This quest is lacking just now on suitable ideas for angles, and in particular how to compute them given coordinates.
This note aspires to elementary geometry on the compact Lie groups, or perhaps spherical geometry on Sn will do.
Geometry on S2 is adequately covered in high school but generalizations are not obvious.
The set of rays emanating from the vertex is divided into those the initially go into the body, and those that don’t.
If we put a small sphere about the vertex then some fraction of its surface will intersect these inward rays.
We need to know that fraction, but that is not enough.
We need to know the content of the supplement of this set.
We have a vertex on the boundary of an n-complex in a Euclidian n-space.
We imagine a small sphere (Sn−1) about the vertex.
Let P be the portion of that sphere is in the body.
We seek the content of P relative to the content of Sn−1.
The boundary of P is composed of segments where each segment is the intersection of Sn−1 and an (n−1)-simplex.
These segments come together at points on Sn−1.
Strategy:
I am looking for the analog to calculating the content of a polygonal patch on S2 by merely summing the external angles at the polygon’s corners.
That sum is like the sum of Gaussian curvature of the boundary of a convex polyhedron in n-space for which we at least have a good definition for higher dimensions.
I see a computational recursion on dimensionality in the fog.
Perhaps an accessible derivation: (dead end just now)
This is a derivation of the excess exterior angle theorem for the area of convex polygons on a sphere.
It is worded so that it works in n dimensions.
Draw a convex polygon on a sphere.
Consider the set U of all points in the enclosed ball that between a point inside the polygon and the center of the ball.
U is a convex set and its total curvature its boundary is 4π.
The curvature of its surface resides in three components:
- a 0D contribution at the center of the ball,
- a 1D integral over the edge of the polygon,
- a 2D integral over the smooth surface of the sphere inside the polygon.
The only novel part of this computation is the 1D integral.
The 0D and 2D contributions are supplemental but without a known sum.
Another pattern:
Hyperspherical Geometry
For S2 we can say that 2π − (the total extrinsic curvature of the boundary) = the deficit = total intrinsic curvature of interior.
The 2π comes from the total extrinsic curvature of the boundary of any convex area in flat 2-space.
This can serve as an exercise.
The 3-content of the interior of our initial room is ((π2/2)/16)*4 = π2/8 since:
- π2/2 is the volume of the unit 4-ball,
- there are 16 3D ‘rooms’ in S2,
- (the surface to volume ratio of a unit n-ball) = n, and here n=4.
We first try κ1 extrinsic curvature.
κ1 is 0 for the walls, π/2 per unit length of the edges where the walls come together, and 2 per unit volume (3-content) for the interior.
The total length of the edges is 6*π/2.
The total κ1 for the surface of a unit 4 cube is π2 from the 16 corners, and 32*π2 from the 32 unit edges.
κ1 is the wrong analog for it is scale sensitive.
The κ2 curvature for the surface of the 3-cube (or any convex body) is 8*(π/2)=4π in flat space.
For our room in S3 there are 4 corners, each locally congruent to a corner of our 3-cube whose κ2 curvature is π/2.
The walls and edges are κ2 flat and the deficit is thus 2π.
The 3-content of the interior of the room is ((π2/2)/16)*4 = π2/8.
We have no analog yet.
Notes on simplex angles,
We encroach on this.
Working this thru seems to require access to some of the results of generalized Gauss-Bonnet which I consider here.
I hope to continue this quest when I have made progress there.