Abstract
This paper exposes a very geometrical yet directly computational way of working with conformal motions in 3D. With the increased relevance of conformal structures in architectural geometry, and their traditional use in CAD, its results should be useful to designers and programmers. In brief, we exploit the fact that any 3D conformal motion is governed by two well-chosen point pairs: the motion is composed of (or decomposed into) two specific orthogonal circular motions in planes determined by those point pairs. The resulting orbit of a point is an equiangular spiral on a Dupin cyclide. These results are compactly expressed and programmed using conformal geometric algebra (CGA), and this paper can serve as an introduction to its usefulness. Although the point pairs come in different kinds (imaginary, real, tangent vector, direction vector, axis vector and ‘flat point’), causing the great variety of conformal motions, all are unified both algebraically and computationally as 2-blades in CGA, automatically producing properly parametrized simple rotors by exponentiation. An additional advantage of using CGA is its covariance: conformal motions for other primitives such as circles are computed using exactly the same formulas, and hence the same software operations, as motions of points. This generates an interesting class of easily generated shapes, like spatial circles moving conformally along a knot on a Dupin cyclide.
Highlights
Conformal transformations are mappings that locally preserve angles
We will focus on conformal motions, i.e., conformal transformations that can be performed by degrees
A principled way to construct conformal mappings is by composing them from simple mappings at the origin, in some fixed standard order: this is the extension of viewing Euclidean motions as a rotation around an axis through the origin followed by a translation, which appears natural in a homogeneous coordinate representation
Summary
Conformal transformations are mappings that locally preserve angles. objects transforming conformally retain much of their shape locally, and over a wider range undergo acceptable but interesting distortions, while remaining recognizable. The resulting orbit of x under the motion C is in general non-circular, as the figure shows clearly; we will find that it is an equiangular spiral on this network of circles Such 2D pictures may be found in pictorial accounts of conformal motions in the plane, such as the excellent [11]. They are not limited to 2D: Fig. 1b shows a very similar 3D motion C, split into a B and G, where the blue circles are again passing from a source to a sink in a conformal scaling, and the green circles perform a conformal rotation in an orthogonal direction. The theory in that paper will form the technical basis of our explanation here
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