Constrained fitting with free-form 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 CAD/CAM 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 free-form curves and surfaces is also necessary in engineering practice; this motivated our work.We numerically optimize complex systems of non-linear 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 B-spline surfaces.After formulating equations for a representative set of free-form 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.