# How to find an angle in spherical geometry.

• yungman
In summary, the conversation discusses finding the great circle angle between two points specified by their longitude and latitude angles in spherical geometry. The suggested method is to first find the great circle distance between the two points, and then divide it by the radius of the sphere. A link to a resource for more information is also provided.
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

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.

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?

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.

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,

## 1. How do I find the measure of an angle in spherical geometry?

To find the measure of an angle in spherical geometry, you will need to use the formula θ = r / ρ, where θ is the angle measure, r is the arc length, and ρ is the radius of the sphere. This formula is based on the fact that the circumference of a sphere is equal to 2π times its radius, and the measure of a central angle is equal to the ratio of the arc length to the radius.

## 2. What is the difference between measuring angles in spherical geometry and Euclidean geometry?

The main difference between measuring angles in spherical geometry and Euclidean geometry is that in spherical geometry, angles are measured along the surface of a sphere rather than on a flat plane. This means that the sum of the angles of a triangle in spherical geometry will always be greater than 180 degrees, unlike in Euclidean geometry where it is always exactly 180 degrees.

## 3. Do I need to use trigonometry to find angles in spherical geometry?

Yes, trigonometry is an essential tool for finding angles in spherical geometry. Spherical trigonometry is a specialized branch of trigonometry that deals specifically with the relationships between angles and sides on the surface of a sphere. Some common trigonometric functions used in spherical geometry include the sine, cosine, and tangent.

## 4. Can I use the Pythagorean theorem to find angles in spherical geometry?

No, the Pythagorean theorem only applies to right triangles in Euclidean geometry. In spherical geometry, triangles are not necessarily right triangles, and the Pythagorean theorem does not hold. Instead, you will need to use specialized formulas and relationships specific to spherical geometry to find angle measures.

## 5. Are there any real-world applications for finding angles in spherical geometry?

Yes, there are many real-world applications for spherical geometry, such as navigation and astronomy. For example, sailors and pilots use spherical geometry to determine their location and to plot the most efficient course between two points on a globe. Astronomers also use spherical geometry to map and study celestial bodies in space.

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