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Reflections in conics, quadrics and hyperquadrics via Clifford algebra. (English) Zbl 1380.15018

The author presents the quadric geometric algebra model introduced by J. Zamora-Esquivel [Adv. Appl. Clifford Algebr. 24, No. 2, 493–514 (2014; Zbl 1298.15033)]. This model is a generalization of the conformal geometric algebra of C. Doran and A. Lasenby [Geometric algebra for physicists. Cambridge: Cambridge University Press (2003; Zbl 1078.53001)]. The benefit of this model is that inversions and reflections with respect to conics, quadrics and even hyperquadrics in any dimension can be computed explicitly. All necessary concepts are defined in this paper. As an example, calculations are carried out in two and three dimensional cases.

MSC:

15A66 Clifford algebras, spinors
51B99 Nonlinear incidence geometry
51M15 Geometric constructions in real or complex geometry
51N15 Projective analytic geometry

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References:

[1] Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science. Morgan Kaufmann Publishers, Burlington (2007)
[2] Dorst, L., Fontijne, D.: 3D Euclidean Geometry Through Conformal Geometric Algebra (a GAViewer tutorial) (2005). http://www.science.uva.nl/ga/files/CGAtutorial_v1.3.pdf
[3] Fontijne, D.: Clifford Algebras and the Classical Groups. Morgan Kaufmann Publishers, Burlington (2007)
[4] Gallier, J.: Clifford Algebras, Clifford Groups, and a Generalization of the Quaternions: The Pin and Spin Groups, unpublished (2011). http://www.cis.upenn.edu/ cis610/clifford.pdf
[5] Hestenes, D., Sobczyk, G.: Clifford Algebra to Geometric Calculus. Kluwer Academic Publishers, Dordrecht (1984) · Zbl 0541.53059
[6] Perwass, C.: Geometric Algebra with Applications in Engineering. Springer, Berlin (2009) · Zbl 1179.15025
[7] Porteous, I.R.: Clifford Algebras and the Classical Groups. Cambridge University Press, Cambridge (1995) · Zbl 0855.15019
[8] Selig, J.M.: Geometric Fundamentals of Robotics, 2nd edn. Springer, New York (2005) · Zbl 1062.93002
[9] Zamora-Esquivel, J.: \[G_{6,3}\] G6,3 Geometric algebra, description and implementation. Adv. Appl. Clifford Algebras 24(2), 493-514 (2014) · Zbl 1298.15033
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