An Elementary Course in Synthetic Projective Geometry

by Derrick Norman Lehmer

CHAPTER VIII - INVOLUTION

Fig. 32

121. Fundamental theorem. The important theorem concerning two complete quadrangles (§ 26), upon which the theory of four harmonic points was based, can easily be extended to the case where the four lines KL, K'L', MN, M'N' do not all meet in the same point A, and the more general theorem that results may also be made the basis of a theory no less important, which has to do with six points on a line. The theorem is as follows:

Given two complete quadrangles, K, L, M, N and K', L', M', N', so related that KL and K'L' meet in A, MN and M'N' in A', KN and K'N' in B, LM and L'M' in B', LN and L'N' in C, and KM and K'M' in C', then, if A, A', B, B', and C are in a straight line, the point C' also lies on that straight line.

The theorem follows from Desargues's theorem (Fig. 32). It is seen that KK', LL', MM', NN' all [pg 72] meet in a point, and thus, from the same theorem, applied to the triangles KLM and K'L'M', the point C' is on the same line with A and B'. As in the simpler case, it is seen that there is an indefinite number of quadrangles which may be drawn, two sides of which go through A and A', two through B and B', and one through C. The sixth side must then go through C'. Therefore,

122. Two pairs of points, A, A' and B, B', being given, then the point C' corresponding to any given point C is uniquely determined.

The construction of this sixth point is easily accomplished. Draw through A and A' any two lines, and cut across them by any line through C in the points L and N. Join N to B and L to B', thus determining the points K and M on the two lines through A and A', The line KM determines the desired point C'. Manifestly, starting from C', we come in this way always to the same point C. The particular quadrangle employed is of no consequence. Moreover, since one pair of opposite sides in a complete quadrangle is not distinguishable in any way from any other, the same set of six points will be obtained by starting from the pairs AA' and CC', or from the pairs BB' and CC'.

123. Definition of involution of points on a line.

Three pairs of points on a line are said to be in involution if through each pair may be drawn a pair of opposite sides of a complete quadrangle. If two pairs are fixed and one of the third pair describes the line, then the other also describes the line, and the points of the line are said to be paired in the involution determined by the two fixed pairs.

Fig. 33

124. Double-points in an involution. The points C and C' describe projective point-rows, as may be seen by fixing the points L and M. The self-corresponding points, of which there are two or none, are called the double-points in the involution. It is not difficult to see that the double-points in the involution are harmonic conjugates with respect to corresponding points in the involution. For, fixing as before the points L and M, let the intersection of the lines CL and C'M be P (Fig. 33). The locus of P is a conic which goes through the double-points, because the point-rows C and C' are projective, and therefore so are the pencils LC and MC' which generate the locus of P. Also, when C and C' fall together, the point P coincides with them. Further, the tangents at L and M to this conic described by P are the lines LB and MB. For in the pencil at L the ray LM common to the two pencils which generate the conic is the ray LB' and corresponds to the ray MB of M, which is therefore the tangent line to the conic at M. Similarly for the tangent LB at L. LM is therefore the polar of B with respect to this conic, and B and B' are therefore harmonic conjugates with respect to the double-points. The same discussion applies to any other pair of corresponding points in the involution.

Fig. 34

125. Desargues's theorem concerning conics through four points. Let DD' be any pair of points in the involution determined as above, and consider the conic passing through the five points K, L, M, N, D. We shall use Pascal's theorem to show that this conic also passes through D'. The point D' is determined as follows: Fix L and M as before (Fig. 34) and join D to L, giving on MN the point N'. Join N' to B, giving on LK the point K'. Then MK' determines the point D' on the line AA', given by the complete quadrangle K', L, M, N'. Consider the following six points, numbering them in order: D = 1, D' = 2, M = 3, N = 4, K = 5, and L = 6. We have the following intersections: B = (12-45), K' = (23-56), N' = (34-61); and since by construction B, N, and K' are on a straight line, it follows from the converse of Pascal's theorem, which is easily established, that the six points are on a conic. We have, then, the beautiful theorem due to Desargues:

The system of conics through four points meets any line in the plane in pairs of points in involution.

126. It appears also that the six points in involution determined by the quadrangle through the four fixed [pg 75] points belong also to the same involution with the points cut out by the system of conics, as indeed we might infer from the fact that the three pairs of opposite sides of the quadrangle may be considered as degenerate conics of the system.

127. Conics through four points touching a given line. It is further evident that the involution determined on a line by the system of conics will have a double-point where a conic of the system is tangent to the line. We may therefore infer the theorem

Through four fixed points in the plane two conics or none may be drawn tangent to any given line.

Fig. 35

128. Double correspondence. We have seen that corresponding points in an involution form two projective point-rows superposed on the same straight line. Two projective point-rows superposed on the same straight line are, however, not necessarily in involution, as a simple example will show. Take two lines, a and a', which both revolve about a fixed point S and which always make the same angle with each other (Fig. 35). These lines cut out on any line in the plane which does not pass through S two projective point-rows, which are not, however, in involution unless the angle between the lines is a right angles. For a point P may correspond to a point P', which in turn will correspond to some other point [pg 76] than P. The peculiarity of point-rows in involution is that any point will correspond to the same point, in whichever point-row it is considered as belonging. In this case, if a point P corresponds to a point P', then the point P' corresponds back again to the point P. The points P and P' are then said to correspond doubly. This notion is worthy of further study.

Fig. 36

129. Steiner's construction. It will be observed that the solution of the fundamental problem given in § 83, Given three pairs of points of two protective point-rows, to construct other pairs, cannot be carried out if the two point-rows lie on the same straight line. Of course the method may be easily altered to cover that case also, but it is worth while to give another solution of the problem, due to Steiner, which will also give further information regarding the theory of involution, and which may, indeed, be used as a foundation for that theory. Let the two point-rows A, B, C, D, ... and A', B', C', D', ... be superposed on the line u. Project them both to a point S and pass any conic κ through S. We thus obtain two projective pencils, a, b, c, d, ... and [pg 77] a', b', c', d', ... at S, which meet the conic in the points α, β, γ, δ, ... and α', β', γ', δ', ... (Fig. 36). Take now γ as the center of a pencil projecting the points α', β', δ', ..., and take γ' as the center of a pencil projecting the points α, β, δ, .... These two pencils are projective to each other, and since they have a self-correspondin ray in common, they are in perspective position and corresponding rays meet on the line joining (γα', γ'α) to (γβ', γ'β). The correspondence between points in the two point-rows on u is now easily traced.

130. Application of Steiner's construction to double correspondence. Steiner's construction throws into our hands an important theorem concerning double correspondence: If two projective point-rows, superposed on the same line, have one pair of points which correspond to each other doubly, then all pairs correspond to each other doubly, and the line is paired in involution. To make this appear, let us call the point A on u by two names, A and P', according as it is thought of as belonging to the one or to the other of the two point-rows. If this point is one of a pair which correspond to each other doubly, then the points A' and P must coincide (Fig. 37). Take now any point C, which we will also call R'. We must show that the corresponding point C' must also coincide with the point B. Join all the points to S, as before, and it appears that the points α and π' coincide, as also do the points α'π and γρ'. By the above construction the line γ'ρ must meet γρ' on the line joining (γα', γ'α) with (γπ', γ'π). But these four points form a quadrangle inscribed in the conic, and we know by § 95 that the tangents at the opposite [pg 78] vertices γ and γ' meet on the line v. The line γ'ρ is thus a tangent to the conic, and C' and R are the same point. That two projective point-rows superposed on the same line are also in involution when one pair, and therefore all pairs, correspond doubly may be shown by taking S at one vertex of a complete quadrangle which has two pairs of opposite sides going through two pairs of points. The details we leave to the student.

Fig. 37

Fig. 38

131. Involution of points on a point-row of the second order. It is important to note also, in Steiner's construction, that we have obtained two point-rows of the second order superposed on the same conic, and have paired the points of one with the points of the other in such a way that the correspondence is double. We may then extend the notion of involution to point-rows of the second order and say that the points of a conic are paired in involution when they are corresponding [pg 79] points of two projective point-rows superposed on the conic, and when they correspond to each other doubly. With this definition we may prove the theorem: The lines joining corresponding points of a point-row of the second order in involution all pass through a fixed point U, and the line joining any two points A, B meets the line joining the two corresponding points A', B' in the points of a line u, which is the polar of U with respect to the conic. For take A and A' as the centers of two pencils, the first perspective to the point-row A', B', C' and the second perspective to the point-row A, B, C. Then, since the common ray of the two pencils corresponds to itself, they are in perspective position, and their axis of perspectivity u (Fig. 38) is the line which joins the point (AB', A'B) to the point (AC', A'C). It is then immediately clear, from the theory of poles and polars, that BB' and CC' pass through the pole U of the line u.

132. Involution of rays. The whole theory thus far developed may be dualized, and a theory of lines in involution may be built up, starting with the complete quadrilateral. Thus,

The three pairs of rays which may be drawn from a point through the three pairs of opposite vertices of a complete quadrilateral are said to be in involution. If the pairs aa' and bb' are fixed, and the line c describes a pencil, the corresponding line c' also describes a pencil, and the rays of the pencil are said to be paired in the involution determined by aa' and bb'.

133. Double rays. The self-corresponding rays, of which there are two or none, are called double rays of the involution. Corresponding rays of the involution are harmonic conjugates with respect to the double rays. To the theorem of Desargues (§ 125) which has to do with the system of conics through four points we have the dual:

The tangents from a fixed point to a system of conics tangent to four fixed lines form a pencil of rays in involution.

134. If a conic of the system should go through the fixed point, it is clear that the two tangents would coincide and indicate a double ray of the involution. The theorem, therefore, follows:

Two conics or none may be drawn through a fixed point to be tangent to four fixed lines.

135. Double correspondence. It further appears that two projective pencils of rays which have the same center are in involution if two pairs of rays correspond to each other doubly. From this it is clear that we might have deemed six rays in involution as six rays which pass through a point and also through six points in involution. While this would have been entirely in accord with the treatment which was given the corresponding problem in the theory of harmonic points and lines, it is more satisfactory, from an aesthetic point of view, to build the theory of lines in involution on its own base. The student can show, by methods entirely analogous to those used in the second chapter, that involution is a projective property; that is, six rays in involution are cut by any transversal in six points in involution.

136. Pencils of rays of the second order in involution. We may also extend the notion of involution to pencils of rays of the second order. Thus, the tangents to a conic are in involution when they are corresponding rays of two protective pencils of the second order superposed upon the same conic, and when they correspond to each other doubly. We have then the theorem:

137. The intersections of corresponding rays of a pencil of the second order in involution are all on a straight line u, and the intersection of any two tangents ab, when joined to the intersection of the corresponding tangents a'b', gives a line which passes through a fixed point U, the pole of the line u with respect to the conic.

138. Involution of rays determined by a conic. We have seen in the theory of poles and polars (§ 103) that if a point P moves along a line m, then the polar of P revolves about a point. This pencil cuts out on m another point-row P', projective also to P. Since the polar of P passes through P', the polar of P' also passes through P, so that the correspondence between P and P' is double. The two point-rows are therefore in involution, and the double points, if any exist, are the points where the line m meets the conic. A similar involution of rays may be found at any point in the plane, corresponding rays passing each through the pole of the other. We have called such points and rays conjugate with respect to the conic (§ 100). We may then state the following important theorem:

139. A conic determines on every line in its plane an involution of points, corresponding points in the involution [pg 82] being conjugate with respect to the conic. The double points, if any exist, are the points where the line meets the conic.

140. The dual theorem reads: A conic determines at every point in the 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.