Journal for Geometry and Graphics, Vol. 4, No. 2, pp. 169-180 (2000)

On Curves and Surfaces in Illumination Geometry

Gunter Weiss, Horst Martini

Institute of Geometry, Dresden University of Technology
Zellescher Weg 12-14, D-01062 Dresden, Germany

Abstract: A point like light source in $R^d$ induces a certain illumination intensity at hypersurface elements of $R^d$. Manifolds of such elements with the same intensity of illumination are called isophotic. A uniformly radiating light source causes isophotic strips along sinusoidal spirals. In the present paper this investigation is extended in two directions. First all isophotic $C^2$-hypersurfaces are found, and also manifolds of hypersurface elements which are isophotic with respect to two and more central illuminations are discussed. It suggests itself to treat such illumination problems also in non-Euclidean spaces. The second part of the paper deals with the generating curves of isophotic strips. They belong to the well-known families of Clairaut curves and sinusoidal spirals. Their known relations to each other and to other curve families (such as Ribaucour curves and roses) are extended by some perhaps new aspects.

Keywords: cardioid, cassinoid, central illumination, Clairaut curve, cycloid (of higher order), hyperbolic geometry, inversion, isophotic strips and surfaces, isotropic geometry, (isotropic) logarithmic spiral, projections (of spatial curves), pedal curve, rhodonea, Ribaucour curve, rose, sinusoidal spiral, Steiner's three-cusped hypocycloid

Classification (MSC2000): 53A07; 53A04

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