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Curves

curve, equation, line, algebraic, system, arbitrary and ay

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CURVES, Higher Plane. A curve can be looked upon in many ways; geometrically as the intersection of two surfaces, as the locus of a moving point, or envelope of a moving line; analytically as a representation of an equation in point- or line-co-ordinates, and therefore as yielding a singly infinite system of points or lines. But if this view be adopted, the °curve') must not necessarily be regarded as identical with the system of points; for most curves (not all) have tangents, hence a curve yields also a system of lines, of equal importance with the system of points; a defini tion that lays stress on one system to the ex clusion of the other is incomplete. This was recognized by Plucker ((Theorie der alge braischen Curven,' 1839, p. 200), in his state ment of the dual generation of a curve: °If a point continually moves along a straight line, while the line continually rotates about the point, one and the same curve is enveloped by the line and described by the point Clifford ((Math. Papers,' pp. 40-42) treats the true curve as an undefined entity, of which the as semblages of points and lines are two distinct manifestations. The present tendency is to ward this view, at least as regards an algebraic curve—a curve whose equation, whether in point- or line-co-ordinates, is algebraic. A curve whose Cartesian equation cannot be re duced to an algebraic form, e.g., y= sin x, is non-algebraic or transcendental. It is conven ient to treat first of algebraic curves.

If x, y satisfy an algebraic equation dy f (x, 0, it can be shown that has a defi a a tiitevalue( --- : ay af ---) f unless — and z ax a f vanish. There is therefore a tangent, dy ,„. , a f , , " ,af „ (X - x) dx ay in homogeneous co-ordinates ,,af „af , af— r ax ay, azThe co-ordinates of the tangent are: ; the elimination of x, y, ax ay fig from these equations and f(x, y, gives an algebraic equation (f. n, C) the line equation (tangential equation) of the curve. Thus the curve has two equations, both alge braic, a point equation of degree m, a line equa tion of degree n; these two numbers m, n, the order and class of the curve, are the number of points that lie on an arbitrary line, the num ber of tangents that pass through an arbitrary point; they belong respectively to the point sys tem and the line system, not to the curve itself.

A number that is more intimately associated with the curve is the genus, p, to be defined later.

An algebraic curve cannot break off; the tangent cannot suffer a sudden change in di rection; no finite part of the curve can coin cide with a straight line. Thus the normal character of an arc of an algebraic curve ex presses gradual and continuous change of posi tion (motion of point), gradual and continuous change of direction (motion of tangent), as stated by Plucker.

The number of terms in the equation f(x, y, z)= 0 is + 1) (m + 2) ; the num ber of disposable constants is therefore %(n + 1)(m + 2) — I, i.e., Y2m ( m + 3). Hence passage through + 3) arbitrary points determines the curve; while if the curve passes through /m 3)—q arbitrary points, the coefficients can be expressed linearly in terms of q parameters, and the curve has q degrees of freedom. If the points are not arbi trary, the curve may have mobility greater than q, or it may break up into curves of lower order. The theory is really that of the inter sections of curves. Two curves of orders in, m' intersect in mm' points, for the elimination of z from the equations produces an equation of degree mm' for x:y. The m' intersections of two rn-ics, u, v, do not determine an m-ic (although + 3)), for all curves is + kv —0 pass through the points; the vie points impose precisely + 3)-1 condi tions. Similarly, the mm intersections of C,,, and C„,, do not impose independent conditions on all curves; e. g., the 20 intersections of C. and C. impose 14 conditions on a quartic, 17 on a quintic, 19 on a sextic, but 20 on all higher curves. The most convenient statement is: the quartic excess is 6, the quintic excess is 3, the sextic excess is 1 (Macaulay). The first notice of theorems of this character is due to Maclaurin (1720) ; the first explanation was given by Euler and Cramer (1748-50). From these has arisen the whole modern theory of groups of points on a curve (geometry on a curve). (Scott, 'Bull. Am. Math. Soc.,' 2d series, v. 4, pp. 260-273; 1897-98).

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