Curve

Curve, in Geometry, means any line that is not straight. Any line, whether straight or curved, may be understood to be a succession of points ranged at an infinitely small distance from each other. Two consecutive points on the curve determine its direction; the change in direction as we proceed along the curve is called the curvature. The curvature is great if the change in direction is rapid, as in the case of a circle of small radius. The curvature is slight if the rate of change of direction is slow, as in the case of a circle of large radius, or of a line nearly straight. A straight line has no curvature, i.e. the radius of curvature is of infinite length. The line joining two consecutive points is known as the tangent to the curve. Three consecutive points determine two consecutive tangents, and therefore determine the curvature or rate of change of direction; through these three points a circle may be drawn, which is called the circle of curvature.

The reciprocal of the radius of this circle measures the curvature. A curve might also be regarded as the envelope of a series of lines, all of which are tangents to the curve. From these principles curves may be classified. Thus, curves of the second degree can be cut by any plane in two points only; curves of the second order can have only two tangents drawn to them from any point. Examples of such curves are the ordinary conic-sections, the circle, ellipse, parabola, hyperbola, line-pair, etc. Curves of higher degree can be cut in more points than two; curves of higher order can have more than two tangents from any one point, as, for example, the cycloid, the cissoid, the conchoid, etc. A curve that is not confined to one plane is said to be tortuous, such as the screw, or spiral. The more important curves are noticed separately.

The curvature of any surface is determined by that of any two sections of the surface at right angles to each other. It is constant in the case of a sphere. If the centre of curvature is on the side exterior to the surface, this is called concave; if interior, the surface is convex. Hence the terms concave and convex as applied to mirrors and lenses.