OSCULATION

, in Geometry, denotes the contact between any curve and its osculatory circle, that is, the circle of the same curvature with the given curve, at the point of contact or of Osculation. If AC be the evolute of the involute curve AEF, and the tangent CE the radius of curvature at the point E, with which, and the centre C, if the circle BEG be described; this circle is said to osculate or kiss the curve AEF in the point E, which point E Mr. Huygens calls the point of Osculation, or kissing point.

The line CE is called the osculatory radius, or the radius of curvature; and the circle BEG the osculatory or kissing circle.|

The evolute AC is the locus of the centres of all the circles that osculate the involute curve AEF.

Osculation also means the point of concourse of two branches of a curve which touch each other. For example, if the equation of a curve be , it is easy to see that the curve has two branches touching one another at the point where x = 0, because the roots have each the signs + and -.

The point of Osculation differs from the cusp or point of retrocession (which is also a kind of point of contact of two branches) in this, that in this latter case the two branches terminate, and pass no farther, but in the former the two branches exist on both sides of the point of Osculation. Thus, in the second figure above, the point B is the Osculation of the two branches ABD, EBF; but C, though it is also a tangent point, is a cusp or point of retrocession, of AC and AB, the branches not passing beyond the point A.

OSCULATORY Circle, or Kissing Circle, is the same as the circle of curvature; that is, the circle having the same curvature with any curve at a given point. See the foregoing article, Osculation, where BEG, in the last figure but one, is the Osculatory circle of the curve AEF at the point E; and CE the Osculatory radius, or the radius of curvature.

This circle is called Osculatory, or kissing, because that, of all the circles that can touch the curve in the same point, that one touches it the closest, in such manner that no other such tangent circle can be drawn between it and the curve; so that, in touching the curve, it embraces it as it were, both touching and cutting it at the same time, being on one side at the convex part of the curve, and on the other at the concave part of it.

In a circle, all the Osculatory radii are equal, being the common radius of the circle; the evolute of a circle being only a point, which is its centre. See some properties of the Osculatory circle in Maclaurin's Algebra, Appendix De Linearum Geometricarum Proprietatibus generalibus Tractatus, Theor. 2, § 15 &c, treated in a pure geometrical manner.

Osculatory Parabola. See Parabola.

Osculatory Point, the Osculation, or point of contact between a curve and its Osculatory circle.

OSTENSIVE Demonstrations, such as plainly and directly demonstrate the truth of any propofition. In which they stand distinguished from Apagogical ones, or reductions ad absurdum, or ad impossibile, which prove the truth proposed by demonstrating the absurdity or impossibility of the contrary