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Advances in Applied Clifford Algebras

, Volume 24, Issue 1, pp 71–88 | Cite as

Boosted Surfaces: Synthesis of Meshes using Point Pair Generators as Curvature Operators in the 3D Conformal Model

  • Pablo Colapinto
Article

Abstract

This paper introduces a new technique for the formulation of parametric surfaces. Applying translation operations to tangent vectors \({n_{\circ} {\bf \upsilon}}\) results in null point pairs \({\tau}\). We treat these null point pairs as surface and mesh curvature control points which can be interpolated and exponentiated to construct continuous topological transformations \({\mathcal{K}}\) of the form \({e^{{-}\frac{\tau}{2}}}\) 2. Some basic algorithms are proposed, including the boost which bends a line to a circle of curvature \({\kappa}\), and the twisted boost which generates the Hopf fibration. We investigate methods to control curvature in two orthogonal directions u and v and examine a few distance-based and linear weighting techniques for synthesizing surface patches using multiple curvature control points. We consider the expressivity of the technique in manipulating meshes, and find that applying these rotors to mesh points provides a novel and computationally efficient method for creating boosted forms.

Keywords

Conformal geometric algebra computer graphics parametric surfaces point pair special conformal transformation surface topology Hopf fibration 

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Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.University of California at Santa Barbara, Media Arts and Technology ProgramSanta BarbaraUSA

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