📝 Abstract:
This thesis develops a framework for constructing cubic Bézier splines on polygon meshes. Our method replaces the affine interpolation steps of de Casteljau's construction with interpolation along continuous Dijkstra-like geodesic paths. Recursive evaluation at a selected finite depth produces a face-conforming piecewise-linear parametric curve trace for later geometry processing operations. While the interpolatory control points at two ends are provided by the user or by an algorithm, we use extrapolation to construct the other two interior control points of each cubic Bezier curve segment. We have a special treatment to handle the ends of an open spline using solely the ending curves themselves, while closed splines use cyclic neighbors. Furthermore, two interior control points at a shared user defined control point are aligned in a local polyhedral representation in order to impose a C1 continuity in a discrete sense. The face-conforming piecewise-linear parametric curve trace of the generated cubic Bezier spline supports three geometry processing applications. Firstly, in automatic surface filling; instead of a user, the control points can be distributed to the mesh automatically and their traversal order can be determined. The control points, then, can be connected with geodesics as well as the cubic Bezier splines. Secondly, interactive cubic Bézier spline untangling keeps the user defined control points fixed and uses the Separation–Direction-Coherence objective function energy minimization with coordinate descent and multiple trial directions. The user controls when each untangling run starts, pauses, continues, and stops. Thirdly, curve-conforming mesh adaptation inserts the face-conforming piecewise-linear parametric curve trace points of the closed cubic Bezier splines as new mesh vertices and adds required mesh edges for triangulation. Also, performs local interior triangles' selection of the closed Bezier splines, namely surface painting. Our framework supports any polygon input mesh, e.g., triangle and quadrilateral meshes. We fan-triangulate higher-degree faces, e.g., quads, as all subsequent operations use a triangular mesh.