Polypoint Transformation Dependency on the Polyfiber Configuration

Applying polypoint transformation to a triangle mesh is a promising technique that deserves extensive exploration. Unlike traditional deformation techiques used in 3D animation that are based on spatial interpolation, polypoint transformation transforms not a set of points or mesh vertices, but a polyfiber — a set of planes that form an object of transformation. This can be beneficial in practical applications such as computer animation, deformative modeling in CAD, or deformation prediction in additive manufacturing since, with polypoint transformation, the object of deformation includes the topological information about the mesh and not just its vertex positions. The way a polyfiber can form a triangle mesh is, however, an understudied problem. This problem gets easier to study if we start from a 2D case where a structure that corresponds to a triangle mesh in 3D is a polyline or a conotur made of line segments. For this structure we can decompose the study into a set of explorable questions one of them being: how the configuration of contour forming polyfiber affects the results of transformation, or, more specifically, how an angle between two lines forming a vertex affects the translation of that vertex under a polypoint transformation? Would this dependency prohibit the application of polypoint transformation to 3D mesh deformation? This article answers both questions.