CS 445 / 645 Introduction to Computer GraphicsACM ElectionsSplines – Old SchoolRepresentations of CurvesSpecifying Curves (hyperlink)Parametric CurvesCubic PolynomialsSlide 8Piecewise Curve SegmentsContinuity of CurvesParametric Cubic CurvesParametric Cubic SplinesCoefficientsSlide 14Slide 15Hermite Cubic SplinesSlide 17Slide 18Hermite Cubic SplineSlide 20Slide 21Slide 22Slide 23Hermite SpecificationSolve Hermite MatrixSpline and Geometry MatricesResulting Hermite Spline EquationDemonstrationSample Hermite CurvesBlending FunctionsHermite Blending FunctionsSlide 32CS 445 / 645Introduction to Computer GraphicsLecture 22Lecture 22Hermite SplinesHermite SplinesLecture 22Lecture 22Hermite SplinesHermite SplinesACM ElectionsShould take 15 minutesShould take 15 minutesTaking place now in OLS 120Taking place now in OLS 120Hurry BackHurry BackI’m showing movies that you can watch after class I’m showing movies that you can watch after class today (or some other day)today (or some other day)Should take 15 minutesShould take 15 minutesTaking place now in OLS 120Taking place now in OLS 120Hurry BackHurry BackI’m showing movies that you can watch after class I’m showing movies that you can watch after class today (or some other day)today (or some other day)Splines – Old SchoolRepresentations of CurvesProblems with series of points used to model a Problems with series of points used to model a curvecurve•Piecewise linear - Does not accurately model a smooth linePiecewise linear - Does not accurately model a smooth line•It’s tediousIt’s tedious•Expensive to manipulate curve because all points must be Expensive to manipulate curve because all points must be repositionedrepositionedInstead, model curve as piecewise-polynomialInstead, model curve as piecewise-polynomial•x = x(t), y = y(t), z = z(t) x = x(t), y = y(t), z = z(t) –where x(), y(), z() are polynomials where x(), y(), z() are polynomials Problems with series of points used to model a Problems with series of points used to model a curvecurve•Piecewise linear - Does not accurately model a smooth linePiecewise linear - Does not accurately model a smooth line•It’s tediousIt’s tedious•Expensive to manipulate curve because all points must be Expensive to manipulate curve because all points must be repositionedrepositionedInstead, model curve as piecewise-polynomialInstead, model curve as piecewise-polynomial•x = x(t), y = y(t), z = z(t) x = x(t), y = y(t), z = z(t) –where x(), y(), z() are polynomials where x(), y(), z() are polynomialsSpecifying Curves (hyperlink)Control PointsControl Points•A set of points that influence the A set of points that influence the curve’s shapecurve’s shapeKnotsKnots•Control points that lie on the curveControl points that lie on the curveInterpolating SplinesInterpolating Splines•Curves that pass through the control Curves that pass through the control points (knots)points (knots)Approximating SplinesApproximating Splines•Control points merely influence shapeControl points merely influence shapeControl PointsControl Points•A set of points that influence the A set of points that influence the curve’s shapecurve’s shapeKnotsKnots•Control points that lie on the curveControl points that lie on the curveInterpolating SplinesInterpolating Splines•Curves that pass through the control Curves that pass through the control points (knots)points (knots)Approximating SplinesApproximating Splines•Control points merely influence shapeControl points merely influence shapeParametric CurvesVery flexible representationVery flexible representationThey are not required to be functionsThey are not required to be functions•They can be multivalued with respect to any dimensionThey can be multivalued with respect to any dimensionVery flexible representationVery flexible representationThey are not required to be functionsThey are not required to be functions•They can be multivalued with respect to any dimensionThey can be multivalued with respect to any dimensionCubic Polynomialsx(t) = ax(t) = axxtt33 + b + bxxtt22 + c + cxxt + dt + dxx•Similarly for y(t) and z(t)Similarly for y(t) and z(t)Let t: (0 <= t <= 1)Let t: (0 <= t <= 1)Let T = [tLet T = [t33 t t22 t 1] t 1]Coefficient Matrix CCoefficient Matrix C Curve: Q(t) = T*CCurve: Q(t) = T*Cx(t) = ax(t) = axxtt33 + b + bxxtt22 + c + cxxt + dt + dxx•Similarly for y(t) and z(t)Similarly for y(t) and z(t)Let t: (0 <= t <= 1)Let t: (0 <= t <= 1)Let T = [tLet T = [t33 t t22 t 1] t 1]Coefficient Matrix CCoefficient Matrix C Curve: Q(t) = T*CCurve: Q(t) = T*C zzyyxxzyxzyxdcdcdcbbbaaattt 123Parametric CurvesHow do we find the tangent to a curve?How do we find the tangent to a curve?•If f(x) = xIf f(x) = x22 – 4 – 4–tangent at (x=3) is f’(x) = 2 (x) – 4 = 2 (3) - 4tangent at (x=3) is f’(x) = 2 (x) – 4 = 2 (3) - 4How do we find the tangent to a curve?How do we find the tangent to a curve?•If f(x) = xIf f(x) = x22 – 4 – 4–tangent at (x=3) is f’(x) = 2 (x) – 4 = 2 (3) - 4tangent at (x=3) is f’(x) = 2 (x) – 4 = 2 (3) - 4Derivative of Q(t) is the tangent vector at t:Derivative of Q(t) is the tangent vector at t:•d/dt Q(t) = Q’(t) = d/dt T * C = [3td/dt Q(t) = Q’(t) = d/dt T * C = [3t22 2t 1 0] * C 2t 1 0] * CDerivative of Q(t) is the tangent vector at t:Derivative of Q(t) is the tangent vector at t:•d/dt Q(t) = Q’(t) = d/dt T * C = [3td/dt Q(t) = Q’(t) = d/dt T * C = [3t22 2t 1 0] * C 2t 1 0] * CPiecewise Curve SegmentsOne curve constructed by connecting many smaller One curve constructed by connecting many smaller segments end-to-endsegments end-to-endContinuity describes the jointContinuity describes the joint•CC11 is tangent continuity (velocity) is tangent continuity (velocity)•CC22 is 2 is 2ndnd derivative continuity (acceleration) derivative continuity (acceleration)One curve constructed by connecting many smaller One curve constructed by connecting many smaller segments end-to-endsegments end-to-endContinuity describes the jointContinuity describes the joint•CC11 is tangent continuity (velocity) is tangent continuity (velocity)•CC22 is 2 is 2ndnd derivative continuity (acceleration) derivative continuity (acceleration)Continuity of CurvesIf direction (but not necessarily magnitude) of tangent If direction (but not necessarily magnitude) of tangent matchesmatches•GG11 geometric continuity geometric
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