Finding parametric equations for curve of intersection

Question

I am having difficulty finding parametric equations of the curve of intersection of $z = x^2 - y^2$ and cylinder $x^2 + y^2 = 1$

I am aware that the first equation represents a hyperboloid, and the second equation represents a cylinder. I think for the second one I can just do $x = cos(t)$ and $y = sin(t)$. Not sure about the other one.

Answer 1

As you point out, the second equation implies that $x = \cos(t)$ and $y = \sin(t)$. Substituting these into the first equation yields $$ z = x^2 - y^2 = \cos^2(t) - \sin^2(t) \, . $$ Thus the curve is parametrized by $$ (x(t), y(t), z(t)) = (\cos(t), \sin(t), \cos^2(t) - \sin^2(t)) = (\cos(t), \sin(t), \cos(2t)) \, . $$

Here's a SageMathCell illustrating the surfaces and curve of intersection, which produces the following plot.

$\hspace 4cm$

Answer 2

For the first one use that $$\cosh^2(t)-\sinh^2(t)=1$$ And since$$x^2=z+y^2$$ we get

$$z+2y^2=1$$ so.... $$z=1-2y^2$$

Answer 3

We have the two surfaces

$$ x^2-y^2 = z\\ x^2+y^2 = 1 $$

Solving for $x^2, y^2$ we have

$$ x^2 = \frac 12(1+z)\\ y^2 = \frac 12(1-z) $$

so a parametric representation can be read as

$$ (x(s),y(s),z(s)) = \left(\pm\sqrt{\frac 12(1+s)},\pm\sqrt{\frac 12(1-s)},s\right) $$

for $-1\le s \le 1$

Attached a plot with the four leafs