Imposing angle boundary conditions on B-spline/NURBS surfaces

• We study the construction of a B-spline surface satisfying prescribed angle distribution of tangent planes along its boundary curve.
• We prove that for a given B-spline curve, the exact solution exists only in very special cases (for a special form of an angle function).
• We propose an algorithm for finding an approximate solution, derive a bound on its approximation error and study the approximation order of the proposed algorithm.

In this paper, we study the construction of a B-spline surface satisfying prescribed angle distribution (with respect to a chosen vector) of tangent planes along its boundary curve. This problem arises e.g. in a creation of a parametric geometric model of a Pelton turbine bucket, where specific angle distributions along a splitter and an outlet curve have to be fulfilled in order to control the flow of water into and out of the bucket. We prove that for a given B-spline curve c(t), t∈[0,1]t∈[0,1], the exact solution exists only in very special cases (for a special form of an angle function f(t)f(t)). Further, we formulate an algorithm for finding an approximate solution. We also derive a bound on its approximation error and give a numerical evidence that the approximation order of the proposed algorithm is four. Finally, the method is demonstrated on several examples.

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