大三05计算机图形学and spline curves surfaces

1、Bezier and Spline Curves and Surfaces1Angel: Interactive Computer Graphics 5E Addison-Wesley 2009原著Ed AngelProfessor of Computer Science, Electrical and Computer Engineering, and Media ArtsUniversity of New Mexico编辑 武汉大学计算机学院图形学课程组2Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Objective

2、sIntroduce the Bezier curves and surfacesDerive the required matricesIntroduce the B-spline and compare it to the standard cubic Bezier3Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Beziers IdeaIn graphics and CAD, we do not usually have derivative dataBezier suggested using the same 4

3、data points as with the cubic interpolating curve to approximate the derivatives in the Hermite form 4Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Approximating Derivativesp0p1p2p3p1 located at u=1/3p2 located at u=2/3slope p(0)slope p(1)u5Angel: Interactive Computer Graphics 5E Addiso

4、n-Wesley 2009Equationsp(0) = p0 = c0p(1) = p3 = c0+c1+c2+c3p(0) = 3(p1- p0) = c0p(1) = 3(p3- p2) = c1+2c2+3c3Interpolating conditions are the sameApproximating derivative conditionsSolve four linear equations for c=MBp6Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Bezier Matrixp(u) = uT

5、MBp = b(u)Tpblending functions7Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Blending FunctionsNote that all zeros are at 0 and 1 which forcesthe functions to be smooth over (0,1)8Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Bernstein PolynomialsThe blending functions are

6、a special case of the Bernstein polynomialsThese polynomials give the blending polynomials for any degree Bezier formAll zeros at 0 and 1For any degree they all sum to 1They are all between 0 and 1 inside (0,1) 9Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Convex Hull PropertyThe prope

7、rties of the Bernstein polynomials ensure that all Bezier curves lie in the convex hull of their control pointsHence, even though we do not interpolate all the data, we cannot be too far awayp0p1p2p3convex hullBezier curve10Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Bezier PatchesUsi

8、ng same data array P=pij as with interpolating formPatch lies inconvex hull11Angel: Interactive Computer Graphics 5E Addison-Wesley 2009AnalysisAlthough the Bezier form is much better than the interpolating form, we have the derivatives are not continuous at join pointsCan we do better?Go to higher

9、order BezierMore workDerivative continuity still only approximateSupported by OpenGLApply different conditions Tricky without letting order increase12Angel: Interactive Computer Graphics 5E Addison-Wesley 2009B-SplinesBasis splines: use the data at p=pi-2 pi-1 pi pi-1T to define curve only between p

10、i-1 and piAllows us to apply more continuity conditions to each segmentFor cubics, we can have continuity of function, first and second derivatives at join pointsCost is 3 times as much work for curvesAdd one new point each time rather than threeFor surfaces, we do 9 times as much work 13Angel: Inte

11、ractive Computer Graphics 5E Addison-Wesley 2009Cubic B-splinep(u) = uTMSp = b(u)Tp14Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Blending Functionsconvex hull property15Angel: Interactive Computer Graphics 5E Addison-Wesley 2009B-Spline Patchesdefined over only 1/9 of region16Angel: I

12、nteractive Computer Graphics 5E Addison-Wesley 2009Splines and BasisIf we examine the cubic B-spline from the perspective of each control (data) point, each interior point contributes (through the blending functions) to four segmentsWe can rewrite p(u) in terms of the data points asdefining the basi

13、s functions Bi(u)17Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Basis FunctionsIn terms of the blending polynomials18Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Generalizing SplinesWe can extend to splines of any degree Data and conditions to not have to given at equally

14、 spaced values (the knots)Nonuniform and uniform splinesCan have repeated knotsCan force spline to interpolate pointsCox-deBoor recursion gives method of evaluation19Angel: Interactive Computer Graphics 5E Addison-Wesley 2009NURBSNonuniform Rational B-Spline curves and surfaces add a fourth variable w to x,y,zCan interp