CURVATURE. The C. of a plane curve at a point is its tendency to depart from a tangent to the curve at that point. In the circle, this tendency is the same throughout, for the curve is perfectly symmetrical round its center; in other words, the C. of a circle is constant. In different circles, the C. is inversely as the radius—i.e., it diminishes as the radius increases. The reciprocal of the radius is accordingly assumed as the measure of C. of a circle. A straight line, which has no C., may be considered part of a circle whose radius equals infinity as the reciprocal of infinity, measures the C., and is = 0.
The constancy of C. in the circle suggests an absolute measure of C. at any point in any other curve; for whatever be the C. at that point, we can always find a circle of the same curvature. The radius of the circle which has the same C. at any point in a curve as the curve itself at that point, is called the radius of C. of the curve for that point;
and the circle itself is called the osculating circle. If we know the radius of C. of a curve at different points, we can compare its C. at those points. We have thus the means also of comparing degrees of C. in different curves.
The problem of measuring the C. of a curve at any point is the same, then, with that of finding its radius of curvature. In some simple cases, as in the conic sections, this may be done geometrically; it is usually necessary, however, to employ the calcu lus. If the curve be referred to rectangular co ordinates, and x, y be a point in it, then de/ /211 it can be shown that radius of C. = . If the curved line, instead of being plane, twists in space, it is called a curve of double curvature. See CONTACT and OSCULATION.