Show that the hyperbolic paraboloid can be represented parametrically as $$r(u,v)=\langle a(u+v), b(u-v), uv\rangle$$
Find the curves $u$ is constant and $v$ is constant.
I guess I need to use the hyperbolic paraboloid equation. But I cannot solve this. Please help me doing it. Thank you very much
Best Answer
You've written $$r(u,v)=\langle a(u+v), b(u-v), uv\rangle$$ and that's the same thing as saying $\langle x,y,z\rangle = \langle a(u+v), b(u-v), uv\rangle$, so that $$ \begin{align} x & = a(u+v), \\ y & = b(u-v), \\ z & = uv. \end{align} $$ So $$ \frac{y^2}{b^2} - \frac{x^2}{a^2} = \frac{(b(u-v))^2}{b^2} - \frac{(a(u+v))^2}{a^2}. $$ You can cancel $a$ and $b$ and then do routine simplifications and see if it turns into $-4uv$, which then turns into $-4z$, so that $\frac{y^2}{b^2} - \frac{x^2}{a^2} = \frac{z}{c}$ with $c = -\frac14$.
That shows that the parametrized surface you've got is a subset of the surface defined by the equations involving $x$, $y$, and $z$. Then you need to show that it's the whole set. One way to do that would be by solving for $u$ and $v$ in terms of $x$ and $y$, thereby showing that all $(x,y,z)$ points on the surface actually appear within the surface parametrized by $u$ and $v$.