The surface integral of the Gaussian curvature over some region of a surface is called the total curvature. The total curvature of a geodesic triangle equals the deviation of the sum of its angles from π. The sum of the angles of a triangle on a surface of positive curvature will exceed π, while the sum of the angles of a triangle on a surface of negative curvature will be less than π. On a surface of zero curvature, such as the Euclidean plane, the angles will sum to precisely π.
A more general result is the Gauss–Bonnet theorem.
Read more about this topic: Gaussian Curvature
Famous quotes containing the word total:
“When we suffer anguish we return to early childhood because that is the period in which we first learnt to suffer the experience of total loss. It was more than that. It was the period in which we suffered more total losses than in all the rest of our life put together.”
—John Berger (b. 1926)