How to tell if two spherical coordinates lie on the same plane

Question

I have the $\rho$, $\theta$, and $\phi$ values of two points, how can one tell that two vectors are normal to the same plane by looking at their spherical coordinates?

Answer 1

Any two points lie on some plane through the origin, and on any plane that contains the line between them. So the answer to your original question is "they always lie on some plane."

To answer the question as it's now phrased:

1. Ignore $r$, unless $r = 0$ for one of the points, in which case the normal plane is undefined, so "they don't determine the same plane".

2. Assuming $-\pi/2 \le \phi \le \pi/2$ is the latitude coordinate and $0 \le \theta < 2\pi$ is longitude...

The vectors determine the same plane if

1. $|\phi_1| = |\phi_2| = \pi/2$ OR

2. $0 < |\phi_1| < \pi/2, 0 < |\phi_2| < \pi/2$, AND

   a. $\phi_1 = \phi_2$ and $\theta_1 = \theta_2$ OR

   b. $\phi_1 = -\phi_2$ and $\theta_1 = \theta_2 + \pi \bmod 2\pi$. OR

3. $\phi_1 = \phi_2 = 0$ and either

   a. $\theta_1 = \theta_2$ OR

   b. $\theta_1 = \theta_2 + \pi \bmod 2\pi$.

Case 3 is really covered by case 2, but it's nice to call it out separately.

It seems just possible to me that you really meant to ask the following:

Suppose $C$ is the center of a polar coordinate system, and $P$ and $Q$ are points different from $C$. Then there is a plane $U$ containing $P$ and normal to the vector $P-C$, and a plane $V$ containing $Q$ and normal to $Q - C$. Under what conditions on the polar coordinates of $P$ and $Q$ are the planes $U$ and $V$ identical as planes?

I believe that that question just leads to a horrible mess, and the best answer is "just convert to rectangular coords and write the formulas there."

Answer 2

Suppose the coordinates are $(\rho_1,\theta_1,\varphi_1)$ and $(\rho_2,\theta_2,\varphi_2)$ Two vectors are normal to the same plane if and only if they are nonzero multiples of each other. One case when this occurs is when $\theta_1=\theta_2$ and $\varphi_1=\varphi_2$. This takes care of the case where the vectors are positive multiples are each other.

It is also possible that the vectors could be negative multiples of each other. The formulas for $\rho,\theta,\varphi$ in terms of $x,y,z$ are $$\rho=\sqrt{x^2+y^2+z^2}$$ $$\theta=\arccos\left(\frac{z}{\sqrt{x^2+y^2+z^2}}\right)$$ $$\varphi=\arctan\left(\frac{y}{x}\right)$$ If we negate all of $x,y,z$ then $\rho$ and $\varphi$ stay the same, but theta becomes $\pi-\theta$. Hence the two possibilities are $$\theta_1=\theta_2\mbox{ and }\varphi_1=\varphi_2$$ or $$\theta_1=\pi-\theta_2\mbox{ and }\varphi_1=\varphi_2$$ As long as $\rho$ is not zero, it is irrelevant to the discussion.