Chapter 3 curves 1

What are examples of analytical curve entities?

-Points

-Lines

-Arcs

-Circles

-Fillets

-Chamfers

-Conics

How is a conic curve generated?

When a plane intersects a cone

What’s a non-parametric representation of a curve?

A curve can be represented using an equation such as:

F(x,y,z) = 0

G(x,y,z) 0

Equations must be solved to find roots y, z, if a certain value of X is given

What’s a parametric representation of a curve?

Each point on the curve is expressed as a function of parameter u:

P(u) = [x(u) y(u) z(u)]^T

Where, P(bolded) is position vector of point P

What’s the parametric equation of a circle of radius R in the xy-plane?

x = Rcos(theta)

y = Rsin(theta)

z = 0

Why use parametric representation?

-Easily defines closed and multi-valued functions

-Replaces slopes with tangent vectors

-Uses polynomial functions rather then equations involving roots

-Well suited for computation and display

What are disadvantages of non-parametric curve equations?

-Vertical / near-vertical slopes can become infinite or very large, making computation difficult

-Curve shape can depend on coordinate system, when it should depend on relationships between data points

What are the 4 synthetic curbes?

-Hermite cubic spline (cubic spline)

-Bezier

-B-spline

-NURBS curves

What is a Bezier curve?

A synthetic curve defined using controls points and Bernstein polynomials

What is the degree of a Bezier curve?

Degree = n

What is the tangent relationship at the ends of a Bézier curve?

The curve is tangent to the 1st and final segments of the control polygon

How can the shape of a Bezier curve be changed?

-Moving / pulling one or more vertices of the characteristics polygon

-Increasing the number of coincident points at a vertex

How can a Bezier curve be made closed?

By making the starting and ending points coindcide

What is a Bezier curve enclosed by?

It is enclosed within the convex hull by the vertices of the polygon

Bezier curve disadvantages

-Will not pass through control points

-No local control of the curve: shape of the curve changes if one control points is changed