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The construction of 3D conformal motions. (English) Zbl 1341.65006

Summary: 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 computer aided design, 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.

MSC:

65D17 Computer-aided design (modeling of curves and surfaces)

Software:

GAviewer
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References:

[1] Colapinto, P.: Articulating space: geometric algebra for parametric design—symmetry, kinematics, and curvature. PhD thesis, University of California at Santa Barbara (2016) · Zbl 1290.68121
[2] Coolidge J.: A Treatise on the Circle and the Sphere. Clarendon Press, Oxford (1916) · JFM 46.0921.02
[3] Dorst L., Fontijne D., Mann S.: Geometric Algebra for Computer Science: An Object-oriented Approach to Geometry. Morgan Kaufman, San Francisco (2009)
[4] Dorst, L.: Conformal geometric algebra by extended Vahlen matrices. In: GraVisma 2009 workshop proceedings (2009), Skala V., Hildenbrandt H., (Eds.), pp. 72-79
[5] Dorst, L.; Dorst, L. (ed.); Lasenby, J. (ed.), Tutorial appendix: structure preserving representation of Euclidean motions through conformal geometric algebra, 435-452 (2011), London · Zbl 1290.68120
[6] Dorst, L.; Valkenburg, R.; Dorst, L. (ed.); Lasenby, J. (ed.), Square root and logarithm of rotors in 3D conformal geometric algebra using polar decomposition, 81-104 (2011), London · Zbl 1290.68121
[7] Fontijne, D., Dorst, L.: GAViewer, free download for various platforms. http://www.geometricalgebra.net/gaviewer_download.html (2007)
[8] Hestenes, D., Rockwood, A., Li, H.: System for encoding and manipulating models of objects. US Patent 6,853,964, granted, 8 Feb 2005
[9] Hestenes, D., Sobczyk, G.: Clifford Algebra to Geometric Calculus. Reidel (1984) · Zbl 0541.53059
[10] Li, H.; Hestenes, D.; Rockwood, A.; Sommer, G. (ed.), Generalized homogeneous coordinates for computational geometry, 27-59 (1999), Heidelberg
[11] Needham T.: Visual Complex Analysis. Clarendon Press, Oxford (1997) · Zbl 0893.30001
[12] Weber, O., Gotsman, C.: Controllable conformal mappings for shape deformation. ACM Trans. Graphics, vol. 4 (2010) · Zbl 1290.68121
[13] Wallner, J., Pottmann, H. (2011) Geometric computing for freeform architecture. J. Math. Ind. 1, 4-19 · Zbl 1269.00008
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