Rational Offset Surfaces and Their Modeling Applications

This survey discusses rational surfaces with rational offset surfaces in Euclidean 3-space. These surfaces can be characterized by possessing a field of rational unit normal vectors, and are called Pythagorean normal surfaces. The procedure of offsetting curves and surfaces is present in most modern 3d-modeling tools. Since piecewise polynomial and rational surfaces are the standard representation of parameterized surfaces in CAD systems, the rationality of offset surfaces plays an important role in geometric modeling. Simple examples show that considering surfaces as envelopes of their tangent planes is most fruitful in this context. The concept of Laguerre geometry combined with universal rational parametrizations helps to treat several different results in a uniform way. The rationality of the offsets of rational pipe surfaces, ruled surfaces and quadrics are a specialization of a result about the envelopes of one-parameter families of cones of revolution. Moreover a couple of new results are proved: the rationality of the envelope of a quadratic two-parameter family of spheres and the characterization of classes of Pythagorean normal surfaces of low parametrization degree.