1.

Evaluate the curve in Example 10.2 at the midpoint of [ u ₃, u ₄] by sketching the polygon, intermediate control points, knot sequence, and the spans involved in each step of the de Boor algorithm. There is no need to compute numbers.

2.

What is the multiplicity vector for the following knot sequence

If this knot sequence belongs to a quadratic curve, how many segments are there? How many are there if the knot sequence belongs to a cubic curve?

3.

What is the derivative at u = 1.5 of the B-spline curve from Example 10.5?

4.

What is the derivative of the same curve at u = 0?

5.

Consider the knot sequence 0, 0, 3, 4, 6, 6. Sketch the B-spline N ² ₂. (Hint: First, find the Greville abscissae.)

6.

Over the knot sequence 0, 0, 0, 3, 4, 6, 6, 6 sketch N ³ ₁.

7.

Using the knot sequence from Exercise 5, write the function y = 3 u as a quadratic B-spline function.

8.

Write the function y = 3 u as a cubic B-spline function with knot sequence 0, 0, 0, 3, 4, 6, 6, 6.

9.

The first and second derivative formulas for a B-spline curve in (10.11) and (10.13) follow a particular pattern. In the same manner, write down the third derivative. How would this be implemented via the de Boor algorithm?

10.

List the condition(s) for...