Reducibility of offsets to algebraic curves

Computing offset curves and surfaces is a fundamental operation in many technical applications. This paper discusses some issues that are encountered during the process of designing offsets, especially the problems of their reducibility and rationality (which are closely related). This study is crucial especially for formulating subsequent algorithms when the number and quality of offset components must be revealed. We will formulate new algebraic and geometric conditions on reducibility of offsets and demonstrate how they can be applied. In addition, we will present that our investigations can also serve to better understand the varieties fulfilling the Pythagorean conditions (PH curves/PN surfaces). A certain analogy of the PH condition for parameterized curves (or general parameterized hypersurfaces) will be presented also for implicitly given (not necessarily rational) curves (or hypersurfaces).

► An analysis of reducibility and rationality of offsets of algebraic curves (hypersurfaces) is discussed. ► Two new conditions (geometric and algebraic) on reducibility of offsets are presented. ► The obtained results are related to the varieties fulfilling the Pythagorean conditions (Pythagorean hodograph curves and surfaces with Pythagorean normals). ► The results from Arrondo et al. (1997), for rational curves end their generalized offsets, are extended.