Cylindrical and Spherical Coordinates
The Essentials
The conversions between rectangular coordinates and cylindrical coordinates are:
\[
r^2=x^2+y^2
\]
\[
x=r\cos(\theta)
\]
\[
\tan(\theta)=\frac{y}{x}
\]
\[
y=r\sin(\theta)
\]
\[
z=z
\]
The conversions between rectangular and spherical coordinates are:
\[
x=\rho\sin(\phi)\cos(\theta)
\]
\[
y=\rho\sin(\phi)\sin(\theta)
\]
\[
z=\rho\cos(\phi)
\]
\[
\rho^2=x^2+y^2+z^2
\]
\[
\tan(\theta)=\frac{y}{x}
\]
\[
\cos(\phi)=\frac{z}{\sqrt{x^2+y^2+z^2}}
\]
The conversion between cylindrical and spherical coordinates are:
\[
r=\rho\sin(\phi)
\]
\[
\theta=\theta
\]
\[
z=\rho\cos(\phi)
\]
\[
\rho=\sqrt{r^2+z^2}
\]
\[
\cos(\phi)=\frac{z}{\sqrt{r^2+z^2}}
\]
Practice
1) Convert the point \( (9,\frac{\pi}{2},5) \) from cylindrical \( (r,\theta,z) \) to rectangular coordinates
2) Convert the point \( (1,2,3) \) from rectangular coordinates to spherical coordinates \( (\rho,\theta,\phi) \).
Solutions:
1) \( (0,9,5) \)
2) \( (3.74,1.107,0.641) \)