A novel projection of open geometry into rectangular domain for 3D shape parameterization

Multi-patch shape models of engineering objects, while adequate for representation purposes, may introduce problems related to cross-partition continuity and changing composition of individual partitions in shape optimization. Single-patch integral parameterizations may consequently be the preferred option in cases where the shape can change significantly. In absence of any topological preparation, fitting of parametric surfaces to point clouds may be satisfactory for simple shapes. However, complex objects involving concave portions and irregular shapes require additional numerical procedures to obtain adequate structured geometry matrices for fitting which need to preserve the original topology. The overall goal of this paper is to develop a procedure that efficiently handles difficulties related to single-partition parametric surface models of complex-shaped engineering objects. This aspiration involves obtaining a structured geometry matrix even for irregular and concave shapes and developing a dedicated method of projection into a rectangular domain. The performance of the resulting procedure is verified using real engineering objects of complicated 3D shape. Moreover, cases of numerically generated surfaces representative for changing shape compositions with emerging and/or dissolving edges are also included.