CURVE. A curve is most easily thought of as the path of a point moving continuously as to both position and direction, ex cept at special points where discontinuities of any kind may occur. Formerly the word line was used to include both curves and straight lines; in modern use it always means a straight line, and is a particular case of the more general term curve.
A curve may also be thought of as given in its entirety, a single infinity of points all present at once ; this is perhaps the artistic rather than the scientific point of view.
In either case, the idea becomes more definite when we regard all the points of the curve, whether given successively or simul taneously, as obeying some sort of law. If we adopt the device of co-ordinates (see ANALYTIC GEOMETRY), the law is expressed by one or more equations between the co-ordinates, involving cer tain functions of them which vanish. The curve is said to be algebraic or transcendental according to the nature of its equations in Cartesian co-ordinates. If it is algebraic, it has only a finite number of exceptional points, but a transcendental curve can have an infinite number (for an example, see article CURVES, SPECIAL) and if its equation involves one of the highly discontinuous func tions known to modern analysis it may lose all or nearly all of the properties usually associated with the idea of curve.
If P is an ordinary point of a plane curve, and Q is a neighbour ing point, then as Q moves up to P, the chord PQ tends to a definite limit called the tangent at P. The line through P per pendicular to the tangent is the normal. Three points P,Q,R of the curve determine a circle, whose limit as Q and R tend to P is the circle of curvature at P. The more sharply the curve bends, the smaller is this circle, and the reciprocal of its radius is the curvature at P, and is equal to the rate of rotation of the tangent (or normal) per unit length of arc described by P. The centre of curvature is also the limit of the intersection of adjacent normals.
In the immediate neighbourhood of P, to a first approximation the curve can be replaced by the tangent, and to a second approxi mation by the circle of curvature.
By the principle of duality (q.v.) a curve can also be regarded as the envelope of its tangents; we may think of the tangent as a line moving continuously in one plane according to some law; its intersection with a neighbouring tangent has a definite limit, which is the point of contact. Thus the tangent replaces the point as the element which generates the curve, and the point of con tact replaces the tangent.
It is useful to think of the tangent, normal and binormal as a rigid frame of rectangular axes, with P as origin, moving forward and rotating as P describes the curve with unit velocity. At any moment, the angular velocity of the frame about the binormal is the curvature, and that about the tangent is the torsion; there is no instantaneous rotation about the normal.
The degree of an algebraic plane curve is that of its equation, and is the number of its intersections with a general line of the plane; and its class is the number of its tangents through a gen eral point. The degree of an algebraic twisted curve is the number of its points that lie in a general plane; and its class is the number of its osculating planes through a given point, and it has another important characteristic, the rank, which is the number of its tangent lines that meet a general line.
In space, the figure dual to a curve locus is a singly infinite set of planes obeying some given law. These form a developable surface. Any plane of the set with any other one determines a line, and with any other two it determines a point. As the second and third planes tend to coincide with the first, this point has a definite limit, whose locus is a curve associated with the devel opable surface, called its edge of regression. The limit of the line is its tangent, and the original plane is its osculating plane. Con versely, the osculating planes of a given curve form the associated developable, of which the curve is the edge of regression.
For the equation of a transcendental curve, there is often a simple expression in polar or other co-ordinates (for example, see article CURVES, SPECIAL : Spiral, Cardioid, Trisectrix). In other cases (see Cycloid in the same article), the simplest form gives each co-ordinate in terms of a common parameter. The parametric expression is often useful for algebraic curves also. For example, by a proper choice of a system of homogeneous co-ordinates (see CO-ORDINATES) any twisted cubic can be represented by x: y: z: w= s : t: Yet another form of expression is in terms of the length of any arc and the angle between the tangents at its extremities; this is called the intrinsic equation, since it depends on no frame of reference unconnected with the curve (see Catenary, in the article CURVES, SPECIAL) .