05计算机图形学双语课程parametric cubic curves

1、Designing Parametric Cubic Curves1Angel: 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 2009ObjectivesInt

2、roduce the types of curvesInterpolatingHermiteBezierB-splineAnalyze their performance3Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Matrix-Vector Formdefinethen4Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Interpolating Curvep0p1p2p3Given four data (control) points p0 , p1

3、 ,p2 , p3determine cubic p(u) which passes through themMust find c0 ,c1 ,c2 , c35Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Interpolation Equationsapply the interpolating conditions at u=0, 1/3, 2/3, 1p0=p(0)=c0p1=p(1/3)=c0+(1/3)c1+(1/3)2c2+(1/3)3c3p2=p(2/3)=c0+(2/3)c1+(2/3)2c2+(2/3)

4、3c3p3=p(1)=c0+c1+c2+c3or in matrix form with p = p0 p1 p2 p3Tp=Ac6Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Interpolation MatrixSolving for c we find the interpolation matrixc=MIpNote that MI does not depend on input data andcan be used for each segment in x, y, and z7Angel: Interac

5、tive Computer Graphics 5E Addison-Wesley 2009Interpolating Multiple Segmentsuse p = p0 p1 p2 p3Tuse p = p3 p4 p5 p6TGet continuity at join points but notcontinuity of derivatives 8Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Blending FunctionsRewriting the equation for p(u)p(u)=uTc=uTM

6、Ip = b(u)Tpwhere b(u) = b0(u) b1(u) b2(u) b3(u)T isan array of blending polynomials such thatp(u) = b0(u)p0+ b1(u)p1+ b2(u)p2+ b3(u)p3b0(u) = -4.5(u-1/3)(u-2/3)(u-1)b1(u) = 13.5u (u-2/3)(u-1)b2(u) = -13.5u (u-1/3)(u-1)b3(u) = 4.5u (u-1/3)(u-2/3)9Angel: Interactive Computer Graphics 5E Addison-Wesley

7、 2009Blending FunctionsShape of the blending functions10Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Interpolating PatchNeed 16 conditions to determine the 16 coefficients cijChoose at u,v = 0, 1/3, 2/3, 111Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Matrix FormDefine v

8、= 1 v v2 v3T C = cij P = pij p(u,v) = uTCvIf we observe that for constant u (v), we obtaininterpolating curve in v (u), we can showp(u,v) = uTMIPMITv C=MIPMI12Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Blending PatchesEach bi(u)bj(v) is a blending patchShows that we can build and ana

9、lyze surfaces from our knowledge of curves13Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Other Types of Curves and SurfacesHow can we get around the limitations of the interpolating formLack of smoothnessDiscontinuous derivatives at join pointsWe have four conditions (for cubics) that

10、we can apply to each segmentUse them other than for interpolationNeed only come close to the data14Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Hermite Formp(0)p(1)p(0)p(1)Use two interpolating conditions andtwo derivative conditions per segmentEnsures continuity and first derivativeco

11、ntinuity between segments15Angel: Interactive Computer Graphics 5E Addison-Wesley 2009EquationsInterpolating conditions are the same at endsp(0) = p0 = c0p(1) = p3 = c0+c1+c2+c3Differentiating we find p(u) = c1+2uc2+3u2c3 Evaluating at end pointsp(0) = p0 = c1p(1) = p3 = c1+2c2+3c316Angel: Interacti

12、ve Computer Graphics 5E Addison-Wesley 2009Matrix FormSolving, we find c=MHq where MH is the Hermite matrix 17Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Blending Polynomialsp(u) = b(u)TqAlthough these functions are smooth, the Hermite formis not used directly in Computer Graphics and

13、 CAD because we usually have control points but not derivativesHowever, the Hermite form is the basis of the Bezier form18Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Parametric and Geometric ContinuityWe can require the derivatives of x, y, and z to each be continuous at join points (

14、parametric continuity)Alternately, we can only require that the tangents of the resulting curve be continuous (geometry continuity)The latter gives more flexibility as we need satisfy only two conditions rather than three at each join point19Angel: Interactive Computer Graphics 5E Addison-Wesley 200

15、9ExampleHere the p and q have the same tangents at the ends of the segment but different derivativesGenerate different Hermite curvesThis techniques is usedin drawing applications20Angel: Interactive Computer Graphics 5E Addison-Wesley 2009Higher Dimensional ApproximationsThe techniques for both interpolating and Hermite curves can be used with higher dimensional parametric polynomialsFor