 curvature

/kerr"veuh cheuhr, choor'/, n.1. the act of curving or the state of being curved.2. a curved condition, often abnormal: curvature of the spine.3. the degree of curving of a line or surface.4. Geom.a. (at a point on a curve) the derivative of the inclination of the tangent with respect to arc length.b. the absolute value of this derivative.5. something curved.[13751425; late ME < L curvatura, equiv. to curvat(us) ptp. of curvare to bend, CURVE + ura URE. See ATE^{1}]
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Measure of the rate of change of direction of a curved line or surface at any point.In general, it is the reciprocal of the radius of the circle or sphere of best fit to the curve or surface at that point. This notion of best fit derives from the principle that only one circle can be drawn though any three points not on the same line. The radius of curvature at the middle point is approximately equal to the radius of that one circle. This calculation becomes more exact the closer the points are. The precise value is found using a limit. Because a straight line can be thought of as an arc of a circle of infinite radius, its curvature is zero.* * *
▪ geometryin mathematics, the rate of change of direction of a curve with respect to distance along the curve. At every point on a circle, the curvature is the reciprocal of the radius; for other curves (and straight lines, which can be regarded as circles of infinite radius), the curvature is the reciprocal of the radius of the circle that most closely conforms to the curve at the given point (see figure—>).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 at any given point can be determined by suitable sectioning planes. The most useful planes are two that both contain the normal (the line perpendicular to the tangent plane) to the surface at the point (see figure—>). 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 socalled 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). The total (or Gaussian) curvature (see differential geometry: Curvature of surfaces (differential geometry)) is the product of the principal curvatures.* * *
Universalium. 2010.