Matching functions to pictures

answered 2014-07-01 11:52:40 -0600

Tiffany
656 ●3 ●18 ●34

The easiest way for me to picture these, is to break them up individually. For instance with the equation you are looking at, we have $\vec p(t) = 2\langle\cos(t), \sin(t), 0 \rangle + \langle \cos(t), \sin(t), 0\rangle\sin(22t) + \langle 0,0,1\rangle \cos (22t)$

so if we break that up and multiply out , our first portion is $\langle 2\cos(t), 2\sin(t), 0 \rangle$ from this we can tell it is a circle in the xy plane that has a radius of 2.

next we have $\langle \sin(22t)\cos(t), \sin(t)\sin(22t), 0\rangle$ which is a ellipse in the x coordinate, with a $sin^2$ wave in the y coordinate. This forms that slinky look you see in the picture.

The last portion is the $\langle0,0,\cos(22t)\rangle$ this takes the above two and extends them into the z coordinate following a sped up cosine wave.

By combining them, we see that we start with a circle in the xy plane, we add the slinky feature from part two, and then make it three dimensional following the cosine wave from part three. which all adds up to the top right picture from the in class assignment.