# Finding an equation to the surface S that is bounded between $z=x^2-y^2$ inside the cylinder $x^2+y^2=1$

How to find a parametric equation to the surface S that is bounded between $z=x^2-y^2$ inside the cylinder $x^2+y^2=1$, and while C be the the Boundary of that surface.

While reading the solution of one of the questions they said that a parametric equation for surface S is:

$\vec r(\rho$,$\theta$) = $\rho$cos$\theta$$\hat i + \rhosin\theta$$\hat j$ + $\rho^2$cos$2\theta$$\hat k while 0\le \theta \le 2\pi , 0\le \rho \le 1 and a parametric equation for curve/boundary C is: \vec r(\theta)= cos\theta$$\hat i$ + sin$\theta$$\hat j + cos2\theta$$\hat k$ while $0\le$ $\theta \le 2\pi$

but how did they reach these equations? I would appreciate any kind of help.

any point inside the cylinder $x^2 + y^2 = 1$ can be represented by $$x = r \cos t, y = r \sin t,\space 0 \le r \le 1, 0 \le t \le 2\pi.$$ then $$z = x^2 - y^2 = r^2 (\cos^2 t - \sin^2 t) = r^2 \cos (2t).$$