cg17_2013 - Curves and Surfaces 2 Computer Graphics Lecture...

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Curves and Surfaces 2 Computer Graphics Lecture 17 Taku Komura
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Today More about Bezier and Bsplines ■ de Casteljau’s algorithm ■ BSpline : General form ■ de Boor’s algorithm ■ Knot insertion – NURBS – Subdivision Surface
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De Casteljau’s Algorithm • A method to evaluate (sample points in) or draw the Bezier curve • The Bezier curve of any degree can be handled • A precise way to evaluate the curves
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de Casteljau’s Algorithm
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Why does this result in the polynomial?
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Why do we need this? • The explicit representation (monomial form) that I presented last week can result in some instability • Say the control points are randomly changed for 0.001. • The curve computed by the de Casteljau’s algorithm stays almost the same. • The curve by the polynomial basis form can deviate from the original curve if the degree is high
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Connecting many Bezier Patches in the polynomial form • The same story applies to surfaces • The degree of surface can easily go high, as they are the multiplication of two curves • Bicubic → 6 • The error of 16 control points will be accumulated
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Today – More about Bezier and Bsplines ■ de Casteljau’s algorithm ■ BSpline ± General form ■ de Boor’s algorithm ■ Knot insertion – NURBS – Subdivision Surface
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* ( )spline is a parametric curve composed of a linear combination of basis )±splines > Q±W < L ± Q)²±ォ±R ² the control points 7TUZY ± subdivide the domain of the )± spline curve into a set of knot spans Bt i ² t i³´ µ ;he )±splines can be defined by B-Spline : A General Form
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Bspline Basis
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Bspline Basis (2)
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±²³±²³´²²µ 3ecture ¶ 7roducing (urves by )·:plines • ;he basis functions are multiplied to the control points and to define arbitrary curves ¸ t m¹± ²
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Knots • The knots produce a vector that defines the domain of the curve • The knots must be in the increasing order • But not necessarily uniform • If uniformly sampled and the degree is 3 uniform cubic bspline ± ² ³ ´ µ · ¸ ¹ º ²± ²² ²³ ²´ knots
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Knots Here is an example of non-uniform knots http://i33www.ira.uka.de/applets/mocca/html/noplugin/BSplineBasis/AppBSplineBasis/index.html
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