Conic Section

CONIC SECTION.) For certain sets of values of the parameters, the left-hand side of the equation falls into factors, each of which by itself gives a separate curve. The product vanishes whenever any factor van ishes by itself, and the equation is satisfied by the co-ordinates of a point on any one of these curves, and so represents their ag gregate. In many cases it is convenient still to regard this as a single curve, which is then said to be degenerate. Thus a pair of intersecting lines is a degenerate conic. A curve which does not break up is said to be proper.

In the plane; any aggregate of curves can be regarded as a degenerate curve, of degree equal to the sum of their separate degrees. In space this is not so unless the components satisfy certain conditions of incidence : a pair of skew lines is not a degenerate conic.

A conic and a line meeting it twice, and therefore lying in its plane, form a degenerate plane cubic ; a conic and a line, not in its plane, but meeting it once, form a degenerate twisted cubic ; a conic and a line not meeting it, but meeting its plane in a point not lying on the conic, cannot be regarded as a single degenerate curve.

The common points of two curves of degrees are those whose co-ordinates satisfy both equations, and their number is equal to that of the solutions of the eliminant. This con tinues to hold if one or both of the curves break up. In any case, we must reckon imaginary and infinite solutions as corresponding to intersections of the same natures. The points at infinity of the plane must be considered to lie on a line with which the curve k has n intersections, real or imaginary.

A point of k at infinity lies on a branch stretching to infinity; the tangent at a point retreating along this branch may tend to a limiting position not wholly at infinity, its one point at infinity being that point through which the branch of k passes. Such a tangent is called an asymptote (q.v.), and k has in general n of them; their directions are given by the terms of highest degree in its Cartesian equation. But it may happen that the limit of the tangent is the line at infinity itself ; then there is no corresponding linear asymptote.

Now an asymptote has only one point at infinity, the same towards whichever end of the line we retreat ; thus a curve approaches its asymptote at both ends, and in general on opposite sides, as viewed from the finite part of the plane. This appears strange at first sight ; but two parallel lines have a simple inter section at infinity, and the second lies on the same side of the first at both ends. A curve touches its asymptote at infinity, having two intersections with it there ; so if it and a given parallel to the asymptote lie on the same side of the asymptote at one end, they must lie on opposite sides at the other.

Singularities.

The interest of any curve depends largely on its exceptional points, where the ordinary properties fail. A singu lar point of a curve k is one where there is not a unique, definite tangent. The simplest is an ordinary double point or node 0, where two separate branches of k cross without touching. Here there is not one definite tangent, but two, one belonging to each branch. A moving point describ ing k comes to 0 twice, on differ ent occasions and in different di rections. A general line 1, passing near 0 but not through it, meets k in two points near 0, one on each branch, both of which tend to 0 when 1 moves so as to pass through 0. If 1 is ultimately one of the tangents at 0, three such intersections are absorbed, two on the branch touched, and one on the other.

In general, the multiplicity s of a singular point 0 is the number of intersections with a general line absorbed there. Through 0 there are always s lines, distinct or coincident, on which more than s intersections are absorbed, and which are the tangents at 0. If the s tangents are distinct, the multiple point 0 is ordinary. If 0 is s1-fold on one curve and s,-fold on another, it absorbs just s162 of their points of intersection if they have no common tangent at 0; if they touch, it may absorb any greater number.

If any of the s tangents coincide, 0 is an extraordinary singu larity. The simplest is a cusp of first species, a variety of double point, which may be thought of as the limit of a loop drawn together to its node.

There is only one branch of k at 0, and one distinct tangent meeting k in three coincident points at 0. A point P describing k continuously comes to rest at 0 and reverses its direction of motion; for this reason a cusp is often called a stationary point. The tangent at P rotates in the same sense without reversal.

A cusp is something like the path of an engine running past catch-points to rest, and backing on to another pair of rails; that is, running down on one branch of a Y and backing up on the other.

A set of different but corresponding singularities presents itself when a curve is regarded as an envelope. A singular tangent is one which has not a unique, definite point of contact. The line singu larity which answers to an ordi nary double point is a double tan gent, with two distinct, definite points of contact, each a simple point of the curve. That which answers to a cusp is an inflexion, when the two points of contact coincide, and the tangent meets the curve in three coincident points. A tangent describing the envelope comes to rest and reverses its direction of rotation, but a point describing the curve goes on without reversing, for the inflexion is a simple point of the curve.

Every singularity of a curve, which is of higher multiplicity or more complicated nature than the four elementary kinds described above, is, from a great many points of view, equivalent to a cer tain set of distinct double points, cusps, double tangents and inflexions, and can usually be regarded as the limit of this set when certain points and lines come to coincide. Thus a triple point can arise from three double points.

Transformations; Genus.—Two curves, whether plane or twisted, are said to be in r, z correspondence, or transformations of each other, or represented upon each other, if each ordinary point of either corresponds to one and only one point of the other. If the co-ordinates of the first point are given, those of the second are one-valued and therefore rational functions of these, and the equations expressing this are rationally reversible, so that the co-ordinates of the first point are also expressible as rational functions of those of the second. Any curve can be thus trans formed into a plane curve having no multiple points except or dinary double points, or into a twisted curve having no singular points at all.

A plane curve of given degree cannot have more double points or cusps than (n— I) (n— 2), or their equivalent, without break ing up. Thus the only nodal conic is a pair of lines. A proper cubic can have one node; if it had two, the line joining them would have four intersections with the cubic, which is impossible: the curve would break up into the line and a conic through the two points. The number p= by which the equivalent number 6-1--K of double points and cusps falls short of this maximum is called its deficiency or more usually its genus. The fundamental property of the genus is that it is unaltered by any I, I transformation of the curve.

If p = o, the curve is rational or unicursal, and can be trans formed into a line. The co-ordinates of its general point can be expressed as rational functions of a single parameter, the co ordinate of the corresponding point on the line.

A I, i correspondence is set up between a twisted curve

k and its projection k' from any vertex V upon any plane. The singulari ties of k' are the projections of those of k, and, in addition, a cer tain number of ordinary double points lying on rays through V which meet k in two distinct points. These are called apparent double points of k, and their num ber is one of its essential charac teristics, for it reduces the genus, being an additional number of singularities of k', whose genus is the same as that of k.

Plucker's Equations.—All plane curves other than lines and conics possess singularities of some sort. If they are free from multiple points, they are bound to have definite numbers of double tangents and inflexions, or their equivalent in higher singular tan gents. If multiple points are present, the numbers of multiple tangents are reduced. Between the degree n, class n', and the equivalent numbers 6, K, S,'K' of double points, cusps, double tan gents and inflexions there exists a remarkable set of relations known as Plucker's equations, by which any three of the six num bers can be calculated in terms of the other three: 2 (8' —a) = (n' — n) (n'+n— 9) Thus the cubic with no node is of class 6 and has no double tangents and 9 inflexions: the nodal cubic is of class 4 with 3 inflexions; and the cuspidal cubic is of class 3 with one inflexion. The quartic with no singular points has 28 double tangents and 24 inflexions.

The points of contact of the

n' tangents which can be drawn to k from any point P of the plane are the intersections, other than the multiple points of k, with a certain derived curve of degree n— I, called the first polar of P with regard to k. This passes through each double point of k, meeting it in two points there, and touches k at each cusp, meeting it in three points there. The first of Pliicker's equations expresses that the total number of inter sections of k with the first polar is n(n— I).