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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 $C_0(u) = (\cos(u),\,\sin(u))$, $0\leq u \leq \pi/2$ defines the first quadrant of the unit circle. Show that

\begin{displaymath}C_1(t):\:x(t) = \frac{1-t^2}{1+t^2},\,y(t) = \frac{2t}{1+t^2},\,\quad 0\leq t \leq 1\end{displaymath}

is an alternative representation of the quarter circle.
2.
(5 pts) Evaluate the above trigonometric form of the circle at $u=\pi/4$ 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

\begin{eqnarray*}x(u,\,v) & = & \sin(u)\cos (v) \\
y(u,\,v) & = & \sin(u)\sin ...
... = & \cos(u) (v) \quad 0\leq u \leq \pi,\,0\leq v \leq 2\pi.\\
\end{eqnarray*}


5.
(10 pts) Consider the xy planar cubic Bézier curve C(u) given by the control points:

\begin{displaymath}P_0=(0,\,6),\,P_1=(3,\,6),\,P_2=(6,\,3),\,P_3=(6,\,0).\end{displaymath}

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 $[0,\,1]$. Given the Bézier curve

\begin{displaymath}C(u) = \sum_{0}^{n} B_i^n(u) P_i\quad u\in [0,\,1]\end{displaymath}

where

\begin{displaymath}B_i^n(u) = {n \choose i} (1-u)^{n-i}u^i\end{displaymath}

are the Bernstein polynomials and Pi are the control points, let $v = [a,\,b]$ where u=(v-a)/(b-a). Derive the reparameterized curve

\begin{displaymath}C(v) = \frac{1}{(b-a)^n}\sum_{0}^{n} {n \choose i} (b-v)^{n-1}(v-a)^i P_i.\end{displaymath}

7.
(10 pts) Rational Bézier curves are defined by appending a (homogeneous) weight wi to each point Pi and performing a (homogeneous) divide

\begin{displaymath}C(u) = \frac{\sum_{0}^{n} B_i^n(u) P_i}{\sum_{0}^{n} B_i^n(u) w_i}\quad u\in [0,\,1].\end{displaymath}

Compute the point C(2/3) on the rational cubic Bézier curve with control points:

\begin{displaymath}P_0=(0,\,6),\,P_1=(3,\,6),\,P_2=(6,\,3),\,P_3=(6,\,0)\end{displaymath}

and corresponding weights

\begin{displaymath}w_0=4,\,w_1=1,\,w_2=1,\,w_3=4.\end{displaymath}

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 $S(u,\,v)$ defined by the control net:

\begin{displaymath}\{P_{i,0}\} = \{(0,\,0,\,0),\,(3,\,0,\,3),\,(6,\,0,\,3),\,(9,\,0,\,0)\},\end{displaymath}


\begin{displaymath}\{P_{i,1}\} = \{(0,\,2,\,2),\,(3,\,2,\,5),\,(6,\,2,\,5),\,(9,\,2,\,2)\},\end{displaymath}


\begin{displaymath}\{P_{i,2}\} = \{(0,\,4,\,0),\,(3,\,4,\,3),\,(6,\,4,\,3),\,(9,\,4,\,0)\}.\end{displaymath}

10.
(10 pts) Find the bilinear surface that defined by the points

\begin{displaymath}P_0=(0,\,6,\,2),\,P_1=(3,\,6,\,5),\,P_2=(6,\,3,\,4),\,P_3=(6,\,0,\,6)\end{displaymath}

11.
(30 pts) Let a cubic B-spline be defined by

\begin{displaymath}C(u) = \sum_{0}^{7}N_i^{3}(u)P_i\end{displaymath}

and the knot vector $\langle 0,\,0,\,0,\,0,\,1/4,\,1/4,\,2/3,\,3/4,\,1,\,1,\,1,\,1\rangle$.
12.
(10 pts) Determine the functional representation of the B-spline basis for the knot sequence $\langle 0,\,0,\,0,\,0,\,0,\,1,\,1,\,1,\,1,\,1\rangle$. Explain the results.

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William Shoaff
1999-02-22