. The principles of projective geometry applied to the straight line and conic . 5 is a self-corresponding point. Therefore s the axis ofperspective passes through A and B and is therefore a commonchord. Theorems eoncernlmj Tno Copies 279 If Q^ is taken as corresponding to Q, the points D and C are self-corresponding points and DC is the axis of perspective. If the conies do not intersect in real points, let the point Q correspondto the point Q. Then the tangents at Q and Q, being correspondinglines, will intersect at a point R on s. Draw the other tangents RL andRL from R to
. The principles of projective geometry applied to the straight line and conic . 5 is a self-corresponding point. Therefore s the axis ofperspective passes through A and B and is therefore a commonchord. Theorems eoncernlmj Tno Copies 279 If Q^ is taken as corresponding to Q, the points D and C are self-corresponding points and DC is the axis of perspective. If the conies do not intersect in real points, let the point Q correspondto the point Q. Then the tangents at Q and Q, being correspondinglines, will intersect at a point R on s. Draw the other tangents RL andRL from R to the two conies. These are corresponding lines. There-fore the chords of contact QL and QU are also corresponding linesand intersect in R on s. R and R are a pair of common conjugatepoints on s with respect to the conies. Similarly any number of pairsof common conjugate points may be determined on s. Hence the twoconies determine on s the same involution, and s is one of the commoninvolution chords of the conies, and passes through two of theirimaginary points of Similarly, if Q, Qi be taken as corresponding to each other, a secondreal chord passing through two other imaginary points of intersection ofthe conies is obtained as the axis of perspective. 280 Principles of Projective Geometrif Generally there are six points of intersection of the four commontangents and six common chords joining the four points of intersectionof the two conies. Corresponding to each of the six centres of perspective there aretwo axes of perspective, making in all twelve perspectives. Each of the six centres of perspective has two axes of perspectiveand each of the six axes has two centres of perspective. The chords of contact of the common tangents to two conies pass fourhy four through tJie three vertices of their common self-conjugate triangle,that is, through the diagonal points of the quadrangle formed hy theirfour points of intersection. In the figure, page 278, the chords of con
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
