CS 445/645 Fall 2001Specifying CurvesPiecewise Curve SegmentsParametric Cubic CurvesParametric Cubic SplinesCoefficientsHermite Cubic SplinesSlide 8Slide 9Hermite Cubic SplineSlide 11Slide 12Slide 13Slide 14Hermite SpecificationSolve Hermite MatrixSpline and Geometry MatricesResulting Hermite Spline EquationDemonstrationSample Hermite CurvesBlending FunctionsHermite Blending FunctionsBézier CurvesSlide 24Bézier vs. HermiteSlide 26Bézier Basis and Geometry MatricesBézier Blending FunctionsSlide 29Slide 30Slide 31Slide 32Slide 33CS 445/645Fall 2001Hermite and Bézier SplinesSpecifying Curves•Control Points–A set of points that influence the curve’s shape•Knots–Control points that lie on the curve•Interpolating Splines–Curves that pass through the control points (knots)•Approximating Splines–Control points merely influence shapePiecewise Curve Segments•One curve constructed by connecting many smaller segments end-to-end•Continuity describes the jointParametric Cubic Curves•In order to assure C2 continuity, curves must be of at least degree 3•Here is the parametric definition of a spline in two dimensionshgtftetydctbtatx2323Parametric Cubic Splines•Can represent this as a matrix toohgtftetydctbtatx2323 hgfedcbatttyx 123Coefficients•So how do we select the coefficients?–[a b c d] and [e f g h] must satisfy the constraints defined by the knots and the continuity conditionsHermite Cubic Splines•An example of knot and continuity constraintsHermite Cubic Splines•One cubic curve for each dimension•A curve constrained to x/y-plane has two curves: dcbatttdctbtattfx1)(2323 hgfettthgtftettfy1)(2323Hermite Cubic Splines•A 2-D Hermite Cubic Spline is defined by eight parameters: a, b, c, d, e, f, g, h•How do we convert the intuitive endpoint constraints into these eight parameters?•We know:–(x, y) position at t = 0, p1–(x, y) position at t = 1, p2–(x, y) derivative at t = 0, dp/dt–(x, y) derivative at t = 1, dp/dtHermite Cubic Spline•We know:–(x, y) position at t = 0, p1 xpdfdcbadcbafxx12323)0(1000000)0( yphfhgfehgfefyy12323)0(1000000)0(Hermite Cubic Spline•We know:–(x, y) position at t = 1, p2 xpdcbafdcbadcbafxx22323)1(1111111)1( yphgfefhgfehgfefyy22323)1(1111111)1(Hermite Cubic Splines•So far we have four equations, but we have eight unknowns•Use the derivatives dcbatttfcbtattfdctbtattfxxx0123)(23)()(2223 hgfetttfgftettfhgtftettfyyy0123)(23)()(2223Hermite Cubic Spline•We know:–(x, y) derivative at t = 0, dp/dt dtdpcfdcbacbafxxx122)0(0102030203)0( dtdpgfhgfegfefyyy122)0(0102030203)0(Hermite Cubic Spline•We know:–(x, y) derivative at t = 1, dp/dt dtdpcbafdcbacbafxxx12223)1(0112131213)1( dtdpgfefhgfegfefyyy12223)1(0112131213)1(Hermite Specification•Matrix equation for Hermite Curvet = 0t = 1t = 0t = 1t3 t2 t1 t0p1p2r p1r p2dtdpdtdpppdtdpdtdppphgfedcbayyyyxxxx212111210123010011111000Solve Hermite Matrixhgfedcbad td pd tdpppdtd pdtd pppyyyyxxxx2121112110123010011111000Spline and Geometry Matriceshgfedcbadtd pd tdpppd tdpd tdpppyyyyxxxx2121112110001010012331122MHermite GHermiteResulting Hermite Spline EquationDemonstration•HermiteSample Hermite CurvesBlending Functions•By multiplying first two components, you have four functions of ‘t’ that blend the four control parametersHermite Blending Functions•If you plot the blending functions on the parameter ‘t’Bézier Curves•Similar to Hermite, but more intuitive definition of endpoint derivatives•Four control points, two of which are knotsBézier Curves•The derivative values of the Bezier Curve at the knots are dependent on the adjacent points•The scalar 3 was selected just for this curveBézier vs. Hermite•We can write our Bezier in terms of Hermite–Note this is just matrix form of previous equationsBézier vs. Hermite•Now substitute this in for previous HermiteBézier Basis and Geometry Matrices•Matrix Form•But why is MBezier a good basis matrix?Bézier Blending Functions•Look at the blending functions•This family of polynomials is calledorder-3 Bernstein Polynomials–C(3, k) tk (1-t)3-k; 0<= k <= 3–They are all positive in interval [0,1]–Their sum is equal to 1Bézier Blending Functions•Thus, every point on curve is linear combination of the control points•The weights of the combination are all positive•The sum of the weights is 1•Therefore, the curve is a convex combination of the control pointsBézier Curves•Will always remain within bounding region defined by control pointsBézier Curves•Can form an approximating spline for n points–p(u) = nkpkBEZk,n(u), 0<=u<=1•BEZk,n(u)=C(n,k)uk(1-u)n-k•Alternatively, piecewise combination of lower-degree polysBézier Curves•Can model interesting shapes by repeating points–Three in a row = straight line–First = last = closed curve–Two in a row = higher weightBézier
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