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Advanced Computer Graphics (Spring 2005) COMS 4162, Lecture 13: NURBs, Spline Surfaces Ravi Ramamoorthi http://www.cs.columbia.edu/~cs4162 Some material courtesy Szymon Rusinkiewicz

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To Do / Motivation Questions?? One of written questions in assignment deals with material in this lecture Otherwise, mainly intended for completeness, since splines are very common modeling tool This lecture discusses extensions to rational (beyond polynomial) splines, NURBs and surfaces

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Outline Rational Splines Quadratic rational splines NURBs (briefly) Parametric patches (briefly)

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Parametric Polynomial Curves A parametric polynomial curve of order n: Advantages of polynomial curves Easy to compute Infinitely differentiable everywhere

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Rational Splines Can’t represent certain shapes (e.g. circles) with piecewise polynomials Wider class of functions: rational functions Ratio of polynomials Can represent any quadric (e.g. circles) exactly Mathematical trick: homogeneous coordinates Ratio of 2 polynomials in 3D equivalent to single polynomial in 4D

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Rational Splines Example: creating a circular arc with 3 control points (0,0) (0,1)(1,1) Polynomial spline: parabolic arc (0,0;1) (1,1;1) Rational spline: circular arc (???)

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Standard deCasteljau Standard deCasteljau A B

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Rational deCasteljau A B

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Homogeneous deCasteljau Non-rational splines simply have all weights set to 1 instead

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Outline Rational Splines Quadratic rational splines NURBs (briefly) Parametric patches (briefly)

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Evaluation: Rational Splines Add an extra weight coordinate Multiply standard coords of control point by weight (essentially the same as the use of homogeneous coords) Apply standard deCasteljau or other evaluation alg. Divide by final value of weight coordinate Essentially the same as perspective division/dehomogenize Rational because of final division: Rational polynomial at the end (ratio of two polynomials)

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Quadratic Bezier Curve Example: quarter-circle arc, weight midpoint Smaller w: pulled away from middle control point Larger w: pulled towards middle control point (0,0;1) (0,1;1)(1,1;1) Polynomial spline: parabolic arc (0,0;1) (1,1;1) Rational spline: result depends on w (0,1;w)

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Quadratic Bezier Curve General Bezier Formula for quadratic? (0,0;1) (0,1;1)(1,1;1) Polynomial spline: parabolic arc (0,0;1) (1,1;1) Rational spline: result depends on w (0,1;w)

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Dependence of curve on w For w very large, curve pulled toward middle control point, get a section of a hyperbola For w = 1 (standard spline), standard parabola For w < 1, positive, curve moves away from middle control point For w = 0, curve becomes a straight line When is curve part of a circle? (homework)

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Mid-Point (t = ½ ?)

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W very large (0,0;1) (0,1;>>1)(1,1;1)

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W = 1 (non-rational spline) (0,0;1) (0,1;1)(1,1;1) Polynomial spline: parabolic arc

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W < 1 (0,0;1) (1,1;1) Rational spline: result depends on w (0,1;w)

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W = 0 (0,0;1) (1,1;1) Straight Line (0,1;w)

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Outline Rational Splines Quadratic rational splines NURBs (briefly) Parametric patches (briefly)

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NURBS Non-uniform (vary time interval per segment) Rational B-Splines Can model a wide class of curves and surfaces Same convenient properties of B-Splines Still widely used in CAD systems

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Polar Forms: Cubic Bspline Curve -2 –1 0 –1 0 1 0 1 2 1 2 3 For Uniform B-splines, uniform knot vector (below) For non-uniform, only require non-decreasing, not necessarily uniform (can be arbitrary) Uniform knot vector: -2, -1, 0, 1, 2,3 Labels correspond to this

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NURBS

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Outline Rational Splines Quadratic rational splines NURBs (briefly) Parametric patches (briefly)

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Parametric Patches Each patch is defined by blending control points FvDFH Figure 11.44

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Parametric Patches Point Q(u,v) on the patch is the tensor product of curves defined by the control points Watt Figure 6.21 Q(u,v) Q(0,0) Q(1,0) Q(0,1) Q(1,1)

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Parametric Bicubic Patches Point Q(u,v) defined by combining control points with polynomial blending functions: Where M is a matrix describing the blending functions for a parametric cubic curve (e.g., Bezier, B-spline, etc.)

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Summary Splines still used commonly for modeling Start with simple spline curves (Bezier, uniform non- rational B-splines) Discussed extension to rational curves (add homogeneous coordinate for rational polynomial) Brief discussion of NURBs: widely used Brief discussion of Parametric patches for modeling surfaces (rather than curves)

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