Projective Geometry |
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Common terms and phrases
A₁ ABCD angles asymptotes axes axis B₁ bisects Brianchon theorem bundle C₁ chain of perspectivity chord circle circle of Apollonius coincide complete quadrangle cone confocal conjugate diameters conjugate lines construct coplanar corresponding elements corresponding points corresponding rays cross-ratio curve of second D₁ determined diagonal double polarity dual elementary forms ellipse envelope of second Exercises given line given point harmonic conjugate harmonic pencil harmonic range harmonically separated Hence hexagon hyperbola ideal line ideal point involution k₁ KLMN locus meet P₁ pair of conjugate pairs of corresponding pairs of points parabola parallel perpendicular perspective position point of contact polar line projective geometry projectively related point-rows projectively related sheaves projector quadric quadric transformation reciprocal regulus ruled surface S₁ second class second order segment self-polar triangle sheaf of planes sheaf of rays sides sponding straight line superposed surface of second tangents theorem u₁ vertex vertices
Popular passages
Page 32 - Find the limit of the position of a point, the ratio of whose distances from two fixed points is less than a given ratio.
Page 3 - fundamental assumption: On every straight line there is one and only one ideal or infinitely distant point. This point makes the line continuous from any one point on it to any other point on it in either direction. Through a given point there can be drawn one and only one line parallel to a given line. This parallel intersects the given line in the ideal or infinitely distant point.
Page 61 - ... intersect the tan- the vertices of a triangle cirgents at the opposite vertices cumscribed to a conic to the in points of one straight line. points of contact of the opposite sides intersect in one point. 62. Pascal's theorem yields itself at once to the construction of a conic of which there are given five points, or four points and the tangent at one of them, or three points and the tangents at two of them. In the case of five points being given, if these are A, B, C, D, E, and they are joined...
Page 86 - If a triangle is inscribed in a conic, any line conjugate to one side meets the other two sides in conjugate points.
Page 100 - Eliminating t we obtain .(8) which is the equation of an ellipse referred to a pair of conjugate diameters. Moreover, since any point on the path may be regarded as the starting point, the formula (7) shews that the velocity at any point P varies as the length of the semidiameter (OD, say) conjugate to OP (cf. Art. 21, Ex. 2). In other words, the hodograph is similar to the locus of D, ie to the elliptic...
Page 87 - PRR', which is the tangent at P ; that is to say, the polar of a point on the conic is the tangent at that point.
Page 150 - Show that the circumscribed circles of the four triangles formed by the sides of a complete quadrilateral pass through one...
Page iii - THE present volume is based on a course of lectures given by the author for a number of years at the University of Illinois. It is intended as an...
Page 177 - It is easily shown that these three radical axes pass through a common point ; this point is called the radical center of the three circles.
Page 78 - CONIC, THE TANGENTS AT THE VERTICES MEET THE OPPOSITE SIDES IN THREE COLLINEAR POINTS. 74. 1.3 3 I'.