Computing normal vectors and derivatives at regular and singular points of parametric surfaces

The paper describes a generalization of the Weingarten formulae to find the partial derivatives of arbitrary order of a unit surface normal vector at a given point on a parametric surface. It also proposes methods for computing the normal vectors and its partial derivatives at singular points on the parametric surface. It considers the singular points where two surface base vectors in the tangent plane of the surface are linearly dependent or where at least one of them is zero. It provides algorithms for calculating the normal vectors and its partial derivatives at regular and singular points. This algorithm can be used in computer-aided design and manufacturing systems and integrated into geometry libraries for working with offset surfaces or their generalizations, the description of surface growth processes, the generation of tool paths for numerical control machining applications, geometric modeling of changes in surface shape when multilayer fabric draping, filament winding and tape laying, and access space representations in robotics.