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Today while avoiding studying things I will actually be evaluated on I was pondering different things I could try to parameterize. Having no formal exposure to parameterization these types of tasks are fun to see if I can figure out. General objects like cones and cylinders are easy enough, but after centering a cylinder on all three different coordinate axis I thought,

How would I parameterize the unit cylinder centered on the line $$\vec{v} = <x,y,z> = <t,t,t>, \text{ for } -\infty \leq t \leq \infty \text{ ...?}$$

I played around for a while, first just trying to parameterize a unit circle centered on the line at the origin. Once I arrived at that it was easy to turn into a cylinder.

Attached below is two photo's of the image for the parameterization I came up with and the explicit parameterization I used. Of course as shown it is only a segment of a cylinder, I bounded it so the line I was attempting to center on would be visible.

enter image description here enter image description here

My question / the reason I am posting is:

(1)Does anyone have a cleaner parameterization of this object? Clearly I chose to go cartesian, perhaps a different coordinate system would be nicer?

(2)Does anyone have advice for general efficient methods to approach these types of problems? A clearer way of thinking than brute force approach I took?

(3)Although I am confident and feel I completed my task, I actually havent justified that this object actually is a cylinder. When I arrived at the circle (same parametric equations without the $u$ term) I showed that the arc length was $2\pi$ and that $x^2 + y^2 + z^2 = 1$ which was enough evidence for me that my object was a circle, but does anyone have any advice for justifying that the final object is infact a cylinder? Or perhaps disprove that it is a cylinder?

(4)Can even more unnatural objects be parameterized? Could I make the cylinder "turn a corner"? If I made it continuously "turn a corner" I suppose it would just end up a torus?

Sorry for so much writing and maybe informal or unclear writing.. or maybe for making a mountain out of a mole hole with my elementary little parameterization problems!

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