Principal Patches for Computational Geometry

This thesis presents a new class of surface patches for the creation of sculptured surfaces in mechanical engineering applications of computer aided design. Ideas from differential geometry are used to create principal patches whose edges are lines of curvature. This is done by ensuring that the sides obey two equations called the frame matching and position matching conditions. It is shown that these conditions are both sufficie~t and necessary to create a patch based on lines of curvature. A solution to the frame matching equation is given for all cases where the patch boundaries are plane curves. Continuity between principal patches is examined. The method is illustrated by means of two well understood surface types which are of engineering importance: generalised cylinders and surfaces of revolution. The method is then used to create patches whose lines of curvature are circular arcs. Such patches come from surfaces known as Dupin's cyclides. These patches are demoostra ted to have one degree of freedom, assuming two sides of the patch are known, and a discussion of how to use this choice is given. Means of finding points in the patch interior are given. It is shown that cycl ide patches are a subset of rational biquadratic patches, and that the sphere, torus, cone, cylinder and plane are special cases. Another useful property is that the offset surface to a cyclide is another cyclide. The need for non four sided regions of surface is examined. These occur at umbilics when using principal patches. The behaviour of surfaces near umbilics is reviewed, and suggestions are made for dealing with such regions using principal patches. Some ideas for future work are recorded.