Understanding the Formulas for Area of Spherical Triangles

[denni89627]

I've been reading Penrose's Road to Reality where he presents two formulas for area of shperical triangles. the first is Lamberts which is
pi-(A+B+C)=area (where A,B,C are angles of triangle)

the other is Harriot's which is
Area=R^2(A+B+C-Pi)

What I'm trying to figure out is if the radius must be the same for each segment for these formulas to work. In other words, if you extrapolated each curved side of the triangle into a complete circle, do they all have to be the same size circles? I'm guessing they do, and increasing R would just change the scaling. But Lambert's formula does not require a radius so would that formula work for a triangle created from cirlces of different radius?

Hope I worded this in an understandable way.

dennis

 

[mathman]

The radius being used is the radius of the sphere. Also the sides of a spherical triangle (by definition) are all arcs of great circles, which all have the same radius as the sphere.

 

[Hurkyl]

Incidentally, this is spherical geometry, not hyperbolic geometry.

Area, like any other measurement, is a relative quantity... the area of something depends on what you define to be a unit area!


Spherical geometry has some intrinsic measures (e.g. the length of any "line", or the total area of the entire "plane"), so it is natural to define your unit of measure relative to those. For example, I might decide to degree that the length of a line is 2 pi, and the area of the plane is 4 pi.

In that case, the area of a triangle is A + B + C - pi.


On the other hand, we might be might really be interested in the geometry of a Euclidean sphere. (which is a model of spherical geometry) In that case, we might want to speak about Euclidean lengths and areas, rather than the natural units of spherical geometry.

In that case, the area of a triangle is R^2 (A + B + C - pi).


There's a nice compromise; use a different word for the natural units of spherical geometry. Measure lengths in "radians", and areas in "steradians". Then, whatever perspective we are using, there are

2 pi radians in a spherical line or a Euclidean great circle.
4 pi steradians in a spherical plane or a Euclidean sphere.

and to convert into Euclidean lengths and areas, we simply multiply by R or R^2 as appropriate.


(note: when doing spherical geometry, people probably use the word "sphere" instead of "plane" -- I was using the latter to emphasize the fact I was talking about spherical geometry, and not just the Euclidean geometry of a Euclidean sphere)