kernel demonstrate the remarkable performance of both methods in computing the kernel much faster. Both approaches promptly and accurately identify an empty kernel for non-star-shaped configurations. In summary, these approaches may open up avenues for solving various problems, such as shape deformation, shape simplification, spherical parametrization, star decomposition, and castable shape reconstruction.

Subject Keywords

3D kernel, Star-shape, Visibility coverage, Mesh guidance

URI

https://hdl.handle.net/11511/110437

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Graduate School of Natural and Applied Sciences, Thesis