Unifying Theory of Pythagorean-Normal Surfaces Based on Geometric Algebra

Pythagorean-normal (PN) surfaces, defined as rational surfaces admitting rational offsets, are important for industrial Computer-aided Design. Traditionally PN surfaces are considered from the point of view of Laguerre geometry, using three main models: cyclographic model, Blaschke cylinder or isotropic model. We propose a unifying formalism to deal with PN surfaces: all these models are embedded into one ambient pseudo-Euclidean space , that is known as a model for Lie sphere geometry. Various relations between different models are described in terms of closed formulas in the geometric algebra and illustrated by examples of applications.