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Journal for Geometry and Graphics, Vol. 3, No. 2, pp. 121-139 (1999)
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Reflections on Spheres and Cylinders of Revolution

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Georg Glaeser

Institute for Architecture, University of Applied Arts of Vienna

Oskar Kokoschka-Platz 2, A 1010 Wien, Austria

email: georg.glaeser@uni-ak.ac.at

**Abstract:** In computer graphics, it is often an advantage to calculate reflections directly, especially when the application is time-critical or when line graphics have to be displayed. We specify formulas and parametric equations for the reflection on spheres and cylinders of revolution. The manifold of all reflected rays is the normal congruence of an algebraic surface of order four. Their catacaustic surfaces are given explicitly. The calculation of the reflex of a space point leads to an algebraic equation of order four. The up to four practical solutions are calculated exactly and efficiently. The generation of reflexes of straight lines is optimized. Finally, reflexes of polygons are investigated, especially their possible overlappings. Such reflexes are the key for the reflection of polyhedra and curved surfaces. We describe in detail how to display their contours.

**Keywords:** Reflection, curved perspectives, caustic, normal congruence, specular points, real-time rendering

**Classification (MSC2000):** 51N05; 53A04; 68U05; 65Y25

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