How to find an angle in spherical geometry.

[yungman]

Hi
I am not familiar with spherical geometry. I am working with elliptical polarization that involves using poincare sphere that present the latitude and longitude angle in spherical geometry. I need to find the great circle angle if given two points that each specified by their longitude angle $2\tau$ and latitude angle $2\chi$.

ie. If I am given the $\chi$ and $\tau$ of $M_{1}(\tau_{1},\chi_{1})$ and $M_{2}(\tau_{2},\chi_{2})$, how can I find the great circle angle between $M_{1}(\tau_{1},\chi_{1})$ and $M_{2}(\tau_{2},\chi_{2})$?

I really don't want to learn the details of spherical geometry, just want to learn the way of finding the angle as this is only a small part of antenna design.

Thanks

Alan

 

[Simon Bridge]

google for "great circle distance". eg. http://mathworld.wolfram.com/GreatCircle.html

If the great circle distance is $d$, then the angle (in radians) between the points is $\theta=d/R$ where R is the radius of the sphere.

 

[yungman]

Thanks for the reply, but what if if I have only the longitude and latitude angle of the two points, how can I find the great circle angle between the two points?

 

[Simon Bridge]

what if if I have only the longitude and latitude angle of the two points, how can I find the great circle angle between the two points?

Step 1: find the great-circle distance between the two points from the long and lat values.
Step 2: divide this by the radius of the sphere.

Anticipating your next question: see link in post #2.

 

[blue_raver22]

Hello Alan,

Thank you for sharing your question with me. Finding angles in spherical geometry can be challenging, but it is an important skill for understanding and designing antennas.

To find the great circle angle between two points on a sphere, you can use the Haversine formula. This formula takes into account the latitude and longitude angles of the two points and calculates the shortest distance between them, which is equivalent to the great circle angle.

Alternatively, you can also use the Law of Cosines to find the great circle angle. This formula involves the sides and angles of a triangle on a sphere and can be used to find any of the angles.

I understand that you may not want to delve too deeply into the details of spherical geometry, but these formulas should help you in finding the great circle angle between two points. I hope this helps and good luck with your antenna design project.

Best,