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Clean up your mesh! Part 1: plane and simplex
Steven De Keninck1, Martin Roelfs2, Leo Dorst1
1Universiteit van Amsterdam , Amsterdam, The Netherlands.
Summary
Plane-based geometric algebra (PGA) offers a unified framework for discrete geometry, simplifying calculations for mesh representations and geometric properties. This approach enhances efficiency and expressiveness in geometric algebra applications.
Area of Science:
- Geometric Algebra
- Discrete Geometry
- Computational Geometry
Background:
- Traditional mesh representations can be complex and computationally intensive.
- Existing methods for calculating geometric properties often lack a unified approach.
Purpose of the Study:
- To explore the efficiency and expressiveness of plane-based geometric algebra (PGA) for discrete geometry.
- To develop unified, coordinate-free formulas for geometric properties using PGA.
Main Methods:
- Representing k-simplices and k-complexes using joins and sums within PGA.
- Deriving k-magnitudes from PGA's Euclidean and ideal norms.
- Extending PGA to derive unified formulas for volume, center of mass, and moments of inertia.
Main Results:
- Compact representation of k-simplices and k-complexes in PGA.
- A single formula for k-magnitudes derived from PGA norms.
- Unified, coordinate-free formulas for classical geometric results.
Conclusions:
- PGA provides an efficient and expressive framework for discrete geometry.
- The developed formulas offer a unified and simplified approach to calculating geometric properties.
- PGA has practical applications in real-world geometric problems.
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