DIAMETERS AND AXES OF CONICS
§ 69. Diameters.—The theorems about the harmonic properties of poles and polars contain, as special cases, a number of important metrical properties of conics. These are obtained if either the pole or the polar is moved to infinity, it being remembered that the harmonic conjugate to a point at infinity, with regard to two points A, B, is the middle point of the segment AB. The most important properties are stated in the following theorems:
The middle points of parallel chords of a conic lie in a line—viz. on the polar to the point at infinity on the parallel chords.
This line is called a diameter.
The polar of every point at infinity is a diameter.
The tangents at the end points of a diameter are parallel, and are parallel to the chords bisected by the diameter.
All diameters pass through a common point, the pole of the line at infinity.
All diameters of a parabola are parallel, the pole to the line at infinity being the point where the curve touches the line at infinity.
In case of the ellipse and hyperbola, the pole to the line at infinity is a finite point called the centre of the curve.
A centre of a conic bisects every chord through it.
The centre of an ellipse is within the curve, for the line at infinity does not cut the ellipse.
The centre of an hyperbola is without the curve, because the line at infinity cuts the curve. Hence also
From the centre of an hyperbola two tangents can be drawn to the curve which have their point of contact at infinity. These are called Asymptotes (§ 59).
To construct a diameter of a conic, draw two parallel chords and join their middle points.
To find the centre of a conic, draw two diameters; their intersection will be the centre.
§ 70. Conjugate Diameters.—A polartriangle with one vertex at the centre will have the opposite side at infinity. The other two sides pass through the centre, and are called conjugate diameters, each being the polar of the point at infinity on the other.
Of two conjugate diameters each bisects the chords parallel to the other, and if one cuts the curve, the tangents at its ends are parallel to the other diameter.
Further
Every parallelogram inscribed in a conic has its sides parallel to two conjugate diameters; and
Every parallelogram circumscribed about a conic has as diagonals two conjugate diameters.
This will be seen by considering the parallelogram in the first case as an inscribed fourpoint, in the other as a circumscribed fourside, and determining in each case the corresponding polartriangle. The first may also be enunciated thus
The lines which join any point on an ellipse or an hyperbola to the ends of a diameter are parallel to two conjugate diameters.
§ 71. If every diameter is perpendicular to its conjugate the conic is a circle.
For the lines which join the ends of a diameter to any point on the curve include a right angle.
A conic which has more than one pair of conjugate diameters at right angles to each other is a circle.
Let AA' and BB' (fig. 24) be one pair of conjugate diameters at right angles to each other, CC' and DD' a second pair. If we draw through the end point A of one diameter a chord AP parallel to DD', and join P to A', then PA and PA' are, according to § 70, parallel to two conjugate diameters. But PA is parallel to DD', hence PA' is parallel to CC', and therefore PA and PA' are perpendicular. If we further draw the tangents to the conic at A and A', these will be perpendicular to AA', they being parallel to the conjugate diameter BB'. We know thus five points on the conic, viz. the points A and A' with their tangents, and the point P. Through these a circle may be drawn having AA' as diameter; and as through five points
, one conic only can be drawn, this circle must coincide with the given conic.
§ 72. Axes.—Conjugate diameters perpendicular to each other are called axes, and the points where they cut the curve vertices of the conic.
In a circle every diameter is an axis, every point on it is a vertex; and any two lines at right angles to each other may be taken as a pair of axes of any circle which has its centre at their intersection.
If we describe on a diameter AB of an ellipse or hyperbola a circle concentric to the conic, it will cut the latter in A and B (fig. 25). Each of the semicircles in which it is divided by AB will be partly within, partly without the curve, and must cut the latter therefore again in a point. The circle and the conic have thus four points A, B, C, D, and therefore
one polartriangle, in common (§ 68). Of this the centre is one vertex, for the line at infinity is the polar to this point, both with regard to the circle and the other conic. The other two sides are conjugate diameters of both, hence perpendicular to each other. This gives
An ellipse as well as an hyperbola has one pair of axes.
This reasoning shows at the same time how to construct the axis of an ellipse or of an hyperbola.
A parabola has one axis,
if we define an axis as a diameter perpendicular to the chords which it bisects. It is easily constructed. The line which bisects any two parallel chords is a diameter. Chords perpendicular to it will be bisected by a parallel diameter, and this is the axis.
§ 73. The first part of the righthand theorem in § 64 may be stated thus: any two conjugate lines through a point P without a conic are harmonic conjugates with regard to the two tangents that may be drawn from P to the conic.
If we take instead of P the centre C of an hyperbola, then the conjugate lines become conjugate diameters, and the tangents asymptotes. Hence
Any two conjugate diameters of an hyperbola are harmonic conjugates with regard to the asymptotes.
As the axes are conjugate diameters at right angles to one another, it follows (§ 23)
The axes of an hyperbola bisect the angles between the asymptotes.
Let 0 be the centre of the hyperbola (fig. 26), t any secant which cuts the hyperbola in C,D and the asymptotes in E,F, then the line OM which bisects the chord CD is a diameter conjugate to the
diameter OK which is parallel to the secant t, so that OK and OM are harmonic with regard to the asymptotes. The point M therefore bisects EF. But by construction M bisects CD. It follows that DF=EC, and ED=CF; or
On any secant of an hyperbola the segments between the curve and the asymptotes are equal.
If the chord is changed into a tangent, this gives
The segment between the asymptotes on any tangent to an hyperbola is bisected by the point of contact.
The first part allows a simple solution of the problem to find any number of points on an hyperbola, of which the asymptotes and one point are given. This is equivalent to three points and the tangents at two of them. This construction requires measurement.
§ 74. For the parabola, too, follow some metrical properties. A diameter PM (fig. 27) bisects every chord conjugate to it, and the pole P of such a chord BC lies on the diameter. But a diameter cuts the parabola once at infinity. Hence
The segment PM which joins the middle point M of a chord of a parabola to the pole P of the chord is bisected by the parabola at A.
§ 75. Two asymptotes and any two tangents to an hyperbola may be considered as a quadrilateral circumscribed about the
Fin. 24.
hyperbola. But in such a quadrilateral the intersections of the diagonals and the points of contact of opposite sides lie in a line (§ 54). If therefore DEFG (fig. 28) is such a quadrilateral, then the diagonals DF and GE will meet on the line which joins the points of contact of the asymptotes, that is, on the line at infinity; hence they are parallel. From this the following theorem is a simple deduction:
All triangles formed by a tangent and the asymptotes of an hyperbola are equal in area.
If we draw at a point P
(fig. 28) on an hyperbola
a tangent, the part HK
between the asymptotes
is bisected at P. The
parallelogram PQOQ'
formed by the asymptotes and lines parallel to them through
P will be half the triangle OHK, and will therefore be con
stant. If we now take the asymptotes OX and OY as oblique
axes 0f coordinates, the lines OQ and QP will be the coordinates of P, and will satisfy the equation xy=const.=a2.
For the asymptotes as axes of coordinates the equation of the hyperbola
is xy=const.
INVOLUTION
§ 76. If we have two projective rows, ABC on u and A'B'C' on u', and place their bases on the same line, then each point in this line counts twice, once as a point in the row u and once as a point in the row u'. In fig. 29 we denote the points as points in the one row by letters above the line A, B, C ..., and as points in the second
row by A', B', C' . . below the
B line. Let now A and B' be the same point, then to A will correspond a point A' in the second,
row. In general these points A' and B will be different. It may, however, happen that they coincide. Then the correspondence is a peculiar one, as the following theorem shows:
If two projective rows lie on the same base, and if it happens that to one point in the base the same point corresponds, whether we consider the
Vint as belonging to the first or to the second row, then the same will happen for every point in the base—that is to say, to every point in the line corresponds the same point in the first as in the second row.
In order to determine the correspondence, we may assume three pairs of corresponding points in two projective rows. Let then
A', B', C', in fig. 30, correspond to
A D 8 C A, B, C, so that A and B', and also B and A',, denote the same point. Let us further denote the point
the first row by D ; then it is to 'be proved that the point D', which corresponds to D, is the same point as C. We know that the crossratio of four points is equal to that of the corresponding row. Hence
(AB, CD) = (A'B', C'D')
but .replacing the dashed letters by those undashed ones which denote the same points, the second crossratio equals (BA, DD'),
which, according to § 15, equals (AB, D'D) ; so that the equation becomes
(AB, CD) = (AB, D'D).
This requires that C and D' coincide.
§ 77. Two projective rows on the same base, which have the above property, that to every point, whether it be considered as a point in the one or in the other row, corresponds the same point, are said to be in involution, or to form an involution of points on the line.
We mention, but without proving it, that any two projective rows may be placed so as to form an involution.
An involution may be said to consist of a row of pairs of points, to every point A corresponding a point A', and to A' again the point A. These points are said to be conjugate, or, better, one point is termed the " mate " of the other.
From the definition, according to which an involution may be considered as made up of two projective rows, follow at once the following important properties:
1. The crossratio of four points equals that of the four conjugate points.
2. If we call a point which coincides with its mate a " focus " or " double point " of the involution, we may sa : An involution has either two foci, or one, or none, and is called respectively a hyperbolic, parabolic or elliptic involution (§ 34).
3. In a hyperbolic involution any two conjugate points are harmonic conjugates with regard to the two foci.
For if A, A' be two conjugate points, Fl, F2 the two foci, then to the points Ft, F2, A, A' in the one row correspond the points Fl, F2, A', A in the other, each focus corresponding to itself. Hence (FIF2, AA') = (FIF2,A'A)—that is, we may interchange the two points AA' without altering the value of the crossratio, which is the characteristic property of harmonic conjugates (§ 18).
4. The point conjugate to the point at infinity is called the " centre " of the involution. Every involution has a centre, unless the point at infinity be a focus, in which case we may say that the centre is at infinity.
In an hyperbolic involution the centre is the middle point between the foci.
5. The product of the distances of two conjugate points A, A' from the centre 0 is constant : OA . OA' =c.
For let A, A' and B, B' be two pairs of conjugate points, 0 the centre, I the point at infinity, then
(AB, 0I) = (A'B', I0),
or
OA . OA'=OB . OB'.
In order to determine the distances of the foci from the centre, we write F for A and A' and get
OF2=c; OF==%Ic.
Hence if c is positive OF is real, and has two values, equal and opposite. The involution is hyperbolic.
If c=o, OF =o, and the two foci both coincide with the centre.
If c is negative, %l c becomes imaginary, and there are no foci. Hence we may write
In an hyperbolic involution, OA . OA'=k2, In a parabolic involution, OA . OA'=o, In an elliptic involution, OA . OA' =k2.
From these expressions it follows that conjugate points A, A' in an hyperbolic involution lie on the same side of the centre, and in an elliptic involution on opposite sides of the centre, and that in a parabolic involution one coincides with the centre.
In the first case, for instance, OA . OA' is positive; hence OA and OA' have the same sign.
It also follows that two segments, AA' and BB', between pairs of conjugate points have the following positions: in an hyperbolic involution they lie either one altogether within or altogether without each other; in a parabolic involution they have one point in common; and in an elliptic involution they overlap, each being partly within and partly without the other.
Proof.—We have OA . OA' =0B . OB' = le in case of an hyperbolic involution. Let A and B be the points in each pair which are nearer to the centre O. If now A, A' and B, B' lie on the same side of 0, and if B is nearer to O than A, so that OB
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