Spherical Coordinates

3D coordinate system: alternative to Cylindrical Coordinates and 3D Cartesian Coordinates.

Spherical coordinates use:

• $\rho$ : Distance from origin to point

  • $\rho \ge 0$

• $\theta$ : Angle between the positive $x$-axis and the line above denoted by $r$

  • No requirements
  • $\theta$ and $r$ are both the same as they are in polar/spherical coordinates

• $\varphi$ : Angle between the positive $z$-axis and the line from the origin to the point

  • $0\le \varphi \le \pi$

---

Conversion from Spherical to Cylindrical

Looking at the triangle above from the front:

Thus we have:

$$\begin{array}{rl}r& =\rho \sin \varphi \\ \theta & =\theta \\ z& =\rho \cos \varphi \end{array}$$

By the Pythagorean theorem we also have ${\rho }^2=r^2+z^2$.

---

Conversion from Spherical to Cartesian

Recall:

Conversion from cylindrical to Cartesian:

$$\begin{array}{rl}x& =r\cos \theta \\ y& =r\sin \theta \\ z& =z\end{array}$$ Link to original

Thus, we have:

$$\begin{array}{rl}x& =\rho \sin \varphi \cos \theta \\ y& =\rho \sin \varphi \sin \theta \\ z& =z\end{array}$$

Since we have $r^2=x^2+y^2$, we also have:

$${\rho }^2=x^2+y^2+z^2$$

---

Single-variable functions and what they represent

• $\rho =a$ — sphere of radius $a$ centered at the origin
• $\varphi =\alpha$ — cone that makes angle $\alpha$ with the positive $z$-axis
• $\theta =\beta$ — vertical plane that makes an angle of $\beta$ with the positive $x$-axis