AngelCG38山东university软件学院图形学教材.ppt

Angel: Interactive Computer Graphics 4E ? Addison-Wesley 2005 Rendering Curves and Surfaces Ed Angel Professor of Computer Science, Electrical and Computer Engineering, and Media Arts University of New Mexico Objectives Introduce methods to draw curves Approximate with lines Finite Differences Derive the recursive method for evaluation of Bezier curves and surfaces Learn how to convert all polynomial data to data for Bezier polynomials Evaluating Polynomials Simplest method to render a polynomial curve is to evaluate the polynomial at many points and form an approximating polyline For surfaces we can form an approximating mesh of triangles or quadrilaterals Use Horner’s method to evaluate polynomials p(u)=c0+u(c1+u(c2+uc3)) 3 multiplications/evaluation for cubic Finite Differences Building a Finite Difference Table p(u)=1+3u+2u2+u3 Finding the Next Values Starting at the bottom, we can work up generating new values for the polynomial deCasteljau Recursion We can use the convex hull property of Bezier curves to obtain an efficient recursive method that does not require any function evaluations Uses only the values at the control points Based on the idea that “any polynomial and any part of a polynomial is a Bezier polynomial for properly chosen control data” Splitting a Cubic Bezier l(u) and r(u) Convex Hulls Equations Efficient Form Every Curve is a Bezier Curve We can render a given polynomial using the recursive method if we find control points for its representation as a Bezier curve Suppose that p(u) is given as an interpolating curve with control points q There exist Bezier control points p such that Equating and solving, we find p=MB-1MI Matrices Example Surfaces Can apply the recursive method to surfaces if we recall that for a Bezier patch curves of constant u (or v) are Bezier curves in u (or v) First subdivide in u Process creates new points Some of the original points are discarded Second Subdivision Normals For rendering we need the normals