CURVATURE

in mathematics, the rate of change of direction of a curve with respect to distance along the curve. If the curve is a section of a surface (that is, the curve formed by the intersection of a plane with the surface), then the curvature of the surface is taken to be that of the curve. The curvature of all plane curves (and straight lines, which can be regarded as circles of infinite radius) varies from point to point; the curvature at any point is defined as the limit of the ratio of the change in direction to the length of arc as the end points of the arc approach the point given. At every point on a circle, the curvature is the reciprocal of the radius; for other curves, the curvature is the reciprocal of the radius of the circle that most closely conforms to the curve at the given point. For determination of the curvature of a surface at a point, an infinite number of sectioning planes might be chosen. The most useful are two that both contain the normal (the line perpendicular to the tangent plane) to the surface at the point. One of these planes produces the section with the greatest curvature among all such sections; the other produces that with the least. These two planes define the two so-called principal directions on the surface at the point; these directions lie at right angles to one another. The curvatures in the principal directions are called the principal curvatures of the surface. The mean curvature of the surface at the point is either the sum of the principal curvatures or half that sum (usage varies among authorities).