An Introductory Account of Certain Modern Ideas and Methods in Plane Analytical Geometry |
Other editions - View all
Common terms and phrases
algebraic angle asymptotes Cartesian centre circle circular points coefficients coincide collinear complete quadrangle concurrent concurrent lines condition consider constant construction correspondence covariant cross-ratio cubic curve of order cusp degenerate Desargues distance double points dualistic transformation envelope equianharmonic example fixed point foci four points geometry given gives harmonic with respect hence homogeneous coordinates homographic hyperbola imaginary points intersection invariant inverse involution isotropic lines line coordinates line equation line infinity line joining line-pair linear transformation locus metric obtained pairs of conjugates pairs of points parabola parallel pass perpendicular plane point and line point coordinates point equation point of contact points at infinity position primary element projection properties quadric quartic range ratios reciprocal regard secondary element self-conjugate shows Similarly straight line tacnode tangents theorem theory three points triangle of reference unicursal values vanishing
Popular passages
Page 127 - ... Two given circles are cut by a third circle in P, Q. and R, S and the chords PQ and RS are produced to meet in T. Prove that T is a point on the radical axis of the two given circles. Hence, give a geometric construction for the radical axis of two circles which do not intersect. Ex.
Page 121 - If a rectangular hyperbola circumscribe a triangle, it passes through the orthocentre.
Page 217 - Werth räumlicher Anschauung. Wenn wir im Texte die räumliche Anschauung als etwas Beiläufiges bezeichnen, so ist dies mit Bezug auf den rein mathematischen Inhalt der zu formulirenden Betrachtungen gemeint. Die Anschauung hat für ihn nur den Werth der Veranschaulichung, der allerdings in pädagogischer Beziehung sehr hoch anzuschlagen ist.
Page 125 - D' is a right angle, and the angle DPF = DPD', and PF = PD', .-. DFP is a right angle. In like manner, DFP' is a right angle ; hence, first, the part of the tangent intercepted between the point of contact and the directrix, subtends a right angle at the focus ; second, the line joining the points of contact of perpendicular tangents always passes through thefocut.
Page 156 - Then since there is a correspondence, x is a function of y, and y is a function of x.
Page 176 - F' (the real foci) : the two perpendicular tangents at P are therefore harmonic with respect to PF, PF', that is, they are the bisectors of the angle FPF ; hence the tangent at any point is equally inclined to the focal distances. Taking a point T not on the conic considered, let TP, TP...
Page 123 - Cartesian equation of the locus of a point P whose distance from a fixed point...
Page 55 - Hence, the hyperbola is often denned as the locus of a point which moves so that the product of its distances from two fixed lines is constant. (The distances here may be the perpendicular distances; or, the distance to each line may be measured parallel to the other.) 41. An important special case is that of the "rectangular...
Page 80 - If fig. 179 were spun about OA, what figure would be generated (i) by the circle, (ii) by AP, (iii) by PQ? Hence find the locus of the points of contact of tangents from a fixed point to a fixed sphere. Ex.
Page 55 - The ellipse (Fig. 101) is a plane curve which is the locus of a point moving so that the sum of its distances (focal radii) from two fixed points (foci) in the plane is a constant (equal to the major axis).