00:02
We're asked to find a parameterization for the hyperboloid of one sheet, which has equation x squared plus y squared minus z squared equals 1.
00:20
In terms of the angle of theta, associated with the circle x squared plus y squared equals r squared, so a circle in the x, y, plane, and the hyperbolic parameter u associated with the hyperbolic function r squared minus z.
00:39
Z squared equals 1.
00:52
So we have that for part a, it would seem that we want x squared plus y squared to be equal to something such that that squared equal to something such that that minus z squared would equal 1.
01:39
So we're going to take w to equal x squared plus y squared, or w squared will be x4 plus y squared.
01:49
So we have that w squared minus z squared equals 1.
01:57
Now, recall that the hyperbolic functions cosine hyperbolic and hyperbolic sign satisfy hyperbolic of u squared minus hyperbolic sine of view squared squared equals 1 for all you.
02:23
So it stands to reason that we should set w squared.
02:31
Well, w should be hyperbolic cosine of you, and z should be hyperbolic sine of view.
02:48
So, in other words, we want to have x squared plus y squared to equal hyperbolic cosine of you squared.
03:01
For this to be the case, we'll have to require that the...
03:10
The radius of the circle x squared plus y squared is going to be a hyperbolic cosine of view...
