A first course in projective geometry . Then, m Figs. 97a and h, PK the required chord of curvaturethrough P jp2 Q\j2 = Lt = Lt , where QV is parallel to TP. TQ PV ^ -p QV2 _ CD^ /upper sign—ellipse \ PV. VP ~ ~CP Vlower „ —hyperbola/where CD is the semi-diameter conjugate to CP. PV CP2 CD2 ^^„ r ^ .X \ ^CD-^ 182 PROJECTIVE GEOMETRY But if PFO be the normal at P and PO the diameter ofcurvature, since CKOF is cyclic, PF. PO - PC. PK = 2CD2 in absolute magnitude. .•. P0 = Fig. 976. conic Hence the radius of curvature at any point P of a central_ sq. of semi-diameter parallel to the tang
A first course in projective geometry . Then, m Figs. 97a and h, PK the required chord of curvaturethrough P jp2 Q\j2 = Lt = Lt , where QV is parallel to TP. TQ PV ^ -p QV2 _ CD^ /upper sign—ellipse \ PV. VP ~ ~CP Vlower „ —hyperbola/where CD is the semi-diameter conjugate to CP. PV CP2 CD2 ^^„ r ^ .X \ ^CD-^ 182 PROJECTIVE GEOMETRY But if PFO be the normal at P and PO the diameter ofcurvature, since CKOF is cyclic, PF. PO - PC. PK = 2CD2 in absolute magnitude. .•. P0 = 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^
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