Surface Area of the unit sphere in the region bounded by three vectors?

Question

Before I get into the meat of the question, I think it would be interesting to explain the intuition behind this question. In two dimensions, you could measure an angle by taking two vectors , using them to cut out a segment of the unit circle, and measuring the length of that segment to find the angle in between the two vectors in terms of radians. Why not apply this analogously to three dimensions? So we have three vectors which cut out a segment of the unit sphere, and this special measurement would be found by measuring the surface area of the cut out segment.

With this intuition in mind, how can we measure this area with any three vectors and how woild this be derived mathematically? Of course, there is the trivial case in which the three vectors all lie on the same plane or axis, in which case the area would be 0. Does this "measuremenr" have a special name, and does it have any interesting mathematical applications?

Answer 1

The three given vectors $a$, $b$, $c$ determine an infinite cone $C$. This cone intersects the unit sphere $S^2$ in a spherical triangle $T$. In order to get the spherical area of $T$ we need its angles $\alpha$, $\beta$, $\gamma$, because $${\rm area}(T)=\alpha+\beta+\gamma-\pi\ .$$ Each angle of $T$ is the angle between two bounding planes of $C$. The normals of these planes are given by $a\times b$, $\>b\times c$, $\>c\times a$, up to sign and normalization. The scalar product between two normals then determines the cosine of the corresponding angle of $T$.

The end result can be found under the subtitle "Tetrahedron" in the wikipedia article Solid angle.

Answer 2

Let the sector angles of your intuition be $( a,b,c).$

Using Sine and Cosine Rule of spherical trigonometry compute $ angles \alpha,\beta,\gamma$ on the sphere surface.There are plenty of references on the Sph Trig topic. E.g., Todhunter's book.

For a full sphere it is $ 4 \pi$ and hemisphere $ 2 \pi$. For a spherical cap with cone angle $\theta$ it is $ 2 \pi ( 1-\cos \theta).$

What you are interested is the general solid angle measured in steredians. $$ \alpha+\beta+\gamma -\pi $$

termed as spherical excess, which is to be multiplied by square of sphere radius to get enclosed area of spherical triangle.

In future if you pursue the 3D study you would find Gauss Bonnet theorem very interesting.