Problems
- B-spline Basis Functions
- Compute and plot all basis functions up to degree 2 for knot vector
U = { 0, 1, 2, 3, 4 }.
- Compute and plot all basis functions up to degree 2 for knot vector
U = { 0, 1, 2, 3, 3, 3, 4, 5, 6 }.
- 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.
- Given knot sequences U_{1} = { 0, 0, 1, 1 } and
U_{2} = { 0, 0, 0, 1, 1, 1 }, use hand calculation to
verify that the B-spline basis functions on U_{1} and
U_{2} are identical to the Bézier basis
functions.
- B-spline Curves
- Show that a clamped B-spline curve passes through the first and last
control points. More precisely, show that
C(0) = P_{0} and
C(1) = P_{n} hold.
- 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-p}P_{i} and
the curve must pass through P_{i}.
Why is this so?
- 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.
- 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:
- Why does this proposition hold?
- On which leg does C(u_{i})
lie?
- Consider a clamped cubic B-spline curve defined by seven control
points P_{0}, ..., P_{6} 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.
- Modify the derivative computation method so that it works for
open and closed B-spline curves.
- 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.
- 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.