# Converting a point to Cylindrical and Spherical Coordinates

How is any point on the Cartesian coordinates converted to cylindrical and spherical coordinates. Taking as an example, how would you convert the point (1,1,1)? Thanks in advance.

Cylindrical coordinates: $(r, \theta, z)$ are given by the relationship $x = r \cos\theta$, $y= r\sin \theta$ and $z=z$. Using these we can solve for $r$ and $\theta$ explicitly: $$x^2 + y^2 = (r\cos\theta)^2+(r\sin\theta)^2= r^2(\cos^2\theta+\sin^2\theta)= r^2$$ $$\cos\theta = \frac{x}{r} = \frac{x}{\sqrt{x^2+y^2}}$$
Spherical coordinates: $(\rho,\theta,\phi)$ can be given in a few different ways, I would check your textbook, but what I teach is $x = \rho\cos(\theta)\sin(\phi)$, $y= \rho\sin(\theta)\sin(\phi)$, $z = \rho \cos(\phi)$. You can use the same tricks as we used for cylindrical coordinates to isolate $\rho,\theta$ and $\phi$.
• Are you sure about your spherical coordinates? If $z=\rho\sin\theta$ then $\theta$ is measured above the plane. Usually the vertical angle is taken from the $z$ axis. Also, if $z=\rho\sin\theta$, wouldn't $y=\rho\cos\theta\sin\phi$? – John Douma Jun 19 '13 at 20:01
cylindrical: $(r, \theta, z) = (\sqrt{2}, \pi/4, 1)$
spherical: $(\rho, \theta, \phi) = (\sqrt{3}, \pi/4, 0.955\text{ radians})$