I Calculation for angular excess

AI Thread Summary
The discussion centers on the calculation of angular excess in spherical trigonometry, particularly in relation to a problem in general relativity. The user seeks clarity on whether angular excess and spherical excess are the same, noting that Girard's theorem relates the area of a spherical triangle to its spherical excess. They derive that for a sphere with radius a, the area equals a² times the spherical excess. Confusion arises regarding the presence of π in the equation for angular excess, leading to speculation that it may differ from spherical excess by a factor of π. The user concludes by questioning the accuracy of the book's equation, considering its publication date in the 1970s.
Haorong Wu
Messages
417
Reaction score
90
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!
 
Mathematics news on Phys.org
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.
 
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.
 
Thread 'Video on imaginary numbers and some queries'
Hi, I was watching the following video. I found some points confusing. Could you please help me to understand the gaps? Thanks, in advance! Question 1: Around 4:22, the video says the following. So for those mathematicians, negative numbers didn't exist. You could subtract, that is find the difference between two positive quantities, but you couldn't have a negative answer or negative coefficients. Mathematicians were so averse to negative numbers that there was no single quadratic...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Thread 'Unit Circle Double Angle Derivations'
Here I made a terrible mistake of assuming this to be an equilateral triangle and set 2sinx=1 => x=pi/6. Although this did derive the double angle formulas it also led into a terrible mess trying to find all the combinations of sides. I must have been tired and just assumed 6x=180 and 2sinx=1. By that time, I was so mindset that I nearly scolded a person for even saying 90-x. I wonder if this is a case of biased observation that seeks to dis credit me like Jesus of Nazareth since in reality...
Back
Top