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Splossoms: blossoming polynomials on the sphere via geometric algebra
1Boulder Graphics LLC , USA.
Summary
We introduce splossoms, a new method for designing spherical curves, extending polynomial blossoming theory to spherical geometry. This approach, grounded in geometric algebra, offers powerful tools for applications in robotics and computer graphics.
Area of Science:
- Computer Graphics
- Geometric Algebra
- Differential Geometry
Background:
- Classical polynomial blossoms offer a powerful framework for curve design.
- Existing methods for spherical curve interpolation, like SLERP, lack the structured approach of blossoms.
- Geometric Algebra (GA) provides a unified framework for geometric computations.
Purpose of the Study:
- To introduce splossoms, the spherical analogue of polynomial blossoms.
- To extend Ramshaw's blossoming theory into spherical geometry.
- To define and analyze spolynomials for spherical curve construction.
Main Methods:
- Spherical reinterpretation of classical blossom axioms.
- Development of splossom properties and their analogues.
- Definition of iterated spherical interpolants (spolynomials).
- Expression of splossoms within the Geometric Algebra (GA) framework.
Main Results:
- Splossoms satisfy specific spherical analogues of blossom axioms.
- Splossoms generalize Shoemake's SLERP interpolation for spherical curves.
- Spolynomials mirror polynomial structures (e.g., Bézier, B-spline) and their continuity is studied.
- Geometric algebra provides a computationally efficient setting for splossom implementation.
Conclusions:
- Splossoms offer a structured and powerful method for designing spherical curves.
- The GA framework simplifies and enhances spherical curve construction.
- Potential applications span robotics, CNC milling, VR/AR, and animation.
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