Problems

• B-spline Basis Functions
  1. Compute and plot all basis functions up to degree 2 for knot vector U = { 0, 1, 2, 3, 4 }.
  2. Compute and plot all basis functions up to degree 2 for knot vector U = { 0, 1, 2, 3, 3, 3, 4, 5, 6 }.
  3. Verify the following propositions with convincing arguments:
     • N_i,p(u) is a degree p polynomial.
     • For all i, p and u, N_i,p(u) is non-negative.
  4. Given knot sequences U₁ = { 0, 0, 1, 1 } and U₂ = { 0, 0, 0, 1, 1, 1 }, use hand calculation to verify that the B-spline basis functions on U₁ and U₂ are identical to the Bézier basis functions.

• B-spline Curves
  1. Show that a clamped B-spline curve passes through the first and last control points. More precisely, show that C(0) = P₀ and C(1) = P_n hold.
  2. In the discussion of forcing a B-spline to pass through a control point, we indicated that if we let P_i = P_i-1 = ... = P_i-p+1, the convex hull collapses to a line segment P_i-pP_i and the curve must pass through P_i. Why is this so?
  3. In the discussion of forcing a B-spline to pass a control point, we indicated that a point on the curve that corresponds to a knot may become identical to the collapsed control point. If we let P_i = P_i-1 = ... = P_i-p+1, will you be able to identify the knot u_k such that P(u_k) becomes identical to P_i? Why? Elaborate your finding.
  4. In the discussion of multiple knots, we mentioned that if a knot u_i has multiplicity k-1, where k is the degree of a B-spline curve, then C(u_i) lies on a leg of the control polyline. Based on your understanding of multiple knots, answer the following questions:
     1. Why does this proposition hold?
     2. On which leg does C(u_i) lie?
  5. Consider a clamped cubic B-spline curve defined by seven control points P₀, ..., P₆ and knot vector U = { 0, 0, 0, 0, 2/5, 3/5, 3/5, 1, 1, 1, 1 }. Find its derivative B-spline curve, its new control points and knot vector.
  6. Modify the derivative computation method so that it works for open and closed B-spline curves.
  7. Use B-spline curves of degree 2 and 3 and hand calculation to verify that de Boor's algorithms reduces to de Casteljau's algorithm.
  8. Suppose we have a clamped B-spline curve of degree p defined by n+1 control points and a knot vector of simple knots except for the first and last knots which are of multiplicity p+1. Derive a relation of the number of control points between the given B-spline curve and the number of control points of its Bézier curve segments.