On a Special Class of Polynomial Surfaces with Pythagorean Normal Vector Fields
Rational shapes with rational offsets, especially Pythagorean hodograph (PH) curves and Pythagorean normal vector (PN) surfaces, have been thoroughly studied for many years. However compared to PH curves, Pythagorean normal vector surfaces were introduced using dual approach only in their rational version and a complete characterization of polynomial surfaces with rational offsets, i.e., a polynomial solution of the well-known surface Pythagorean condition, still remains an open and challenging problem. In this contribution, we study a remarkable family of cubic polynomial PN surfaces with birational Gauss mapping, which represent a surface counterpart to the planar Tschirnhausen cubic. A full description of these surfaces is presented and their properties are discussed.
KeywordsGeometric Design Rational Surface Polynomial Solution Parameter Line Isothermal Surface
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