Page:The New International Encyclopædia 1st ed. v. 05.djvu/788

* CTTBVE. G80 CURVE. circle itself the osculating circle. By means of this radius wc may comiiare the curvatures at (lifVerent points of the same curve or of different curves. In simple cases, as in the conic sections, the measure or radius of curvature may be de- termined geometrically, but it is usually neces- sary to employ the calculus. Tlie expression for the radius of curvature at any point {x, y) of a curve is equation is given in rectangular coordinates, sub- P = [>+(!/]' d-y If the curve, instead of lying in a plane, twists in space, it is sometimes called a gauche curve or a curve of double curvature, and its curvature at any point may be measured by the radius of its osculating sphere at that point. The centre of the osculating circle or sphere is called the centre of curvature. The curvature of surfaces is de- termined similarly to that of curves. Thus the measure of the curvature of the earth, commonly taken as the deviation of the line of apparent level from the line of true level — that is, from a line ever^^vhere parallel to the surface of still water — is approximately eight inches per mile. Relations. The following are some of the more important relations which exist among certain groups of curves: (1) The evolute (q.v.) of a curve is the locus (q.v.) of its centre of curvature. Regarding the evolute as the principal curve, the original curve is called its involute. The normals to any curve are tangents to its evolute. (2) Two curves or surfaces are said to have contact when they touch at two or more consecu- tive points. A contact (q.v.) of the nth order exists between two curves jtj :^^ (.r), 1/2 = "A (^) at the point whose abscissa is a when (a) = ^ a),i^)W=f(n)(^).Ii n is even, the curves cross at the point. No curve which has contact of a lower order can pass between the given curves. Curves which have contact of the first order have a common tangent, and those having contact of the second order have a com- mon radius of curvature at the point of contact. (3) The envelope of a curve is the locus of the ultimate intersections of the individual curves of the .same species, obtained by constantly vary- ing a parameter of the curve. That is, "the en- velope touches all of the intersecting curves thus obtained; e.g. if p is a variable parameter and /■r=0 is the equation of the curve, then the re- sult obtained by eliminating p between f ^0 and -^ = is the equation of the envelope. Every curve may be an envelope, and some are evidently so by definition — e.g. evolutes and caustics ( qq.v. ). (4) The process of replacing each radius vec- tor of a curve by its reciprocal is called inver- sion. The origin is called the centre of inversion and the resulting curve the inverse of the given one. See Circle, Inversion. (5) The locus of the feet of the perpendicu- lars from the origin upon the tangents to a curve is called a pedal curve. The pedal of a pedal is called the second pedal, and so on. Re- versing the order, the curves are envelopes and are called negative pedals. The pedal and re- ciprocal polar are inverse curves. (See Circle.) In general, to find the inverse of a curve whose stitute for a;, y, k'- k-y -^ respectively. 'x^-ty" ar + y^ (fl) A roulette is defined as the locus of a point rigidly connected with a curve which roUa upon a fi.xed line or curve. See Cycloid. CENTRE.S. A point such that every radius vec- tor (see CooRDiKATES) drawn from "it to a point on the curve is matched by another vector of the same length in the opposite direction is called the centre of a curve. See also Circle and the paragraph on Curvature above. When a plane figure moves in any manner in its own plane, the instantaneous centre of rota- tion is the intersection of two lines drawn through two points perpendicular to the direc- tions in which the points are moving. The number of kinds of curves that might be drawn is infinite. A large number are known by specific names, and are objects of great in- terest on account of their beauty, their remark- able properties, or their relation to physical problems. Among those discussed under sepa- rate titles are the conic sections, cissoid, con- choid, lemniscate, cycloid, trochoid, witch, car- dioid, cartesians, Cassinian ovals, caustic curve, tractrix, curve of pursuit, catenary, cun-es of circular functions (e.g. curves of sines), loga- rithmic curves, and spirals. Though the historii' of curves is inseparable from that of geometry, it may roughly be divided into four periods: (1) The synthetical geonietiy of the Greeks, in which the conic sections (q.v.) play an important role; (2) the birth of analytic- geometry, in which the synthetic geometry of Guldin, Desargues, Kepler, and Roberval merged into the coordinate geometry of Descartes and- Fermat ; (.3) the period 1C50 to 1800, character- ized by the application of the calculus to geome- try and including the names of Newton, Leib- nitz, the Bernoullis, Clairaut, Maelaurin, Euler, and Lagrange; (4) the nineteenth centviry, the renaissance of pure geometiy, characterized by the descriptive geometry of Jlonge, the modern s,^Tithetie geometry of Poncelet, Steiner, von Staudt. Cremona, and Pliicker. Descartes's con- tributions were confined to plane curves, but led to the discovery of many general properties. The scientific foundations of the theory of plane curves may be ascribed to Euler (1748) and Cramer (1750K Euler distinguished algebraic from transcendental curves, and Cramer found- ed the theor,v of singularities. Clairaut (1731) attacked the problem of double curvature ; Monge introduced the use of dilTeicntial equations. Miibius (1852) sunmied up the classification of the cubic curve, Zeuthen (1874) did the same for the quartics, and Bobillier (1827) first used trilinear co.'irdinates (q.v.). In 1828 Pliicker published the first volume of his Analt/tisch-geo- metrische Entwiclcelunpen. which introduced abridged notation and marked a new era in analytic geometry. To him is due (1833) the general treatment of foci, a complete classifica- tion of cubics (183.5), and his celebrated 'six- equations' (1S42). Hesse (1844) gave a com- plete theory of inflections, and introduced the so-called Hessian curve as the first instance of a eovariant of a ternarv form. To Chasles (q.v.) is due the method of characteristics developed by Halphen (1S7.5) and Schubert (1R79). and tiie general theory of correspondence (18G4),