Journal for Geometry and Graphics 25 (2021), No. 1, 001--017
Copyright Heldermann Verlag 2021
Isoptic Ruled Surfaces of Developable Surfaces
Boris Odehnal
University of Applied Arts, Vienna, Austria
boris.odehnal@uni-ak.ac.at
The planar notion of isoptics cannot be carried over directly into three-dimensional spaces. Therefore, the isoptic surface of a developable ruled surface will be defined as the set of intersection lines of pairs of tangent planes that enclose a fixed angle. The existence of a one-parameter family of tangent planes of a developable ruled surface guarantees that such a set of lines is a ruled surface, while in the case of any other surface this construction would result in a complex of lines. The isoptics of quadratic cylinders and cones shall be given. Further, the isoptic ruled surfaces of developable ruled surfaces invariant under one-parameter subgroups of the Euclidean and the equiform group of motions shall be described. Moreover, the orthoptic ruled surfaces of developables with a polynomially parametrized curve of regression shall be computed. It turns out that the orthoptic ruled surfaces allow for a projective generation.
Keywords: Isoptic ruled surface, ruled surface, developable ruled surface, helical surface, spiral surface, cylinder, cone, isoptic curve
MSC: 14J26; 53A25, 51N20.
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