CHAPTER XIV.

RECIPROCAL POLARS. PROJECTIONS.

300. If we have any figure consisting of any number of points and straight lines in a plane, and we take the polars of those points and the poles of the lines, with respect to a fixed conic C, we obtain another figure which is called the polar reciprocal of the former with respect to the auxiliary conic C.

When a point in one figure and a line in the reciprocal figure are pole and polar with respect to the auxiliary conic C, we shall say that they correspond to one another.

If in one figure we have a curve S the lines which correspond to the different points of S will all touch some curve S. Let the lines corresponding to the two points P, Q of S meet in T; then T is the pole of the line PQ with respect to C, that is the line PQ corresponds to the point T. Now, if the point Q move up to and ultimately coincide with P, the two corresponding tangents to S will also ultimately coincide with one another, and their point of intersection T will ultimately be on the curve S'. So that a tangent to the curve S corresponds to a point on the curve S', just as a tangent to S corresponds to a point on S. Hence we see that S is generated from S' exactly as S' is from S.

301. If any line L cut the curve S in any number of points P, Q, R...we shall have tangents to S corresponding to the points P, Q, R..., and these tangents will all pass through a point, viz. through the pole of L with respect to the auxiliary conic. Hence as many tangents to S' can be drawn through a point as there are points on S lying on a

straight line. That is to say the class [Art. 240] of S' is equal to the degree of S. Reciprocally the degree of S' is equal to the class of S.

In particular, if S be a conic it is of the second degree, and of the second class. Hence the reciprocal curve is of the second class, and of the second degree, and is therefore also a conic.

302. To find the polar reciprocal of one conic with respect to another.

Let the equation of the auxiliary conic be

ax2 + By2+1=0...........................................(i) ;

and let the equation of the conic whose reciprocal is required be

ax2 + by2+c+2ƒy + 2gx + 2hxy = 0 .......(ii). The line lx+my+ n = 0 will touch (ii) if

Al2 + Bm2 + C'n2 +2Fmn + 2Gnl+2Hlm = 0 (iii). And, if the pole of lx+my+n=0 with respect to (i) be (x', y'), its equation is the same as ax'x +ẞy'y +1=0.

Substitute, in (iii), and we have
Aa** +BB*ý* +C+2FB+2G

+2Haß ý = 0.

Hence the locus of the poles with respect to (i) of tangents to (ii) is the conic whose equation is

Aar +B+C+2FBy+2 Gà +2Haßxy=0.

303. The method of Reciprocal Polars enables us to obtain from any given theorem concerning the positions of points and lines, another theorem in which straight lines take the place of points and points of straight lines. Before proceeding to give examples of such reciprocal theorems we will give some simple cases of correspondence.

Points in one figure correspond to straight lines in the reciprocal figure.

The line joining two points in one figure corresponds to the point of intersection of the corresponding lines in the other.

The tangent to any curve in one figure corresponds to a point on the corresponding curve in the reciprocal figure. The point of contact of a tangent corresponds to the tangent at the corresponding point.

If two curves touch, that is have two coincident points common, the reciprocal curves will have two coincident tangents common, and will therefore also touch.

The chord joining two points corresponds to the point of intersection of the corresponding tangents.

The chord of contact of two tangents corresponds to the point of intersection of tangents at the corresponding points.

Since the pole of any line through the centre of the auxiliary conic is at infinity, we see that the points at infinity on the reciprocal curve correspond to the tangents to the original curve from the centre of the auxiliary conic. Hence the reciprocal of a conic is an hyperbola, parabola, or ellipse, according as the tangents to it from the centre of the auxiliary conic are real, coincident, or imaginary; that is according as the centre of the auxiliary conic is outside, upon, or within the curve.

The following are examples of reciprocal theorems.

If the angular points of two triangles are on a conic, their six sides will touch another conic.

The three intersections of opposite sides of a hexagon inscribed in a conic lie on a straight line.

(Pascal's Theorem).

If the three sides of a triangle touch a conic, and two of its angular points lie on a second conic, the locus of the third angular point is a conic.

If the sides of a triangle touch a conic, the three lines joining an angular point to the point of contact of the opposite side meet in a point.

If the sides of two triangles touch a conic, their six angular points are on another conic.

The three lines joining opposite angular points of a hexagon described about a conic meet in a point. (Brianchon's Theorem).

If the three angular points of a triangle lie on a conic, and two of its sides touch a second conic, the envelope of the third side is a conic.

If the angular points of a triangle lie on a conic, the three points of intersection of a side and the tangent at the opposite angular point lie on a line.

318

RECIPROCATION WITH RESPECT TO A CIRCLE.

The polars of a given point with respect to a system of conics through four given points all pass through a fixed point.

The locus of the pole of a given line with respect to a system of conics through four fixed points is a conic.

The poles of a given straight line with respect to a system of conics touching four given straight lines all lie on a fixed straight line.

The envelope of the polar of a given point with respect to a system of conics touching four fixed lines is a conic.

304. We now proceed to consider the results which can be obtained by reciprocating with respect to a circle.

We know that the line joining the centre of a circle to any point P is perpendicular to the polar of P with respect to the circle. Hence, if P, Q be any two points, the angle between the polars of these points with respect to a circle is equal to the angle that PQ subtends at the centre of the circle. Reciprocally the angle between any two straight lines is equal to the angle which the line joining their poles with respect to a circle subtends at the centre of the circle.

We know also that the distances, from the centre of a circle, of any point and of its polar with respect to that circle, are inversely proportional to one another.

If we reciprocate with respect to a circle it is clear that a change in the radius of the auxiliary circle will make no change in the shape of the reciprocal curve, but only in its size. Hence, if we are not concerned with the absolute magnitudes of the lines in the reciprocal figure, we only require to know the centre of the auxiliary circle. We may therefore speak of reciprocating with respect to a point O, instead of with respect to a circle having 0 for

centre.

305. If any conic be reciprocated with respect to a point O, the points on the reciprocal curve which corre spond to the tangents through O to the original curve must be at an infinite distance.

The directions of the lines to the points at infinity on the reciprocal curve are perpendicular to the tangents from 0 to the original curve; and hence the angle between

RECIPROCATION WITH RESPECT TO A CIRCLE. 319

the asymptotes of the reciprocal curve is supplementary to the angle between the tangents from 0 to the original

curve.

In particular, if the tangents from 0 to the original curve be at right angles, the reciprocal conic will be a rectangular hyperbola.

The axes of the reciprocal conic bisect the angles between its asymptotes. The axes are therefore parallel to the bisectors of the angles between the tangents from O to the original conic.

Corresponding to the points at infinity on the original conic we have the tangents to the reciprocal conic which pass through the origin. Hence the tangents from the origin to the reciprocal conic are perpendicular to the directions of the lines to the points at infinity on the original conic, so that the angle between the asymptotes of the original conic is supplementary to the angle between the tangents from the origin to the reciprocal conic.

In particular, if a rectangular hyperbola be reciprocated with respect to any point 0, the tangents from 0 to the reciprocal conic will be at right angles to one another; in other words O is a point on the director-circle of the reciprocal conic.

306. The reciprocal of the origin is the line at infinity, and therefore the reciprocal of the polar of the origin is the pole of the line at infinity. That is to say, the polar of the origin reciprocates into the centre of the reciprocal conic.

307. As an example of reciprocation take the known theorem-"If two of the conics which pass through four given points are rectangular hyperbolas, they will all be rectangular hyperbolas.' If this be reciprocated with respect to any point 0 we obtain the following, “If the director-circles of two of the conics which touch four given straight lines pass through a point 0, the director-circles of all the conics will pass through 0." Or, what is the same thing, "The director-circles of all conics which touch four given straight lines have a common radical axis."