Study Of Rational Spline & Algebraic Curves And Surfaces Splines

This Ph. D. Thesis is devoted to construct and analysis piecewise rational splines, algebraic curve and surface for interpolation. It is composed of eight chapters.Firstly, we introduce the historical background and the present progress of problems that concern with parametric curves and surfaces, algebraic curves and surfaces. The main works of this paper are concluded as well.The study of Rational interpolation curves, a kind of blending basis with trigonometric and polynomial functions are constructed, the rational curve generated by the basis with local shape handles is investigated, and the influence of each shape handles for spline analyzed. There is a example for the spline contract to Hermite interpolation function. At the same time, we study a piecewise rational interpolation spline being type of [2m＋1, 2m], which is generalized by low degree Geogry rational spline, and the representation of them are given with algebraic method. The degree of approximation and shape preserving are investigated, then the spline is applied to a classical data for interpolation.The study of algebraic curves and surfaces, an A-spline are piecewise algebra curves of fourth order is investigated, which constructed by a control polygon that is sequence of triangles meeting at the vertices. The arc in a given triangle is a segment that joins these vertices and interpolates also the slope and curvature at each endpoint. This arc in each triangle is controlled by four shape handles, and an exterior point control the slope at the vertices point, the individual arcs is singular and always monotone in the affine translation triangle through barycentric coordinate system, and it can be convex by adjusting four shape handles.In Chapter 5, a new method to construct C~1-continuity (normal vector continuity) interpolation surfaces on triangulation is given. Points in a tetrahedron can be represented by space barycentric coordinates system through space affine transform. Both algebraic patches of third order with three variables are constructed in tetrahedron, which represented by algebraic Bernstein-Bezier zero contour. Principalpatch is in a tetrahedron formed on a given triangle with another exterior control vertex, and three vertexes of the triangle are on the patch. Sub-patch is in a tetrahedron formed on the common vertexes of two neighboring triangles and both control vertexes, and the two common vertexes are on the patch. Some shape handles(including control vertexes and Bernstein coefficients) control those patches. Each triangle with a control vertex builds up a Principal-patch in the tetrahedron, and a Sub-patch can be gained between two neighboring Principal-patches. Constructed patches can be joined as C~1-continuity (normal vector continuity) surfaces through solving system of homogeneous linear equations about the Bernstein coefficients.Chapter 6 construct a space approximate spline with intersection of two algebraic surfaces, tetrahedron can be formed with four sequence and non-coplanar points in space. Both regular algebraic patches of cubic and quadratic with three variables, which constructed in the tetrahedron, represented in algebraic Bernstein-Bezier zero contour. The intersection set of both patches form a regular segment. Fixed quadratic patches' Bernstein coefficients, we gain its parametric representation. The cubic patches reduce to three Bernstein coefficients to control shape of curve, two of them are used to join segments into continuous approximate curves for given sequence data in space, the other is used to adjust segment. We can easy generate a G~2-continuity approximate curve for a sequence points similar to cubic Bezier splines with four control points. The method is quite different from popularly algebraic method by joining planar arc into a continuous curve.The final issue is also about space spline, we introduce a new method to construct space spline based on method of Chapter 6, construct a functional for the parametric of anticlastic to generate different spline for quadrilateral analogous to Bezier parametrical spline, but this spline is more freedom for shape controlling, having unified representation, and various functions can be chose for spline. Curves generated by the splines is easy to join as G~2-continuity.