A first course in projective geometry . Fig. 916. It appears that an ellipse always has a real director circle;a hyperbola only when AC > BC. In the rectangular hyperbola AC = BC, and the directorcircle becomes a point-circle at the centre of the curve. In the parabola the circle has an infinite radius, and thusbecomes a straight line. It is easily shown that this line is the directrix. For ifTP, TP are the tangents, SY, SY the focal perpendiculars onthem, Y and Y lie on the tangent at the vertex (Fig. 92). But SYTY is a rectangle and ST is bisected by YY. T musttherefore lie on a line para
A first course in projective geometry . Fig. 916. It appears that an ellipse always has a real director circle;a hyperbola only when AC > BC. In the rectangular hyperbola AC = BC, and the directorcircle becomes a point-circle at the centre of the curve. In the parabola the circle has an infinite radius, and thusbecomes a straight line. It is easily shown that this line is the directrix. For ifTP, TP are the tangents, SY, SY the focal perpendiculars onthem, Y and Y lie on the tangent at the vertex (Fig. 92). But SYTY is a rectangle and ST is bisected by YY. T musttherefore lie on a line parallel to YY and twice as far from the TANGENT AND NORMAL PROPERTIES 175 focus as YY, it is on the directrix. The director circle for aparabola is therefm^e the Fig. 92. § 10. Gaskins Theorem. The circumcircle of a triangle self-conjugate with respectto an ellipse cuts the director circle orthogonally. In Fig. 93, let TRR be a self-con jugate triangle, and let RRcut the conic in Q, Q. Let V be the middle point of CVT is a straight line and CV. CT = CP^, if CT cuts theconic at P. Now QV2 CD2 / = 7^52 (Chap. IX. § 3), CD being the semi- diameter conjugate to CP. Also, since TR is the polar of R, RQRQ is a harmonic QV2 = VR. VR = VK. VT, if CT cuts the circle TRR in PV . VP = CP2 - CV2 = CV . CT - CV2 (Chap. VIII., § 10, Cor. 3)= CV. VT. Cp2CP2 VK. VTCV. VT VKCV 176 But PROJECTIVE GEOMETRY CD2 + CP2 CK , , , —CP2—= cv (coniponendo) = ^^^-^ CP2 = *. = CD2 + CP-
Size: 1612px × 1550px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
