The problem

You will compute the shadows of a helical space curve

$$\vec{r} (t) = \left[ \begin{array}{c} \cos(4 \pi t) \\ \sin(4 \pi t) \\ t \end{array}\right]\ \ \ \ \ 0 \leq t \leq 5.$$

onto the projection planes:

1.

[Easy] z = -1,

2.

[Easy] x = 2,

3.

[Challenging] x + y + 10 z = -10,

4.

[Challenging] x+y+z=-2.

You must find equations for these curves on the planes, and plot the shadows on the projection plane. You may assume that the light source casting the shadow is infinitely far away, that the helix is between the light source and the plane, and that the light is traveling in a direction normal to the plane. You may have to make some additional assumptions to generate the two-dimensional equations and plots. Be sure to explain what assumptions you must make.