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# Osculation

## curve and circle

OSCULATION (Lat. osculatio, a kissing, from oseulari, to kiss, from °seldom, kiss, little mouth, diminutive of os, Skt. asya, mouth). One curve is said to osculate another when the curves have several consecutive points in com mon, and the degree of osculation depends upon the number of points of contact ; that is, the greater the number of consecutive points in con tact, the higher the degree of osculation. The number of possible points of contact is deter mined by the number of independent arbitrary constants contained in the equation of the tan gent curve. The same is true of a straight line and a curve. The equation of a straight line, be ing of the form y = nix c, contain: two arbi trary constants, m and e; hence a straight line can coincide with a curve in two consecutive points, and the contact is said to be of the first order. This straight line is the tangent at the point of contact. When a straight line. not a

tangent. meets a curve, there is section instead of contact, and in that ease only one point is com mon to the straight line and the curve. The general equation of the circle, + y= -F cy f = 0, contains three arbitrary constants. d. e, and f, and therefore a circle can have three consecu tive points in common with a curve, and the tact is of the second order. The circle is known as the circle of curvature or the osculating circle, and has for its radius the radius of curvature of that portion of the curve with which the circle is in contact. No other circle can have so high a degree of contact with a curve at any point as the osculating circle at that point. Surfaces and some twisted curves admit of spheres of oscu lation. See CURVE.