Bezier curves and surfaces: A new approach.

摘要

Based on Grassmann's master piece "Ausdehnungslehre", Ramshaw's recent work "On Multiplying Points: The Paired Algebras of Forms and Sites", and umbral calculus, a new approach to Bezier curves and surfaces is given in the first chapter. Under the new approach a Bezier curve of degree n can be simply denoted as 1-tA+tB n,0≤t≤1 , where An-kBk, (k = 0, 1,..., n) are the n + 1 control points of the Bezier curve, and a triangular Bezier surface of degree n can be simply denoted as uA+vB+wCn u+v+w=1,0≤u,v,w≤1, where AiBjCk, (i + j + k = n) are the control points of the Bezier surface. Using this new approach, many known results of Bezier curves and surfaces, both the statements and the proofs, can be simplified, and many new results can be obtained more easily.; Some classical problems of Bezier curves are studied in Chapter 2. The geometric Hermite interpolation with specified tangent directions, curvature vectors and torsions at the end points is studied in detail. A solution of degree 5 Bezier curve with optimum approximate order (h 8) is given. The general characterization of singular points, inflection points and torsion vanish points is given for both Bezier curves and Bezier rational curves.; In Chapter 3, we first discuss in detail the general case of conversion between triangular Bezier surfaces and rectangular surfaces. Under our new theory, the conversion, which is an important problem in Computer Aided Geometric Design (CAGD), becomes easier and clearer. Secondly, we prove that under some restriction on control points, which is described by a matrix, the conditions of geometric continuity between two triangular Bezier surface patches can be greatly simplified. The matrix itself, we believe, has an important position in characterizing the control points of Bezier patches. The problem of geometric continuity as it appears in the vertex enclosure problem is also discussed.