Calculation for angular excess

In summary, the conversation discusses the concept of angular excess in spherical trigonometry and the relationship between angular excess and spherical excess. The conversation also mentions Girard's theorem and the calculation of spherical excess using the area of a spherical triangle. It is concluded that the equation given in the book may be incorrect or based on an older definition of angular excess.
  • #1
Haorong Wu
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TL;DR Summary
How to calculate the angular excess on a sphere?
Hello. I am not familiar with spherical trigonometry while I am reading a solution in a GR problem book. It reads,
If we construct a coordinate patch from geodesics we can then bisect that coordinate box with a geodesic diagonal, forming two geodesic triangles. The angular excess of a triangle made from great circles is ##\pi [Area/a^2]## where a is the radius of the sphere.

I study spherical trigonometry on Wikipedia and some other sites, but I am still not sure how to calculate the angular excess.

First, is angular excess equivalent to spherical excess? I have not found a clear definition for angular excess. But the definition for spherical excess makes me think that they are the same concept. Maybe angular excess is just an old-fashioned name?

Second, Girard's theorem states that the area of a spherical triangle is equal to its spherical excess.

Then for a sphere with radius ##a##, Girard's theorem gives that ##Area=a^2 \times E## where ##E## is the spherical excess.

So the spherical excess is given by ##E=Area/a^2##.

Now I am not sure where the ##\pi## comes from. Maybe angular excess differs from spherical excess by a factor ##\pi##?

Thanks!
 
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  • #2
Try looking at parts of a sphere where you can calculate these numbers. For example two meridians at right angles and the equator. Area is ##\frac{\pi}{2}## while angle sum is ##\frac{3\pi}{2}## for a=1 (I can never trust my arithmetic - check it). It looks like ##\pi## is already there.
 
  • #3
Thanks, @mathman . I got the same answer. So I am not sure whether the equation given in the book is wrong, or because it was defined so in the old days since the book is published in the 1970s.
 

FAQ: Calculation for angular excess

1. What is angular excess?

Angular excess is a mathematical concept used in geometry to measure the amount by which the sum of the angles of a polygon exceeds the sum of the angles in a regular polygon with the same number of sides.

2. How is angular excess calculated?

To calculate angular excess, you need to subtract the sum of the interior angles of a regular polygon from the sum of the interior angles of the given polygon. The formula for calculating angular excess is (n-2)180° - Σα, where n is the number of sides and Σα is the sum of the interior angles of the polygon.

3. What is the significance of angular excess?

Angular excess is an important concept in geometry as it allows us to determine the shape of a polygon based on the sum of its angles. It also helps us to classify polygons as convex or concave.

4. Can angular excess be negative?

Yes, angular excess can be negative. This happens when the sum of the angles in a polygon is less than the sum of the angles in a regular polygon with the same number of sides. In this case, the polygon is said to have a negative angular excess.

5. How is angular excess used in real life?

Angular excess is used in various fields such as engineering, architecture, and surveying. It helps in determining the angles and shapes of different structures and objects, such as buildings, roads, and bridges. It is also used in navigation and map-making to calculate the angles and distances between different points.

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