Chapter 3 Curves 2

B Spline Curve

-Powerful generalization of Bezier curves, a way to generate curves defined by data points

What are B spline curves used for?

Creating b-spline surfaces, which are used for automobile surface modeling

B Spline curve advantages

-Each control point has its own blending function, so it affects curve only over a limited range of parameters (aka local control)

-Degree can be separated from number of control data points

What is one difference between the parameter u in Bezier and B spline curves?

In bezier u can only go from 0 to 1, in b spline u’s range is not limited to 0 to 1

What’s the relationship between B-spline order k and degree?

Degree = k - 1

What information is needed to define a B-spline curve?

-Order k

-Control points

-Knot vector

What is a knot vector in a B-spline?

A sequence of parameter values [u0, u1, u2, …, un+k]

How is the number of knots determined?

Number of knots = n + k +1

The nonperiodic B-spline curve moves through what control points, if any?

The first and final control points

What are b-spline curves tangent to?

First and last segment of the polygon

What happens to the B-spline curve when its degree increases?

Curve tightens

Under what condition does the B-spline become a Bezier curve?

k = n + 1

What is a Hermite Cubic Spline?

A cubic parametric curve defined using:
-Two endpoint control points P0 and P1
-Tangent vectors P’0 and P’1 at those endpoints

What continuity condition is applied between consecutive cubic spline segments?

C² continuity