A rational curve on a rational surface such that the unit normal vector field of the surface along this curve is rational will be called a curve providing Pythagorean surface normals . These curves represent rational paths on the surface along which the surface possesses rational offset curves . Our aim is to study rational surfaces containing enough PSN curves . The relation with PN surfaces will be also investigated and thoroughly discussed . The algebraic and geometric properties of PSN curves will be described using the theory of double planes . The main motivation for this contribution is to bring the theory of rational offsets of rational surfaces closer to the practical problems appearing in numerical control machining where the milling cutter does not follow continuously the whole offset surface but only certain chosen trajectories on it . A special attention will be devoted to rational surfaces with pencils of PSN curves . One of the basic operations in Computer Aided Design and related technical applications is offsetting . The two sided offset to an irreducible surface is the envelope of the system of spheres centered at the points of with the radius . Because of their high applicability investigating surface offsets has become a popular research field in recent years and many interesting and challenging problems have been thoroughly studied e.g . analysis of their geometric and algebraic properties determining the number and type of components and constructing rational parameterizations see e.g . Arrondo et al . L and Pottmann Sendra and Sendra . Computational techniques for offsets have been surveyed firstly by Pham and later by Maekawa . A serious problem with offsets is that they are often far more complicated than the input generating surfaces . Hence approximation methods are in several real situations not only possible but unavoidable . However offsets to special classes of rational surfaces admit exact rational representations . The notion of rational surfaces with rational offsets the so called Pythagorean Normal vector surfaces was introduced by Pottmann as the surface counterpart to the theory of Pythagorean Hodograph curves which are planar curves possessing rational offset curves introduced by Farouki and Sakkalis . More details about PN surfaces can be found e.g . in papers by Krasauskas L vi ka and Bastl L vi ka and Vr ek Vr ek and L vi ka . For a survey of shapes with Pythagorean normals property see Farouki and for a recent discussion about rational offset surfaces and their modelling application see Krasauskas and Peternell . The aim of this paper is to start an investigation of rational offsets to surfaces from a new point of view and return this research back to one of the practical applications which originally motivated the introduction of PN surfaces in CAGD namely closer to numerical control machining . We recall that PN surfaces were introduced due to their straightforward applicability in CAD CAM as they guarantee that the surface representing the centers of the sphere milling cutter is. 


Summary: An algebraic analysis of components of offsets to irreducible surfaces along rational curves is provided. A simple criterion for deciding if the curve on the surface is proper non proper PSN is formulated. Surfaces with PSN pencils such that a generic fiber is a proper non proper PSN curve are investigated. It is shown that all developable surfaces are PSN pencils but not necessarily PN surfaces . For non developable surfaces a method for generating all real parameterizations respecting PSN pencils is designed.