Lecture 6 and 7 ENGR3030U Zeid Ch 6

# Lecture 6 and 7 ENGR3030U Zeid Ch 6 - Curves Ch. 6...

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ENGR3030U George Platanitis, Ph.D, P.Eng. Fall 2009 Curves – Ch. 6 Objectives: Geometric modelling and modelling entities. Implicit and parametric equations of curves. Curve properties. Analytic and synthetic curves. Curve manipulations.

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ENGR3030U George Platanitis, Ph.D, P.Eng. Fall 2009 Why study geometric modelling? Provides good understanding of terminology in CAD/CAM field and CAD/CAM systems. Users can intelligently decide which entity to use in particular model to meet geometric requirements. Users can interpret unexpected results from CAD/CAM system. Those in decision-making process and evaluation of CAD/CAM systems better equipped with evaluation criteria. Introduction
ENGR3030U George Platanitis, Ph.D, P.Eng. Fall 2009 Analytic – includes points, lines, arcs, circles, fillets, chamfers, and conics (ellipses, parabolas, and hyperbolas). Described by analytic equations Synthetic – includes splines (cubic and B-spline) and Bezier curves. Described by sets of data (control) points. Parametric polynomials usually fit control points. Synthetic curves allow designer ability to control shape of curve by changing position of control points. Curve Entities

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ENGR3030U George Platanitis, Ph.D, P.Eng. Fall 2009 Curve Entities
ENGR3030U George Platanitis, Ph.D, P.Eng. Fall 2009 Curve Entities

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ENGR3030U George Platanitis, Ph.D, P.Eng. Fall 2009 Curve Entities
ENGR3030U George Platanitis, Ph.D, P.Eng. Fall 2009 Curve Entities

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ENGR3030U George Platanitis, Ph.D, P.Eng. Fall 2009 Curve Entities
ENGR3030U George Platanitis, Ph.D, P.Eng. Fall 2009 Curves describing engineering objects generally smooth and well-behaved. Not every form of curve equations efficient for CAD/CAM systems because of computation or programming problems. Example – divide by zero in computing curve slope; inadequate form of equations of intersecting curves to numerically solve. Curve Representation

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ENGR3030U George Platanitis, Ph.D, P.Eng. Fall 2009 Explicit nonparametric representation of general 3D curve: Curve Representation [ ] ( 29 ( 29 [ ] T T x g x f x z y x = = P P is position vector of point P, and above representation is one-to-one. Cannot represent closed curves (ex., circles) or multivalued curves (ex., parabolas). Implicit nonparametric representation solves this, given by intersection of two surfaces: ( 29 ( 29 0 , , 0 , , = = z y x G z y x F Requires root solving for y and z given x .
ENGR3030U George Platanitis, Ph.D, P.Eng. Fall 2009 Disadvantages of nonparametric representations: Ill-defined conditions such as vertical slope leads to difficulties in computation and programming Shapes of engineering objects intrinsically independent of coordinate system; shape determined by relationships between data points themselves Extensive computation to represent curve as series of points or line segments Curve Representation

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ENGR3030U George Platanitis, Ph.D, P.Eng. Fall 2009 Solution? Use parametric form.
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## Lecture 6 and 7 ENGR3030U Zeid Ch 6 - Curves Ch. 6...

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