RELATIONS. The ollowing are sonic 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. the evolute as the principal curve, the original curve is called its involute. The normals to any curve arc tangents to its evolute.
(2) Two curves or surfaces arc 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 (x) at the point whose abscissa is a when q5 (a) = a), fio' (a) =1,/,(0,0(2) (a)=1,1,(2)(a). If a. is even, the curves cross at the point. No curie 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 tonches all of the intersecting curves thus obtained; e.g. if p is a variable parameter and f = 0 is the equation of the curve, then the re sult obtained by eliminating p between f = df and --- 0 is the equation of the envelope.
dp 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 equation is given in rectangular coordinates, sub stitute for .r, . respectively.
y' ± (6) A roulette is defined as the locus of a point rigidly connected with a curve which rolls upon a fixed line or curve. See CYCLOID.