Question

Find a parametric representation g(s,t) of the part of the one-sheeted hy-perboloid x^2/a^2+y^2/a^2−z^2/c^2= 1 that lies between z=−1 and z= 1

Answer 1

If in the form

$${x^2 \over a^2}+{y^2\over b^2}-{z^2 \over c^2}=1$$

then let $z=cv, v$ as one parameter, and then, the equation can be rewritten in the following way:

$${x^2 \over a^2}+{y^2\over b^2}-{c^2v^2 \over c^2}=1$$ $${x^2 \over a^2}+{y^2\over b^2}=1+v^2$$

Let $x=a\sqrt{1+v^2}\cos u, y=b\sqrt{1+v^2}\sin u.$ Then

$${(a\sqrt{1+v^2}\cos u)^2 \over a^2}+{(b\sqrt{1+v^2}\sin u)^2\over b^2}=1+v^2,$$

$$x=a\sqrt{1+v^2}\cos u,$$ $$y=b\sqrt{1+v^2}\sin u,$$ $$z=cv,$$ $$u\in[0, 2\pi), v\in\big [-{ 1\over c} , {1 \over c} \big].$$

Other parameterizations include

$$x=a\cosh{v}\cos{u},$$

$$y=b\cosh{v}\sin{u},$$

$$z=c\sinh{v},$$

$$u\in[0, 2\pi), v\in\big [-\sinh^{-1}({ 1\over c}) , \sinh^{-1}({ 1\over c})].$$