Basis Function Methods For Constructing Interpolating Spline Curve And Surface

Interpolation is an important subject of Computer AidedGeometric Design, for the difficult interpolation problem ofBezier method,B-Spline method and NURBS method, this paperpropose the basis function methods for constructinginterpolating curve and surface. The basis that we constructedcan be calculated easily, and it' s steady. In addition, withthe shape control parameter, we can adjust the interpolatingcurve and surface without modifying the data points.The first chapter, the prolegomenon, we simply introduce theprovenience, development and application of Computer AidedGeometric Design, synthetically analyze all kinds of methodsof constructing interpolating free form curve and surface.Based on such analysisⅠpropose three principles forconstructing interpolating curve and surface. Finally,introduces some interrelated mathematical knowledge.The second chapter, we construct C~1 continuityinterpolating spline curve and surface. Based on theinterpolating properties and the first derivative on end point,we establish the relationship between the four adjacent datapoints and the degree three Bezier curve control points, andthen we find the C~1 continuity interpolating spline basis function——BB basis function, the interpolating curve andsurface can be produced from the basis functions that we havejust found and the data points, the interpolating curve is C~1continuity and shape preserving. Further more, we discuss theshape preserved properties of the interpolating curve and theinfluence of the shape control parameterαto the interpolatingcurve, gave the relative algorithm and numerical examples.In the third chapter, we constructed a group of degree 2uniform interpolating B-Spline basis functions——BE basisfunction, so did the degree 3 uniform interpolating B-Splinebasis functions——BS basis function. Giving a group of datapoints, a segment of B-Splin,e is constructed from four adjacentdata points, its first point and end point are the two middlepoints. A B-Spline segment is constructed from every four datapoints, it's C~1 continuity on the data point, so the wholeinterpolating curve is C~1 continuity. The basis function takesa parameter t which can adjust the derivative direction on thedata point and a tension parameterλ. We can modify theinterpolating curve by adjusting those parameter.The fourth chapter, based on the end point interpolatingproperties and the first derivative on the end point, the authorestablishes the relationship between the four adjacent data points and the degree three rational Bezier curve controlpoints, and then find the C~1 continuity interpolating splinebasis function——RB basis function with tensionparameterαand shape control parameterω, the interpolationcurve can be produced from the basis function that we have justfound and the data points, the interpolating curve is C~1continuity,shape preserving and can be shape modified.The fifth chapter summarizes the whole paper, sum up the mainpoints and conclusion.