CURVE, a line, such that only one straight line can be made to touch, without cutting it, when the straight line is extended on both sides of the point of contact ; or a curve is a line in which, it' a point be taken, only one straight line can pass that point without cutting the line in which the point is taken. The straight line so drawn is called a tangent to the curve.
The circle is the most simple of all curves, depending only upon the arbitrary extension of its radius, which, if given, the circle is determined in magnitude. Its circumference is one uniform curve, or has its curvature everywhere equal, and equally distant from the centre.
In every curve line whatever, it is evident that a very small portion may be taken as a circular arc at any point ; or that there is a certain circular arc at that point which has the same curvature as an indefinitely small portion of the curve, hut a greater curvature than an indefinitely small portion of the curve upon the one side of the point, and also a less curvature than the nearest indefinitely small portion on the other side of the point. The radius of a circle of equal curvature to the curve at any proposed point, is called the radius of curvature at that point, and is the measure of the curvature of all curves. The circle of
equal curvature with the curve, is called the eguicurve circle, or the osculating circle. Hence, if the osculating circle and the curve have a common tangent, no other circle whatever can be drawn between the two curves; and when the curve is continually upon the increase or decrease in its curvature, the are of the osculating circle will be on the concave side of the curve on one side of the point of contact, and on the convex side on the other side of the said point ; or the curve will be between the tangent and the circumference of the osculating circle on the one side of the point of contact, but the circumference of the osculating circle will be between the tangent and the curve on the other side of the point of Contact.
The principal curves that are useful in architecture, are the conic sections, namely, the circle, the ellipsis, the parabola, and the hyperbola; also, the cycloid, the conchoid, the spiral of Archimedes, and the logarithmic spiral. The definitions and the most useful properties, will be found under each word.