Total Curvature

In mathematical study of the differential geometry of curves, the total curvature of an immersed plane curve is the integral of curvature along a curve taken with respect to arclength:

The total curvature of a closed curve is always an integer multiple of 2π, called the index of the curve, or turning number – it is the winding number of the unit tangent about the origin, or equivalently the degree of the map to the unit circle assigning to each point of the curve, the unit velocity vector at that point. This map is similar to the Gauss map for surfaces.

Read more about Total Curvature:  Comparison To Surfaces, Invariance, Generalizations

Famous quotes containing the word total:

    It is not contrary to reason to prefer the destruction of the whole world to the scratching of my finger. It is not contrary to reason for me to choose my total ruin, to prevent the least uneasiness of an Indian, or person wholly unknown to me. It is as little contrary to reason to prefer even my own acknowledged lesser good to my greater, and have a more ardent affection for the former than the latter.
    David Hume (1711–1776)