An Elementary Course in Synthetic Projective Geometry

by Derrick Norman Lehmer

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.

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.

Fig. 28

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.

Fig. 29

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.

Fig. 30

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.

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.

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.

Fig. 31

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,

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.