Gaussian Curvature - Total Curvature

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:

    [The sceptic] must acknowledge, if he will acknowledge any thing, that all human life must perish, were his principles to prevail. All discourse, all action would immediately cease, and men remain in a total lethargy, till the necessities of nature, unsatisfied, put an end to their miserable existence.
    David Hume (1711–1776)