An Elementary Course in Synthetic Projective Geometry

by Derrick Norman Lehmer

CHAPTER IX - METRICAL PROPERTIES OF INVOLUTIONS

Fig. 39

141. Introduction of infinite point; center of involution. We connect the projective theory of involution with the metrical, as usual, by the introduction of the elements at infinity. In an involution of points on a line the point which corresponds to the infinitely distant point is called the center of the involution. Since corresponding points in the involution have been shown to be harmonic conjugates with respect to the double points, the center is midway between the double points when they exist. To construct the center (Fig. 39) we draw as usual through A and A' any two rays and cut them by a line parallel to AA' in the points K and M. Join these points to B and B', thus determining on AK and AN the points L and N. LN meets AA' in the center O of the involution.

142. Fundamental metrical theorem. From the figure we see that the triangles OLB' and PLM are similar, P being the intersection of KM and LN. Also the triangles KPN and BON are similar. We thus have

OB : PK = ON : PN

and

OB' : PM = OL : PL;

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 [pg 86] 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.

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 [pg 87] 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' = OT2 = OF2, 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 analytic geometry indicates that the circle is a special conic section that happens to pass 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 asserted that a system of conics through four points will cut any line in the plane in a point-row in involution.

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 [pg 89] 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 assign 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.

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 [pg 91] 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.

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 [pg 92] 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 [pg 93] 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

A parabola has one and only one focus in the finite part of the plane.

Fig. 44

154. Focal properties of conics. We proceed to develop some theorems which will exhibit the importance of these points in the theory of the conic section. Draw a tangent to the conic, and also the normal at the point of contact P. These two lines are clearly conjugate normals. The two points T and N, therefore, where they meet the axis which contains the foci, are corresponding points in the involution considered above, and are therefore harmonic conjugates with respect to the foci (Fig. 44); and if we join them to the point P, we shall obtain four harmonic lines. But two of them are at right angles to each other, and so the others make equal angles with them (Problem 4, Chapter II). Therefore

The lines joining a point on the conic to the foci make equal angles with the tangent.

It follows that rays from a source of light at one focus are reflected by an ellipse to the other.

155. In the case of the parabola, where one of the foci must be considered to be at infinity in the direction of the diameter, we have

Fig. 45

A diameter makes the same angle with the tangent at its extremity as that tangent does with the line from its point of contact to the focus (Fig. 45).

156. This last theorem is the basis for the construction of the parabolic reflector. A ray of light from the focus is reflected from such a reflector in a direction parallel to the axis of the reflector.

157. Directrix. Principal axis. Vertex. The polar of the focus with respect to the conic is called the directrix. The axis which contains the foci is called the principal axis, and the intersection of the axis with the curve is called the vertex of the curve. The directrix is at right angles to the principal axis. In a parabola the vertex is equally distant from the focus and the directrix, these three points and the point at infinity on the axis being four harmonic points. In the ellipse the vertex is nearer to the focus than it is to the directrix, for the same reason, and in the hyperbola it is farther from the focus than it is from the directrix.

Fig. 46

158. Another definition of a conic. Let P be any point on the directrix through which a line is drawn meeting the conic in the points A and B (Fig. 46). Let the tangents at A and B meet in T, and call the focus F. Then TF and PF are conjugate lines, and as they pass through a focus they must be at right angles to each other. Let [pg 95] TF meet AB in C. Then P, A, C, B are four harmonic points. Project these four points parallel to TF upon the directrix, and we then get the four harmonic points P, M, Q, N. Since, now, TFP is a right angle, the angles MFQ and NFQ are equal, as well as the angles AFC and BFC. Therefore the triangles MAF and NFB are similar, and FA : FM = FB : BN. Dropping perpendiculars AA and BB' upon the directrix, this becomes FA : AA' = FB : BB'. We have thus the property often taken as the definition of a conic:

The ratio of the distances from a point on the conic to the focus and the directrix is constant.

Fig. 47

159. Eccentricity. By taking the point at the vertex of the conic, we note that this ratio is less than unity for the ellipse, greater than unity for the hyperbola, and equal to unity for the parabola. This ratio is called the eccentricity.

Fig. 48

160. Sum or difference of focal distances. The ellipse and the hyperbola have two foci and two directrices. The eccentricity, of course, is the same for one focus as for the other, since the curve is symmetrical with respect to both. If the distances from [pg 96] a point on a conic to the two foci are r and r', and the distances from the same point to the corresponding directrices are d and d' (Fig. 47), we have r : d = r' : d'; (r ± r') : (d ± d'). In the ellipse (d + d') is constant, being the distance between the directrices. In the hyperbola this distance is (d - d'). It follows (Fig. 48) that

In the ellipse the sum of the focal distances of any point on the curve is constant, and in the hyperbola the difference between the focal distances is constant.