B-spline Curve And Surface Interpolation With Tangent Constraint

B-spline curves and surfaces are important tools for geometric modeling and reversing engineering. While the traditional B-spline interpolation algorithm is only for interpolation of sampled points, the interpolating curves may have unwanted undulations. This paper presents an algorithm for B-spline curve interpolation with tangent constraints. By this method, the interpolating curves and surfaces can have higher approximation accuracy and more natural shapes.It is known that the shapes of interpolating B-spline curves depend on the chosen knot vector and the nodes for all input points heavily. For a set of given points, we set the knot vector for the interpolating curve first, and then we choose nodes for the interpolating B-spline curve based on the constraints of given tangent vectors at the points. To guarantee the regularity of the final curve, a set of intervals that do not overlap are set for nodes where the first and the last node is not included. A search algorithm is developed to find the optimal nodes in the node intervals. The proposed algorithm guarantees that the curve interpolates each point. Meanwhile, the total sum of the angles between the given tangent vectors and the tangent vectors on the interpolating curves has been minimized.We also extend the constrained B-spline curve interpolation for B-spline.surface interpolation under tangent constraint. For a B-spline surface interpolation a lattice of points in space, we can add tangent constraint for those points on the boundary. By using the tangent constraint B-spline interpolation for boundary curves, we obtain interpolating B-spline surfaces with tangent constraints.Several interesting examples have been given for B-spline curve and surface interpolation with tangent constraints. The examples show that the interpolating curves and surfaces usually have more natural shapes than those interpolating curves and surfaces with no constraint.