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Curvilinear coordinate systems for 3D design: building cyclidic nets with conformal geometric algebra
1Media Arts and Technology, University of California Santa Barbara , Santa Barbara, California, USA.
Summary
This study presents parametric methods for creating and controlling triply orthogonal curvilinear lattices. These techniques enable 3D practitioners to generate complex geometric forms using principal contact tangents and conformal mappings.
Area of Science:
- Computational geometry
- Differential geometry
- Geometric algebra
Background:
- Curvilinear lattices are essential in various scientific and engineering fields.
- Parametric modeling offers flexibility in design and manipulation.
- Geometric algebra provides a unified framework for geometric computations.
Purpose of the Study:
- To introduce novel parametric techniques for constructing and manipulating triply orthogonal curvilinear lattices.
- To detail the constraints and provide inverse mappings for these lattice systems.
- To demonstrate the application of these methods for 3D practitioners using geometric primitives.
Main Methods:
- Utilizing principal contact tangent elements.
- Applying rationalization of spheres through conformal transformations.
- Developing a 3D geometric modeling approach.
Main Results:
- Successful construction and manipulation of triply orthogonal curvilinear lattices.
- Detailed understanding of lattice constraints and boundary inverse mapping.
- Demonstration of expressing diverse geometric forms using the developed system.
Conclusions:
- The introduced parametric techniques offer a robust framework for creating and controlling complex curvilinear lattices.
- The methods are practical for 3D practitioners, facilitating the design of intricate geometric shapes.
- This work contributes to the theme of modern applications of geometric algebra.
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