A triangle on the sphere

Drag the three gold vertices about the sphere; the sides are arcs of great circles, and the three angles are marked with small white arcs. The first row in the corner is the sum of the angles minus π, in radians; the second is the area, worked out separately from where the vertices are. The curvature of the unit sphere is 1 everywhere, so the two numbers always agree. Bring the vertices close together and the triangle is nearly flat, its excess nearly 0; the more area it encloses, the larger the excess. Drag two vertices onto each other and there is no triangle any more: all three rows show a dash. Now drag “How far round the vector has gone” all the way from 0 to 1. A gold tangent vector sets off from A and is carried along the sides, A to B to C, leaving a row of short white strokes behind it: the vector as it was at each point on the way. Along each side it keeps the same angle to that side; at a corner the side changes direction and the vector does not. Back at A it is out by an angle from the ice-blue vector it started as. A gold arc at A marks that angle, and the third row gives it, measured in the direction of the circuit, so it can be more than π: the same number again. On the unit sphere the angle excess, the area and the turn picked up by parallel transport are one and the same number.