Spring 1999 CSE 5256 Quiz 1
This quiz must be completed and returned by 5:00 pm, Monday, March 15.
- 1.
- (5 pts) The equation
,
defines the first quadrant of the unit circle.
Show that
is an alternative representation of the quarter circle.
- 2.
- (5 pts) Evaluate the above trigonometric form of the circle at
and the rational function form at t=1/2. Explain the results.
- 3.
- (10 pts) Find the velocity vectors C'0(u) and C'1(t).
What is the magnitude of C'0(u)? What is C'1(0) and C'1(1)?
What can you deduce about the speed of a particle traveling along the
two parameterizations of the (quarter) circle?
- 4.
- (10 pts) Consider the (not unique) parameterization of the sphere
- Find a unit length vector normal to the sphere at
.
- Are there points where this parameterization of the sphere does not
provide a well-defined normal vector?
- 5.
- (10 pts) Consider the xy planar cubic Bézier curve C(u) given by the
control points:
Compute C(1/3) using the de Casteljau algorithm.
- 6.
- (10 pts) It is possible to define a Bézier curve on a parameter
interval other than
.
Given the Bézier curve
where
are the Bernstein polynomials and Pi are the control points,
let
where
u=(v-a)/(b-a). Derive the reparameterized curve
- 7.
- (10 pts) Rational Bézier curves are defined by appending a (homogeneous)
weight wi to each point Pi and performing a (homogeneous) divide
Compute the point C(2/3) on the rational cubic Bézier curve with control points:
and corresponding weights
- 8.
- (5 pts) Find the rational Bézier representation of the circular arc
in the second quadrant, i.e., determine Pi and wi.
- 9.
- (10 pts) Consider the nonrational Bézier surface
defined by the
control net:
- Sketch the control net and the surface.
- Use the de Casteljau algorithm to compute the surface point
.
- 10.
- (10 pts) Find the bilinear surface that defined by the points
- Show how to efficiently locate a point
on the surface
given a known point
.
- 11.
- (30 pts) Let a cubic B-spline be defined by
and the knot vector
.
- What are the Greville abscissa for this knot vector?
- Assume the control points are:
and sketch the curve.
- Insert the knot u=1/4 twice to derive the point C(1/4).
- Construct the B-spline basis for this knot sequence using the
de Boor formula
- If P2 is moved, on what subinterval of
is C(u) affected?
- If P5 is moved, on what subinterval of
is C(u) affected?
- 12.
- (10 pts) Determine the functional representation of the B-spline basis
for the knot sequence
.
Explain the results.
William Shoaff
1999-02-22