In an elliptical involution any two segments AA′ and BB′ lie partly within and partly without each other (fig. 32). Hence two circles described on AA′ and BB′ as diameters will intersect in two points E and E′. The line EE′ cuts the base of the involution at a point O, which has the property that OA . OA′ = OB · OB′, for each is equal to OE . OE′. The point O is therefore the centre of the involution. If we wish to construct to any point C the conjugate point C′, we may draw the circle through CEE′. This will cut the base in the required point C′ for OC · OC′ = OA · OA′. But EC and EC′ are at right angles. Hence the involution which is obtained by joining E or E′ to the points in the given involution is circular. This may also be expressed thus:
| Fig. 32. |
Every elliptical involution has the property that there are two definite points in the plane from which any two conjugate points are seen under a right angle.
At the same time the following problem has been solved:
To determine the centre and also the point corresponding to any given point in an elliptical involution of which two pairs of conjugate points are given.
§ 81. Involution Range on a Conic.—By the aid of § 53, the points on a conic may be made to correspond to those on a line, so that the row of points on the conic is projective to a row of points on a line. We may also have two projective rows on the same conic, and these will be in involution as soon as one point on the conic has the same point corresponding to it all the same to whatever row it belongs. An involution of points on a conic will have the property (as follows from its definition, and from § 53) that the lines which join conjugate points of the involution to any point on the conic are conjugate lines of an involution in a pencil, and that a fixed tangent is cut by the tangents at conjugate points on the conic in points which are again conjugate points of an involution on the fixed tangent. For such involution on a conic the following theorem holds:
The lines which join corresponding points in an involution on a conic all pass through a fixed point; and reciprocally, the points of intersection of conjugate lines in an involution among tangents to a conic lie on a line.
| Fig. 33 |
We prove the first part only. The involution is determined by two pairs of conjugate points, say by A, A′ and B, B′ (fig. 33). Let AA′ and BB′ meet in P. If we join the points in involution to any point on the conic, and the conjugate points to another point on the conic, we obtain two projective pencils. We take A and A′ as centres of these pencils, so that the pencils A(A′BB′) and A′(AB′B) are projective, and in perspective position, because AA′ corresponds to A′A. Hence corresponding rays meet in a line, of which two points are found by joining AB′ to A′B and AB to A′B′. It follows that the axis of perspective is the polar of the point P, where AA′ and BB′ meet. If we now wish to construct to any other point C on the conic the corresponding point C′, we join C to A′ and the point where this line cuts p to A. The latter line cuts the conic again in C′. But we know from the theory of pole and polar that the line CC′ passes through P. The point of concurrence is called the “pole of the involution,” and the line of collinearity of the meets is called the “axis of the involution.”
Involution Determined by a Conic on a Line.—Foci