A first course in projective geometry . Fig. 976. conic Hence the radius of curvature at any point P of a central_ sq. of semi-diameter parallel to the tangent at Pcentral perpendicular on this tangent § 16. The Chord of Curvature throug-h the Focus Sat any Point P of a Parabola is 4SP. Let TQQ be parallel to PP, the focal chord (Fig. 98). Lettpt be the tangent parallel to PP, and let it be cut at t and /by the tangents at P and P which intersect at right angles ath on the directrix (§ 9). Now the triangles ^pt, StP being similar (§ 8 (3)), tptP Spsi S[SP /P2 SpSP TANGENT AND NORMAL PROPERTIES
A first course in projective geometry . Fig. 976. conic Hence the radius of curvature at any point P of a central_ sq. of semi-diameter parallel to the tangent at Pcentral perpendicular on this tangent § 16. The Chord of Curvature throug-h the Focus Sat any Point P of a Parabola is 4SP. Let TQQ be parallel to PP, the focal chord (Fig. 98). Lettpt be the tangent parallel to PP, and let it be cut at t and /by the tangents at P and P which intersect at right angles ath on the directrix (§ 9). Now the triangles ^pt, StP being similar (§ 8 (3)), tptP Spsi S[SP /P2 SpSP TANGENT AND NORMAL PROPERTIES 183 Also, by Newtons Theorem, §2, Chap. IX., ^^ ^^, =, . = (from the above tp^ ^ Sj) yp2 gp 3p , chord of curvature through S = Lt —— = —-. Lt TQ = -- . PP. ^ TQ Sjy Sj^. Fig. 98. But PP = 2tt = (§11, Chap. VIII.)= Apk, since tkt is a right angle,= 48^... chord of curvature through S = this expressions for the diameter of curvature may beobtained. Note. It will be subsequently proved that a circle cannotcut a conic in more than four points. Three of these beingcoincident at P in the above figures, it will cut the conic inone other point only. The line joining P to this point iscalled the chord of curvature at P. 184 PROJECTIVE GEOMETRY Historical Note. A flood of light was thrown upon thestudy of Geometry, and in particular on the geometry of theconic, by Keplers enunciation of his doctrine of drawing attention to the fact that the ellipse, parabolaand hyperbola might be obtained in succession by continuouschange, he was enabled to coordinate and unify resultsalready known, particularly in relation to the foci andasymptotes of conies, and to give the whole theory a com-pactness which, up to his time, it had never possessed.
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