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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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* &( ')splinJLe is FHa pFHarFHamJLetriHJc HJcurvJLe HJcomposJLeIKd oKMf FHa linJLeFHar HJcomGIbinFHation oKMf GIbFHasis ')°splinJLes > Q±VSW < L ± Q)²±ォ±UR ° tOhJLe HJcontrol points 7TUZZZYY% ° suGIbIKdiviIKdJLe tOhJLe IKdomFHain oKMf tOhJLe ')° splinJLe HJcurvJLe into FHa sJLet oKMf knot spFHans @@Bt i ± t i²³ ´ 99;OhJLe ')°splinJLes HJcFHan GIbJLe IKdJLeKMfinJLeIKd GIby B-Spline : A General Form
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Bspline Basis
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Bspline Basis (2)
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³µ¶³µ¶·µµ# 113JLeHJcturJLe 557roIKduHJcinNg (*(urvJLes GIby ')°88:plinJLes • 99;OhJLe GIbFHasis KMfunHJctions FHarJLe multipliJLeIKd to tOhJLe HJcontrol points FHanIKd to IKdJLeKMfinJLe FHarGIbitrFHary HJcurvJLes ¸ t m²³ µ
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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 µ ³ · ¸ ¹ ! " # $ ³µ ³³ ³· ³¸ 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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Some Terms • Order [% YXU QZ]LRUW RV MSRQYWR\ URYQYX YXKQY KQVVUMSY YXU XKQ]U\UNT [\KQ\ZU • Degree [°± ²YXU LRKQXYX VZQMSYYRQX KQWU UR\^_QR]YKQ\X RV NTUWWUU [°±³ Control points 3i ± Y("#"´µ µ] ° Knots : YZ , ( Z("#"´µ µ Q ) An important rule : n m = k The domain of function Y [°± Y Y ]¶± – Below, k = 4, m = 9, domain, Y · Y Y ±´ ¸ t m²³ µ
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Clamped Bsplines • The first and last knot values are repeated with multiplicity equal to the order (= degree + 1) • The end points pass the control point • For cubic Bsplines, the multiplicity of the first / last knots must be 4 (repeated four times)
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