I want to know if there is a way to switch between the inner and outer areas of a sphere triangle based on the orientation of the vectors that make it.
For example let's see this picture:
If $A$ is the area of the sphere, then let $a$ be the area of the small red triangle and $b = A - a$ the area of the outer triangle.
If I give you 3 unit vectors $v_1, v_2, v_3$ they will have 2 orientations (left handed vs right handed).
One can then associate the positive handedness with the small triangle, such that if I give you $v_1, v_2, v_3$ you get $a$. And the negative handedness with the large triangle, so that e.g. $v_2, v_2, v_3$ yields $b$. Is there an algebraic operator similar to the signed angle so that I can get this metric?
I haven't thought this all the way through, but it might set you on the right course.
We have points $v_1, v_2, v_3$ on the surface.
$v_1 \times v_2$ gives us the plane that contains these points
and $v_2\times v_3$ gives another plane connecting two points.
$\frac {(v_1 \times v_2)\cdot(v_2\times v_3)}{\|v_1\times v_2\|\|v_2\times v_3\|}$ gives the cosine of the angle between these two planes.
$\arccos \frac {(v_1 \times v_2)\cdot(v_2\times v_3)}{\|v_1\times v_2\|\|v_2\times v_3\|} + \arccos \frac {(v_2 \times v_3)\cdot(v_3\times v_1)}{\|v_2\times v_ 3\|\|v_3\times v_1\|} + \arccos \frac {(v_3 \times v_1)\cdot(v_1\times v_3)}{\|v_3\times v_1\|\|v_1\times v_2\|} $
Is the sum of the angles in our triangular section on the surface of the sphere.
And now the trick.... the degree to which the sum of the angles is in excess of $\pi$ is the area of the surface on a sphere of radius 1.
I am wondering if there is not a nicer representation in geometric algebra, but I don't know GA all that well.