Answer to Question #288857 in Analytic Geometry for Pankaj

Question

Express the following surfaces in spherical coordinates:

y² +z² -x² = 1

Answer 1

In shperical coordinates:

$$x = r\cos \phi \sin \theta\\ y = r\sin \phi \sin\theta\\ z = r\cos \theta$$

Substituting into the equation, obtain:

$$r^2\sin^2 \phi \sin^2\theta +r^2\cos^2\theta-r^2\cos^2 \phi \sin^2\theta=1\\ r^2\sin^2\theta(\sin^2 \phi -\cos^2 \phi )+r^2\cos^2\theta=1\\ r^2\cos^2\theta - r^2\sin^2\theta\cos2\phi=1$$

Answer. $r^2\cos^2\theta - r^2\sin^2\theta\cos2\phi=1$.