Conchoid surfaces of spheres

The conchoid of a surface $F$ with respect to given fixed point $O$ is roughly speaking the surface obtained by increasing the radius function with respect to $O$ by a constant. This paper studies {\it conchoid surfaces of spheres} and shows that these surfaces admit rational parameterizations. Explicit parameterizations of these surfaces are constructed using the relations to pencils of quadrics in $\R^3$ and $\R^4$. Moreover we point to remarkable geometric properties of these surfaces and their construction

Rational Generalized Offsets of Rational Surfaces

The rational surfaces and their offsets are commonly used in modeling and manufacturing. The purpose of this paper is to present relationships between rational surfaces and orientation-preserving similarities of the Euclidean 3-space. A notion of a similarity surface offset is introduced and applied to different constructions of rational generalized offsets of a rational surface. It is shown that every rational surface possesses a rational generalized offset. Rational generalized focal surfaces are also studied