CHAPTER VII - METRICAL PROPERTIES OF THE CONIC SECTIONS
107. Diameters. Center. After what has been said in the last chapter one would naturally expect to get at the metrical properties of the conic sections by the introduction of the infinite elements in the plane. Entering into the theory of poles and polars with these elements, we have the following definitions:
The polar line of an infinitely distant point is called a diameter, and the pole of the infinitely distant line is called the center, of the conic.
108. From the harmonic properties of poles and polars,
The center bisects all chords through it (§ 39).
Every diameter passes through the center.
All chords through the same point at infinity (that is, each of a set of parallel chords) are bisected by the diameter which is the polar of that infinitely distant point.
109. Conjugate diameters. We have already defined conjugate lines as lines which pass each through the pole of the other (§ 100).
Any diameter bisects all chords parallel to its conjugate.
The tangents at the extremities of any diameter are parallel, and parallel to the conjugate diameter.
Diameters parallel to the sides of a circumscribed parallelogram are conjugate.
All these theorems are easy exercises for the student.
[pg 63]110. Classification of conics. Conics are classified according to their relation to the infinitely distant line. If a conic has two points in common with the line at infinity, it is called a hyperbola; if it has no point in common with the infinitely distant line, it is called an ellipse; if it is tangent to the line at infinity, it is called a parabola.
111. In a hyperbola the center is outside the curve (§ 101), since the two tangents to the curve at the points where it meets the line at infinity determine by their intersection the center. As previously noted, these two tangents are called the asymptotes of the curve. The ellipse and the parabola have no asymptotes.
112. The center of the parabola is at infinity, and therefore all its diameters are parallel, for the pole of a tangent line is the point of contact.
The locus of the middle points of a series of parallel chords in a parabola is a diameter, and the direction of the line of centers is the same for all series of parallel chords.
The center of an ellipse is within the curve.
113. Theorems concerning asymptotes. We derived as a consequence of the theorem of Brianchon (§ 89) the proposition that if a triangle be circumscribed about a conic, the lines joining the vertices to the points of contact of the opposite sides all meet in a point. Take, now, for two of the tangents the asymptotes of a hyperbola, and let any third tangent cut them in A and B (Fig. 28). If, then, O is the intersection of the asymptotes,—and therefore the center of the curve,— [pg 64] then the triangle OAB is circumscribed about the curve. By the theorem just quoted, the line through A parallel to OB, the line through B parallel to OA, and the line OP through the point of contact of the tangent AB all meet in a point C. But OACB is a parallelogram, and PA = PB. Therefore
The asymptotes cut off on each tangent a segment which is bisected by the point of contact.
114. If we draw a line OQ parallel to AB, then OP and OQ are conjugate diameters, since OQ is parallel to the tangent at the point where OP meets the curve. Then, since A, P, B, and the point at infinity on AB are four harmonic points, we have the theorem
Conjugate diameters of the hyperbola are harmonic conjugates with respect to the asymptotes.
115. The chord A"B", parallel to the diameter OQ, is bisected at P' by the conjugate diameter OP. If the chord A"B" meet the asymptotes in A', B', then A', P', B', and the point at infinity are four harmonic points, and therefore P' is the middle point of A'B'. Therefore A'A" = B'B" and we have the theorem
The segments cut off on any chord between the hyperbola and its asymptotes are equal.
116. This theorem furnishes a ready means of constructing the hyperbola by points when a point on the curve and the two asymptotes are given.
[pg 65]
117. For the circumscribed quadrilateral, Brianchon's theorem gave (§ 88) The lines joining opposite vertices and the lines joining opposite points of contact are four lines meeting in a point. Take now for two of the tangents the asymptotes, and let AB and CD be any other two (Fig. 29). If B and D are opposite vertices, and also A and C, then AC and BD are parallel, and parallel to PQ, the line joining the points of contact of AB and CD, for these are three of the four lines of the theorem just quoted. The fourth is the line at infinity which joins the point of contact of the asymptotes. It is thus seen that the triangles ABC and ADC are equivalent, and therefore the triangles AOB and COD are also. The tangent AB may be fixed, and the tangent CD chosen arbitrarily; therefore
The triangle formed by any tangent to the hyperbola and the two asymptotes is of constant area.
118. Equation of hyperbola referred to the asymptotes. Draw through the point of contact P of the tangent AB two lines, one parallel to one asymptote and the other parallel to the other. One of these lines meets OB at a distance y from O, and the other meets OA at a distance x from O. Then, since P is the middle point [pg 66] of AB, x is one half of OA and y is one half of OB. The area of the parallelogram whose adjacent sides are x and y is one half the area of the triangle AOB, and therefore, by the preceding paragraph, is constant. This area is equal to xy · sin α, where α is the constant angle between the asymptotes. It follows that the product xy is constant, and since x and y are the oblique coördinates of the point P, the asymptotes being the axes of reference, we have
The equation of the hyperbola, referred to the asymptotes as axes, is xy = constant.
This identifies the curve with the hyperbola as defined and discussed in works on analytic geometry.
119. Equation of parabola. We have defined the parabola as a conic which is tangent to the line at infinity (§ 110). Draw now two tangents to the curve (Fig. 30), meeting in A, the points of contact being B and C. These two tangents, together with the line at infinity, form a triangle circumscribed about the conic. Draw through B a parallel to AC, and through C a parallel to AB. If these meet in D, then AD is a [pg 67] diameter. Let AD meet the curve in P, and the chord BC in Q. P is then the middle point of AQ. Also, Q is the middle point of the chord BC, and therefore the diameter AD bisects all chords parallel to BC. In particular, AD passes through P, the point of contact of the tangent drawn parallel to BC.
Draw now another tangent, meeting AB in B' and AC in C'. Then these three, with the line at infinity, make a circumscribed quadrilateral. But, by Brianchon's theorem applied to a quadrilateral (§ 88), it appears that a parallel to AC through B', a parallel to AB through C', and the line BC meet in a point D'. Also, from the similar triangles BB'D' and BAC we have, for all positions of the tangent line B'C,
B'D' : BB' = AC : AB,
or, since B'D' = AC',
AC': BB' = AC:AB = constant.
If another tangent meet AB in B" and AC in C", we have
AC' : BB' = AC" : BB",
and by subtraction we get
C'C" : B'B" = constant;
whence
The segments cut off on any two tangents to a parabola by a variable tangent are proportional.
If now we take the tangent B'C' as axis of ordinates, and the diameter through the point of contact O as axis of abscissas, calling the coordinates of B(x, y) and of C(x', y'), then, from the similar triangles BMD' and we have
y : y' = BD' : D'C = BB' : AB'.
Also
y : y' = B'D' : C'C = AC' : C'C.
[pg 68]If now a line is drawn through A parallel to a diameter, meeting the axis of ordinates in K, we have
AK : OQ' = AC' : CC' = y : y',
and
OM : AK = BB' : AB' = y : y',
and, by multiplication,
OM : OQ' = y2 : y'2,
or
x : x' = y2 : y'2;
whence
The abscissas of two points on a parabola are to each other as the squares of the corresponding coördinates, a diameter and the tangent to the curve at the extremity of the diameter being the axes of reference.
The last equation may be written
y2 = 2px,
where 2p stands for y'2 : x'.
The parabola is thus identified with the curve of the same name studied in treatises on analytic geometry.
120. Equation of central conics referred to conjugate diameters. Consider now a central conic, that is, one which is not a parabola and the center of which is therefore at a finite distance. Draw any four tangents to it, two of which are parallel (Fig. 31). Let the parallel tangents meet one of the other tangents in A and B and the other in C and D, and let P and Q be the points of contact of the parallel tangents R and S of the others. Then AC, BD, PQ, and RS all meet in a point W (§ 88). From the figure,
PW : WQ = AP : QC = PD : BQ,
or
AP · BQ = PD · QC.
[pg 69]If now DC is a fixed tangent and AB a variable one, we have from this equation
AP · BQ = constant.
This constant will be positive or negative according as PA and BQ are measured in the same or in opposite directions. Accordingly we write
AP · BQ = ± b2.
Since AD and BC are parallel tangents, PQ is a diameter and the conjugate diameter is parallel to AD. The middle point of PQ is the center of the conic. We take now for the axis of abscissas the diameter PQ, and the conjugate diameter for the axis of ordinates. Join A to Q and B to P and draw a line through S parallel to the axis of ordinates. These three lines all meet in a point N, because AP, BQ, and AB form a triangle circumscribed to the conic. Let NS meet PQ in M. Then, from the properties of the circumscribed triangle (§ 89), M, N, S, and the point at infinity on NS are four harmonic points, and therefore N is the middle point of MS. If the coördinates of S are (x, y), so that OM is x and MS is y, then MN = y/2. Now from the similar triangles PMN and PQB we have
BQ : PQ = NM : PM,
[pg 70]and from the similar triangles PQA and MQN,
AP : PQ = MN : MQ,
whence, multiplying, we have
±b2/4 a2 = y2/4 (a + x)(a - x),
where
or, simplifying,
which is the equation of an ellipse when b2 has a positive sign, and of a hyperbola when b2 has a negative sign. We have thus identified point-rows of the second order with the curves given by equations of the second degree.
PROBLEMS
1. Draw a chord of a given conic which shall be bisected by a given point P.
2. Show that all chords of a given conic that are bisected by a given chord are tangent to a parabola.
3. Construct a parabola, given two tangents with their points of contact.
4. Construct a parabola, given three points and the direction of the diameters.
5. A line u' is drawn through the pole U of a line u and at right angles to u. The line u revolves about a point P. Show that the line u' is tangent to a parabola. (The lines u and u' are called normal conjugates.)
6. Given a circle and its center O, to draw a line through a given point P parallel to a given line q. Prove the following construction: Let p be the polar of P, Q the pole of q, and A the intersection of p with OQ. The polar of A is the desired line.
CHAPTER VIII - INVOLUTION
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.
[pg 73]
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.
[pg 74]
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.
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.
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.
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'.
[pg 80]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.
[pg 81]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.
PROBLEMS
1. Two lines are drawn through a point on a conic so as always to make right angles with each other. Show that the lines joining the points where they meet the conic again all pass through a fixed point.
2. Two lines are drawn through a fixed point on a conic so as always to make equal angles with the tangent at that point. Show that the lines joining the two points where the lines meet the conic again all pass through a fixed point.
3. Four lines divide the plane into a certain number of regions. Determine for each region whether two conics or none may be drawn to pass through points of it and also to be tangent to the four lines.
4. If a variable quadrangle move in such a way as always to remain inscribed in a fixed conic, while three of its sides turn each around one of three fixed collinear points, then the fourth will also turn around a fourth fixed point collinear with the other three.
5. State and prove the dual of problem 4.
6. Extend problem 4 as follows: If a variable polygon of an even number of sides move in such a way as always to remain inscribed in a fixed conic, while all its sides but one pass through as many fixed collinear points, then the last side will also pass through a fixed point collinear with the others.
[pg 83]7. If a triangle QRS be inscribed in a conic, and if a transversal s meet two of its sides in A and A', the third side and the tangent at the opposite vertex in B and B', and the conic itself in C and C', then AA', BB', CC' are three pairs of points in an involution.
8. Use the last exercise to solve the problem: Given five points, Q, R, S, C, C', on a conic, to draw the tangent at any one of them.
9. State and prove the dual of problem 7 and use it to prove the dual of problem 8.
10. If a transversal cut two tangents to a conic in B and B', their chord of contact in A, and the conic itself in P and P', then the point A is a double point of the involution determined by BB' and PP'.
11. State and prove the dual of problem 10.
12. If a variable conic pass through two given points, P and P', and if it be tangent to two given lines, the chord of contact of these two tangents will always pass through a fixed point on PP'.
13. Use the last theorem to solve the problem: Given four points, P, P', Q, S, on a conic, and the tangent at one of them, Q, to draw the tangent at any one of the other points, S.
14. Apply the theorem of problem 9 to the case of a hyperbola where the two tangents are the asymptotes. Show in this way that if a hyperbola and its asymptotes be cut by a transversal, the segments intercepted by the curve and by the asymptotes respectively have the same middle point.
15. In a triangle circumscribed about a conic, any side is divided harmonically by its point of contact and the point where it meets the chord joining the points of contact of the other two sides.