Lecture 9 ENGR3030U Zeid Ch 8

Lecture 9 ENGR3030U Zeid Ch 8 - NURBS Ch. 8 Objectives:...

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ENGR3030U George Platanitis, Ph.D, P.Eng. Fall 2009 NURBS – Ch. 8 Objectives: Modelling and basics (knot vectors and weights). Curves, lines, arcs and circles. Surfaces, including bilinear and ruled.
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ENGR3030U George Platanitis, Ph.D, P.Eng. Fall 2009 NURBS – Nonuniform rational B-splines – used almost exclusively by CAD/CAM systems. Provides unified approach to formulate and represent curves and surfaces. Developed in 1970’s, with Boeing’s TigerCAD system. Integrated representations of B-spline curves with rational Bezier curves. Advantages – unified canonical representation of synthetic and analytical curves and surfaces, especially for exchange standards (IGES and STEP). Also are intuitive and flexible to use, related algorithms accurate, and enhance manufacturing and machining accuracy and speed. Introduction
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ENGR3030U George Platanitis, Ph.D, P.Eng. Fall 2009 Disadvantages – simple curves (arcs, circles, and conics) verbose. Ex., need seven parameters to define circle traditionally (centre (3), circular plane normal (3), and radius (1)). NURBS requires 38. Information may be lost for simple shapes (identifying cylinder as a hole to be drilled or bored instead of a surface to be milled). Extra flexibility may cause ill-behaved NURBS. Some algorithms (ex., surface/surface intersection) work better for non-NURBS representation. Introduction
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ENGR3030U George Platanitis, Ph.D, P.Eng. Fall 2009 Rational curve – defined by algebraic ratio of two polynomials. Nonrational curve – defined by one polynomial. Rational curves draw their theories from projective geometry and are invariant under projective transformation. Basics
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ENGR3030U George Platanitis, Ph.D, P.Eng. Fall 2009 We define homogeneous space (4D space) and homogeneous coordinate. In E 3 , point defined by ( x , y , z ) represented by ( x* , y* , z* , h ), where h is scalar factor. For h = 1, we get point in E 3 . Relationship between coordinates obtained by normalizing h to 1. Thus, Basics h z z h y y h x x * , * , * = = =
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ENGR3030U George Platanitis, Ph.D, P.Eng. Fall 2009 •Rational B-spline curve defined by n + 1 control points P i given by: Basics ( 29 ( 29 ( 29 max 0 , 0 u u u R u n i k i i = = P P R i,k ( u ) are rational B-spline basis functions defined by: ( 29 ( 29 ( 29 = = n i k i i k i i k i u N w u N w u R 0 , , , N i,k ( u ) are B-spline basis functions and w i are weights associated with control points of rational B-spline curve.
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ENGR3030U George Platanitis, Ph.D, P.Eng. Fall 2009 Rational B-spline curves have similar properties to their nonrational counterparts: Basics Partition of unity: ( 29 1 0 , = = n i k i u R Positivity: ( 29 0 all if , 0 , i k i w u R Local support: ( 29 [ ] 1 , , if , 0 + + = k i i k i u u u u R Continuity: ( 29 ( 29 able differenti ly continuous times 2 is , - k u R k i
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Lecture 9 ENGR3030U Zeid Ch 8 - NURBS Ch. 8 Objectives:...

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