{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Chapter6

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

This preview shows pages 1–12. Sign up to view the full content.

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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
Y X θ θ cos R θ 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 θ θ θ Each point on circle found at increments of ∆θ . Parametric equation of a circle: Nonparametric equations

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 θ θ Parametric equation of a circle with centre at point other than origin:
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 θ R x z R y y x Therefore: Start with original circle on x-y plane, then apply transformation.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Conic Sections – Ellipse or Elliptic Arc π θ θ θ 2 0 where 0 sin cos = = = z b y a x Y X θ a O b
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 changes in their values.
More meaningful to use vectors with geometric significance.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}