Geometry and interactive design of curved creases

Abstract

This thesis discusses curved creases from a theoretic and an applied point of view: On the one hand, we utilize differential geometry to describe curved creases between developable surfaces by five quantities. We conclude that defining three of them appropriately determines the remaining two except in some special cases. Apart from degenerated folds, we also address two special types of folds: the planar crease, i.e. the rulings of the corresponding developable surfaces are refected on a plane, and creases of constant angle. The latter crease curves are known as pseudo-geodesics in classical differential geometry. By combining these classical results with our approaches, we examine pseudo-geodesics on cylinders and cones. Furthermore, a connection between bi-cylindrical, bi-concial and cylindro-conical creases of constant angle and geodesics on quadrics can be established. The applied approach is based on the work made by Tang et al. on the interactive design of curved creases. We utilize their proposed guided projection algorithm to solve an optimization problem for B-spline representations of developable surfaces with creases, and discuss the needed variables and constraints. Finally, we present some examples obtained from the author's implementation.