Constrained fitting with freeform curves and surfaces Abstract   The accurate reconstruction of engineering parts from measured data is a challenging problem, in particular, when various geometric constraints need to be imposed to meet requirements in downstream CADCAM applications. There is a wide range of constraints including incidence, tangency, orthogonality, parallelism, symmetry and many others. In the majority of cases, only regular surfaces (planes, natural quadrics, sweeps) are involved; however, imposing constraints on objects with freeform curves and surfaces is also necessary in engineering practice; this motivated our work.  We numerically optimize complex systems of nonlinear equations, containing unknown control points and auxiliary geometric entities. We show that auxiliaries are indispensable; their use simplifies the algebra of the constraint equations and speeds up computations. We do not require the auxiliaries to be fixed in advance, rather propose incorporating them into the constraint system. As a result, the procedure will automatically substantiate various entities that have been unknown beforehand, for example, points of intersection or constrained trim curves on Bspline surfaces.  After formulating equations for a representative set of freeform constraints, we will discuss how this technique can be applied in practical reverse engineering. We will present case studies and analyze how surfaces get improved by means of constrained fitting.

Parametrization quantization with free boundaries for trimmed quad meshing The generation of quad meshes based on surface parametrization techniques has proven to be a versatile approach. These techniques quantize an initial seamless parametrization so as to obtain an integer grid map implying a pure quad mesh. Stateoftheart methods following this approach have to assume that the surface to be meshed either has no boundary, or has a boundary which the resulting mesh is supposed to be aligned to. In a variety of applications this is not desirable and nonboundaryaligned meshes or gridparametrizations are preferred. We thus present a technique to robustly generate integer grid maps which are either boundaryaligned, nonboundaryaligned, or partially boundaryaligned, just as required by different applications. We thereby generalize previous work to this broader setting. This enables the reliable generation of trimmed quad meshes with partial elements along the boundary, preferable in various scenarios, from tiled texturing over design and modeling to fabrication and architecture, due to fewer constraints and hence higher overall mesh quality and other benefits in terms of aesthetics and flexibility.