Chapter6

Chapter6 - Representation and Manipulation of Curves Ch. 6...

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Representation and Manipulation of Curves – Ch. 6 Objectives: Types of curve equations. Conic sections. Hermite curves. Bezier and B-spline curves. Nonuniform rational B-spline (NURBS) curve. Interpolation curves and Intersection of curves.
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Parametric equations – relate x , y , and z coordinates by a parameter. Nonparametric equations – directly relates x , y , and z coordinates by a function. Types of Curve Equations
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Y X θ cos R sin R R O Types of Curve Equations ( 29 0 , or 0 , 0 2 0 , 0 , sin , cos 2 2 2 2 2 = - ± = = = - + = = = z x R y z R y x z R y R x π Each point on circle found at increments of ∆θ . Parametric equation of a circle: Nonparametric equations
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Conic Sections – Circle or Circular Arc Y X θ R O X c Y c 0 sin cos = + = + = z Y R y X R x c c Parametric equation of a circle with centre at point other than origin:
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Conic Sections – Circle or Circular Arc – Example 6.1 [ ] ( 29 ( 29 [ ] ( 29 ( 29 ( 29 ( 29 ( 29 1 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 0 1 0 0 0 0 90 cos 0 90 sin 0 0 1 0 0 90 sin 0 90 cos 1 0 90 , 1 , 1 , 0 1 0 * * + + = - - - - - = - = x y y x y x y Rot Trans y x T T Derive parametric equation of circle in y-z plane by applying appropriate transformation matrices. Z Y 1 1 X 1 cos 1 * 1 sin 1 * 0 * + = + = + = + = = θ R x z R y y x Therefore: Start with original circle on x-y plane, then apply transformation.
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Conic Sections – Ellipse or Elliptic Arc π θ 2 0 where 0 sin cos = = = z b y a x Y X a O b
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Conic Sections – Hyperbola 1 2 2 2 2 = - b y a x Implicit equation: Y X b a u b y u a x sinh cosh = = Parametric form using hyperbolic functions: Obtained by using identity: 2 sinh , 2 cosh : note 1 sinh cosh 2 2 u u u u e e u e e u u u - - - = + = = - Range of u determined from end points of portion of hyperbola to be represented.
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Conic Sections – Parabola 2 cy x = Reference parabola symmetric about x -axis: Converted to following parametric equations: Y X u y cu x = = 2 Not unique; any convenient parametric equation can be chosen. Range of u determined from end points of portion of parabola to be represented.
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Equations of degree 3 used to represent curves in CAD. Guarantees that two curves of degree 3 can be combined so that their second derivatives are continuous at connection point. Higher degrees work also, but require more computation. Hermite Curves
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Simplest parametric equation of degree 3: Hermite Curves ( 29 ( 29 ( 29 ( 29 [ ] ( 29 1 0 3 3 2 2 1 0 + + + = = u u u u u z u y u x u a a a a P where a i ’s are vector coefficients of parametric equation and are row vectors with x , y , and z components. Change of curve’s shape cannot intuitively be determined from
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Chapter6 - Representation and Manipulation of Curves Ch. 6...

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