- #1
- 19,447
- 10,037
Definition/Summary
A Lune is a diangle on the surface of a sphere whose two vertices are opposite points.
So the two angles of a Lune are equal, and each of the two sides is half a circumference.
The whole surface is a Lune of angle [itex]2\pi[/itex].
Equations
Area of a Lune: [itex]2 r^2 \theta[/itex]
Area of the whole surface: [itex]4\pi r^2[/itex]
Any spherical triangle whose sides are arcs of great circles generates six Lunes, which cover the whole surface once and the triangle and its opposite twice more.
From that, it is easy to prove that the area of a spherical triangle is: [itex]\pi r^2 E[/itex]
where E is the sum of the angles minus [itex]\pi[/itex].
In particular, E > 0, and so the sum of the angles of a spherical triangle always exceeds [itex]\pi[/itex] (unlike a plane triangle, where it always equals [itex]\pi[/itex]).
Extended explanation
The word "Lune" comes from the Latin for "moon", and from the fact that the visible sunlit region of the moon is a Lune.
The projection onto a plane of a Lune of angle less than a right-angle ([itex]\frac{\pi}{2}[/itex]) is a crescent.
The formula for the area of a spherical triangle can also be proved by showing directly that E for any triangle is the sum of the Es for any two triangles into which it is divided.
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
A Lune is a diangle on the surface of a sphere whose two vertices are opposite points.
So the two angles of a Lune are equal, and each of the two sides is half a circumference.
The whole surface is a Lune of angle [itex]2\pi[/itex].
Equations
Area of a Lune: [itex]2 r^2 \theta[/itex]
Area of the whole surface: [itex]4\pi r^2[/itex]
Any spherical triangle whose sides are arcs of great circles generates six Lunes, which cover the whole surface once and the triangle and its opposite twice more.
From that, it is easy to prove that the area of a spherical triangle is: [itex]\pi r^2 E[/itex]
where E is the sum of the angles minus [itex]\pi[/itex].
In particular, E > 0, and so the sum of the angles of a spherical triangle always exceeds [itex]\pi[/itex] (unlike a plane triangle, where it always equals [itex]\pi[/itex]).
Extended explanation
The word "Lune" comes from the Latin for "moon", and from the fact that the visible sunlit region of the moon is a Lune.
The projection onto a plane of a Lune of angle less than a right-angle ([itex]\frac{\pi}{2}[/itex]) is a crescent.
The formula for the area of a spherical triangle can also be proved by showing directly that E for any triangle is the sum of the Es for any two triangles into which it is divided.
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!