The present paper investigates a class of two-dimensional rational surfaces φ in R4 whose tangent planes satisfy the following property: For any three-space E in R4 there exists a unique tangent plane T of φ which is parallel to E. The most interesting families of surfaces are constructed explicitly and geometric properties of these surfaces are derived. Quadratically parameterized surfaces in R4 occur as special cases. This construction generalizes the concept of LN-surfaces in R3 to two-dimensional surfaces in R4.

Given a cloud P of data points, we present techniques and algorithms to decide whether P could be measurements of a developable surface and to reconstruct the original shape of the surface. We will use the dual approach to developable surfaces, which says that a developable surface can be considered as envelope of a one-parameter family of planes. A classification will be given which detects cones, cylinders, and developable surfaces of constant slope. Finally we discuss the problems occurring in this context.

A geometric approach to the computation of precise or well approximated tolerance zones for CAD constructions is given. We continue a previous study of linear constructions and freeform curve and surface schemes under the assumption of convex tolerance regions for points. The computation of the boundaries of the tolerance zones for curves/surfaces is discussed. We also study congruence transformations in the presence of errors and families of circles arising in metric constructions under the assumption of tolerances in the input. The classical cyclographic mapping as well as ideas from convexity and classical differential geometry appear as central geometric tools.