Quiz #5

Question #1: If I have a vector $v = (v_x, v_y, v_z)$ then $(r v_x/\ell, r v_y/\ell, r v_z/\ell)$ where $\ell = \sqrt{ v_x^2 + v_y^2 + v_z^2 }$ and with $-1/2 \leq r \leq 1/2$ is the parametric equation for a line segment of length $1$ which passes through the origin in the direction of $v$.

Let $c(t) = (x(t), y(t), z(t))$ be a curve in three dimensions. If you take the cross product of the vector from the origin to $c(t)$ with the tangent vector of $c(t)$, then (most of the time) you have a vector which is perpendicular to the curve. There are some rare exceptions when the distance vector of the curve is parallel to the tangent vector.

Write a function ribonify which accepts a curve $c(t) = (x(t), y(t), z(t))$ and returns the equation of a surface such that for each point $c(t_0)$ on the curve, the surface in the range $-1/2 \leq r \leq 1/2$ is a line segment of length $1$, centered at $c(t_0)$, which is perpendicular to the curve and the vector from the origin to $c(t_0)$.

If your program is correct, you should be able to reproduce the example below.

> with(plots);
> swish:=[30*t*(2*Pi-t)*cos(3*t)/(2*Pi)^2,30*t*(2*Pi-t)*sin(3*t)/(2*Pi)^2,10*cos(t/2)];
> spacecurve(swish,t=0..2*Pi);

> plot3d(ribonify(swish), r=-0.5..0.5,t=0..2*Pi,grid=[3,100],scaling=constrained);

Give the example of a curve $c_1(t) = (x_1(t), y_1(t), z_1(t))$ and an interval for a range of $t$ ...
(a) where the curve is non-zero but ribonify is undefined everywhere in that interval
(b) where the curve is non-zero but ribonify is defined except at a single point or a few points in the interval

Explain your answer clearly.

NOTE THAT THE PENALTY FOR HANDING THIS IS LATE IS QUITE STRICT. I WANT YOU TO DO THIS IN THE ALOTTED TIME. You should be able to complete this quiz within the class time. If you finish after the class time your overall grade for this assignment will be reduced by 20% per hour (or part thereof). Make sure that your file is uploaded by 12:30pm. Upload your worksheet to the course moodle.

You are expected to work alone on this assignment. Any indication of academic dishonesty will result in a $0$ for the assignment and possible higher penalties.