Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Please let me know the formula for the coordinate of the midpoint of 2 points in spherical coordinate system . If possible , I want the answer includes the exact formula as , midpoint = point1 + ( point2 - point1 ) / 2 .

Thank you very much

Thank you very much for the answer. Is it necessary to compute the Cartesian coordinate of the midpoint ? No means to compute directly the spherical coordinate of the midpoint ?

Thank you for the second answer . I am happy to know that I must use Cartesian coordinate system for the purpose .

Please someone let me know how to mark the answer as my accepted answer. There is no button for the purpose on my page .

share|cite|improve this question
This will (most-likely) be messy... What you don't like about Cartesian coordinates? – Fabian Apr 30 '11 at 6:32
"No means to compute directly the spherical coordinate of the midpoint?" - Doing coordinate conversions will result in a direct formula. The thing is, spherical coordinates are poorly suited for systems where you need to do translations, segment cuts, and other such operations where Cartesian coordinates are more "natural"... – J. M. Apr 30 '11 at 8:23
FYI - this problem is trivial with Homogeneous Coordinates. – ja72 Apr 30 '11 at 17:19
Hi seven_swodniw: You have two accounts and they must be merged before you can accept an answer. I informed the moderators and they should take care of that soon. These problems will go away when you register. – t.b. Jun 14 '11 at 3:45
To Theo Buehler : Thank you very much . I finally finished to accept the answer . – seven_swodniw Jun 16 '11 at 8:02
up vote 8 down vote accepted

You can use the general formulas for converting between Cartesian and spherical coordinates to do this:


so the midpoint between two points $1$ and $2$ is

$$\frac{1}{2}\left(\begin{array}{c}x_1+x_2\\y_1+y_2\\z_1+z_2\end{array}\right) =\frac{1}{2}\left(\begin{array}{c}r_1\sin\theta_1\cos\phi_1+r_2\sin\theta_2\cos\phi_2\\r_1\sin\theta_1\sin\phi_1+r_2\sin\theta_2\sin\phi_2\\r_1\cos\theta_1+r_2\cos\theta_2\end{array}\right)\;.$$

Then you can substitute this into the expression for the spherical coordinates in terms of the Cartesian coordinates:


share|cite|improve this answer
where arctan still has to be defined properly, such that $\phi \in [0,2\pi]$. – Fabian Apr 30 '11 at 7:10
@Fabian: correct, depending on the signs of $x$ and $y$; on a computer, this would typically be done using the atan2 function. – joriki Apr 30 '11 at 7:11
Thank you very much for the answer. – seven_swodniw Apr 30 '11 at 7:33
Heh, I was talking about this the other day... some people have $\phi$ be the longitude and $\theta$ be the co-latitude, while others (me included) use $\theta$ as the longitude and $\phi$ as the co-latitude. So kids, be careful with using spherical coordinate formulae! :) – J. M. Apr 30 '11 at 7:53

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.