ME325NotesWeek7

ME325NotesWeek7 - Computational Geometry for CAD/CAM, ME325...

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1 Computational Geometry for CAD/CAM, ME325 Pete Woytowitz Class Notes, Week 7 &±²³´µ¶·¸¹ º»»¼ ½ ¾¿¸¿´ ÀÁ ±³¸±Ãµ¸Ä
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2 Curves (Continued) Advantages of Bezier Curves Convex Hull Property - A “rubber band” stretched around the control points (or a balloon in 3D) determines the bounding area/volume of the curve Truncating, subdividing and rendering can be done using efficient recursive algorithm (de Casteljau Algorithm, Appendix F) Can be used as a basis for more useful/flexible curves known as B-Splines and NURBS (Non-Uniform Rational B-Splines) ± ² ³ ´µ¶·¸ ¹·º »µ¼ ½µ·¾¿¼À¶
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3 Curves : Bezier Curves ! ± ² ³ Derivative of Bezier Curves Bezier curve: p (u) = Σ n i =0 B i,n (u) P i , u [0,1] (6.16) Differentiating this curve provides general equation to determine derivative at any location, u (Eqn. 6.2.4) Using Eqn. 6.2.4 one can show that at the endpoints of the curve (u=0, or, u=1) one has d p (u)/du =n( P 1 - P 0 ) at u=0 and d p (u)/du =n( P n - P n-1 ) at u=1 where n=degree of Bezier curve, P 0 is first and P n is last control point p’ (0)=3( P 1 - P 0 ) p’ (1)=3( P 3 - P 2 )
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4 Curves : Bezier Curves ! ± ² ³ The de Casteljau Algorithm for evaluating Bezier curve Illustrate using cubic Bezier, same algorithm holds for any order, but, number of evaluations must be increased appropriately Cubic Bezier: p (u i ) = (1-u i ) 3 P 0 + 3u i (1-u i ) 2 P 1 + 3u i 2 (1-u i ) P 2 + u i 3
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This note was uploaded on 05/01/2008 for the course MECH 202 taught by Professor Ardema during the Winter '07 term at Santa Clara.

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ME325NotesWeek7 - Computational Geometry for CAD/CAM, ME325...

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