Spline surfaces of arbitrary topology with continuous curvature and optimized shape

A shape optimization problem for spline surfaces of arbitrary topology, with curvature continuous and represented by meshes, is considered. The surface is made of bicubic and binonic polynomial patches and it has a fixed boundary and a tangent plane and curvature at each point of the boundary. The optimization criterion has the form of a functional, measuring the surface undulations. A term measuring undulations of the constant parameter curves of the surface parametrization is used to get a well posed problem. Its presence affects the result. After restricting the space, in which the solution is searched, it is possible to eliminate this term. Three numerical algorithms are described, which may be used to optimize surfaces represented by meshes with small or big numbers of vertices. A possibility of imposing constraints contributes to the practical applicability of the construction.

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