An Elementary Course in Synthetic Projective Geometry Part 15

whence

_OB OB' : PK PM = ON OL : PN PL._

In the same way, from the similar triangles _OAL_ and _PKL_, and also _OA'N_ and _PMN_, we obtain

_OA OA' : PK PM = ON OL : PN PL,_

and this, with the preceding, gives at once the fundamental theorem, which is sometimes taken also as the definition of involution:

_OA OA' = OB OB' = __constant__,_

or, in words,

_The product of the distances from the center to two corresponding points in an involution of points is constant._

*143. Existence of double points.* Clearly, according as the constant is positive or negative the involution will or will not have double points.

The constant is the square root of the distance from the center to the double points. If _A_ and _A'_ lie both on the same side of the center, the product _OA OA'_ is positive; and if they lie on opposite sides, it is negative. Take the case where they both lie on the same side of the center, and take also the pair of corresponding points _BB'_. Then, since _OA OA' = OB OB'_, it cannot happen that _B_ and _B'_ are separated from each other by _A_ and _A'_. This is evident enough if the points are on opposite sides of the center. If the pairs are on the same side of the center, and _B_ lies between _A_ and _A'_, so that _OB_ is greater, say, than _OA_, but less than _OA'_, then, by the equation _OA OA' = OB OB'_, we must have _OB'_ also less than _OA'_ and greater than _OA_. A similar discussion may be made for the case where _A_ and _A'_ lie on opposite sides of _O_. The results may be stated as follows, without any reference to the center:

_Given two pairs of points in an involution of points, if the points of one pair are separated from each other by the points of the other pair, then the involution has no double points. If the points of one pair are not separated from each other by the points of the other pair, then the involution has two double points._

*144.* An entirely similar criterion decides whether an involution of rays has or has not double rays, or whether an involution of planes has or has not double planes.

[Figure 40]

FIG. 40

*145. Construction of an involution by means of circles.* The equation just derived, _OA OA' = OB OB'_, indicates another simple way in which points of an involution of points may be constructed. Through _A_ and _A'_ draw any circle, and draw also any circle through _B_ and _B'_ to cut the first in the two points _G_ and _G'_ (Fig. 40). Then any circle through _G_ and _G'_ will meet the line in pairs of points in the involution determined by _AA'_ and _BB'_. For if such a circle meets the line in the points _CC'_, then, by the theorem in the geometry of the circle which says that _if any chord is __ drawn through a fixed point within a circle, the product of its segments is constant in whatever direction the chord is drawn, and if a secant line be drawn from a fixed point without a circle, the product of the secant and its external segment is constant in whatever direction the secant line is drawn_, we have _OC OC' = OG OG' =_ constant. So that for all such points _OA OA' = OB OB' = OC OC'_.

Further, the line _GG'_ meets _AA'_ in the center of the involution. To find the double points, if they exist, we draw a tangent from _O_ to any of the circles through _GG'_. Let _T_ be the point of contact. Then lay off on the line _OA_ a line _OF_ equal to _OT_. Then, since by the above theorem of elementary geometry _OA OA' = OT__2__ = OF__2_, we have one double point _F_. The other is at an equal distance on the other side of _O_. This simple and effective method of constructing an involution of points is often taken as the basis for the theory of involution. In projective geometry, however, the circle, which is not a figure that remains unaltered by projection, and is essentially a metrical notion, ought not to be used to build up the purely projective part of the theory.

*146.* It ought to be mentioned that the theory of a.n.a.lytic geometry indicates that the circle is a special conic section that happens to pa.s.s through two particular imaginary points on the line at infinity, called the _circular points_ and usually denoted by _I_ and _J_. The above method of obtaining a point-row in involution is, then, nothing but a special case of the general theorem of the last chapter (-- 125), which a.s.serted that a system of conics through four points will cut any line in the plane in a point-row in involution.

[Figure 41]

FIG. 41

*147. Pairs in an involution of rays which are at right angles. Circular involution.* In an involution of rays there is no one ray which may be distinguished from all the others as the point at infinity is distinguished from all other points on a line. There is one pair of rays, however, which does differ from all the others in that for this particular pair the angle is a right angle. This is most easily shown by using the construction that employs circles, as indicated above. The centers of all the circles through _G_ and _G'_ lie on the perpendicular bisector of the line _GG'_. Let this line meet the line _AA'_ in the point _C_ (Fig. 41), and draw the circle with center _C_ which goes through _G_ and _G'_. This circle cuts out two points _M_ and _M'_ in the involution. The rays _GM_ and _GM'_ are clearly at right angles, being inscribed in a semicircle.

If, therefore, the involution of points is projected to _G_, we have found two corresponding rays which are at right angles to each other. Given now any involution of rays with center _G_, we may cut across it by a straight line and proceed to find the two points _M_ and _M'_. Clearly there will be only one such pair unless the perpendicular bisector of _GG'_ coincides with the line _AA'_. In this case every ray is at right angles to its corresponding ray, and the involution is called _circular_.

*148. Axes of conics.* At the close of the last chapter (-- 140) we gave the theorem: _A conic determines at every point in its plane an involution of rays, corresponding rays __ being conjugate with respect to the conic.

The double rays, if any exist, are the tangents from the point to the conic._ In particular, taking the point as the center of the conic, we find that conjugate diameters form a system of rays in involution, of which the asymptotes, if there are any, are the double rays. Also, conjugate diameters are harmonic conjugates with respect to the asymptotes. By the theorem of the last paragraph, there are two conjugate diameters which are at right angles to each other. These are called axes.

In the case of the parabola, where the center is at infinity, and on the curve, there are, properly speaking, no conjugate diameters. While the line at infinity might be considered as conjugate to all the other diameters, it is not possible to a.s.sign to it any particular direction, and so it cannot be used for the purpose of defining an axis of a parabola. There is one diameter, however, which is at right angles to its conjugate system of chords, and this one is called the _axis_ of the parabola. The circle also furnishes an exception in that every diameter is an axis. The involution in this case is circular, every ray being at right angles to its conjugate ray at the center.

*149. Points at which the involution determined by a conic is circular.*

It is an important problem to discover whether for any conic other than the circle it is possible to find any point in the plane where the involution determined as above by the conic is circular. We shall proceed to the curious problem of proving the existence of such points and of determining their number and situation. We shall then develop the important properties of such points.

*150.* It is clear, in the first place, that such a point cannot be on the outside of the conic, else the involution would have double rays and such rays would have to be at right angles to themselves. In the second place, if two such points exist, the line joining them must be a diameter and, indeed, an axis. For if _F_ and _F'_ were two such points, then, since the conjugate ray at _F_ to the line _FF'_ must be at right angles to it, and also since the conjugate ray at _F'_ to the line _FF'_ must be at right angles to it, the pole of _FF'_ must be at infinity in a direction at right angles to _FF'_. The line _FF'_ is then a diameter, and since it is at right angles to its conjugate diameter, it must be an axis.

From this it follows also that the points we are seeking must all lie on one of the two axes, else we should have a diameter which does not go through the intersection of all axes-the center of the conic. At least one axis, therefore, must be free from any such points.

[Figure 42]

FIG. 42

*151.* Let now _P_ be a point on one of the axes (Fig. 42), and draw any ray through it, such as _q_. As _q_ revolves about _P_, its pole _Q_ moves along a line at right angles to the axis on which _P_ lies, describing a point-row _p_ projective to the pencil of rays _q_. The point at infinity in a direction at right angles to _q_ also describes a point-row projective to _q_. The line joining corresponding points of these two point-rows is always a conjugate line to _q_ and at right angles to _q_, or, as we may call it, a _conjugate normal_ to _q_. These conjugate normals to _q_, joining as they do corresponding points in two projective point-rows, form a pencil of rays of the second order. But since the point at infinity on the point-row _Q_ corresponds to the point at infinity in a direction at right angles to _q_, these point-rows are in perspective position and the normal conjugates of all the lines through _P_ meet in a point. This point lies on the same axis with _P_, as is seen by taking _q_ at right angles to the axis on which _P_ lies. The center of this pencil may be called _P'_, and thus we have paired the point _P_ with the point _P'_. By moving the point _P_ along the axis, and by keeping the ray _q_ parallel to a fixed direction, we may see that the point-row _P_ and the point-row _P'_ are projective. Also the correspondence is double, and by starting from the point _P'_ we arrive at the point _P_. Therefore the point-rows _P_ and _P'_ are in involution, and if only the involution has double points, we shall have found in them the points we are seeking. For it is clear that the rays through _P_ and the corresponding rays through _P'_ are conjugate normals; and if _P_ and _P'_ coincide, we shall have a point where all rays are at right angles to their conjugates. We shall now show that the involution thus obtained on one of the two axes must have double points.

[Figure 43]

FIG. 43

*152. Discovery of the foci of the conic.* We know that on one axis no such points as we are seeking can lie (-- 150). The involution of points _PP'_ on this axis can therefore have no double points. Nevertheless, let _PP'_ and _RR'_ be two pairs of corresponding points on this axis (Fig.

43). Then we know that _P_ and _P'_ are separated from each other by _R_ and _R'_ (-- 143). Draw a circle on _PP'_ as a diameter, and one on _RR'_ as a diameter. These must intersect in two points, _F_ and _F'_, and since the center of the conic is the center of the involution _PP'_, _RR'_, as is easily seen, it follows that _F_ and _F'_ are on the other axis of the conic. Moreover, _FR_ and _FR'_ are conjugate normal rays, since _RFR'_ is inscribed in a semicircle, and the two rays go one through _R_ and the other through _R'_. The involution of points _PP'_, _RR'_ therefore projects to the two points _F_ and _F'_ in two pencils of rays in involution which have for corresponding rays conjugate normals to the conic. We may, then, say:

_There are two and only two points of the plane where the involution determined by the conic is circular. These two points lie on one of the axes, at equal distances from the center, on the inside of the conic.

These points are called the foci of the conic._

*153. The circle and the parabola.* The above discussion applies only to the central conics, apart from the circle. In the circle the two foci fall together at the center. In the case of the parabola, that part of the investigation which proves the existence of two foci on one of the axes will not hold, as we have but one axis. It is seen, however, that as _P_ moves to infinity, carrying the line _q_ with it, _q_ becomes the line at infinity, which for the parabola is a tangent line. Its pole _Q_ is thus at infinity and also the point _P'_, so that _P_ and _P'_ fall together at infinity, and therefore one focus of the parabola is at infinity. There must therefore be another, so that