Geometric point interpolation method in R3 space with tangent directional constraint

Abstract

This paper discusses a cubic BB-spline interpolation problem with tangent directional constraint in R3R3 space. Given mm points and their tangent directional vectors as well, the interpolation problem is to find a cubic BB-spline curve which interpolates both the positions of the points and their tangent directional vectors. Given the knot vector of the resulting BB-spline curve and parameter values to all of the data points, the corresponding control points can often be obtained by solving a system of linear equations. This paper presents a piecewise geometric interpolation method combining a unclamping technique with a knot extension technique, with which there is no need to solve a system of linear equations. It firstly uses geometric methods to construct a seed curve segment, which interpolates several data point pairs, i.e., positions and tangent directional vectors of the points. The seed segment is then extended to interpolate the remaining data point pairs one by one in a piecewise fashion. We show that a BB-spline curve segment can always be extended to interpolate a new data point pair by adding two more control points. Methods for a curve segment extending to interpolate one more data point pair by adding one more control point are also provided, which are utilized to construct an interpolation BB-spline curve with as small a number of control points as possible. Numerical examples show the effectiveness and the efficiency of the new method.