A first course in projective geometry . aight line arethe ends of the diameter of the circle perpendicular to thestraight line. For it follows from the first prop, of this section thatthe centre of similitude divides the interval between thecircumferences in the ratio of the radii (a ratio which is hereequal to zero). Ex. Prove the theorem of § 9 by the use of Cevas theorem. § 10. Similar Curves. Let O be a fixed point, P any point of a given curve. OPOn OP take a point P so that —- = a constant ratio k. Then, as P describes the given curve, P will trace outanother curve possessing the followi
A first course in projective geometry . aight line arethe ends of the diameter of the circle perpendicular to thestraight line. For it follows from the first prop, of this section thatthe centre of similitude divides the interval between thecircumferences in the ratio of the radii (a ratio which is hereequal to zero). Ex. Prove the theorem of § 9 by the use of Cevas theorem. § 10. Similar Curves. Let O be a fixed point, P any point of a given curve. OPOn OP take a point P so that —- = a constant ratio k. Then, as P describes the given curve, P will trace outanother curve possessing the following properties : (1) The distances between pairs of corresponding points are always in the constant ratio k. (2) The tangents at corresponding points are parallel. Let P and Q be two points on the given curve, P, Q thecorresponding points on the derived locus. Then, since SIMILITUDE 49 —: = —^ and the angle POQ is common to the trianglesOP OQ ^ p,Q, ^ POQ, POQ, these triangles are similar, and -57^=^-- Also PQ is parallel to Fig. 21. Now if P and Q move up to ultimate coincidence, so alsowill P and Q on the derived locus, and in the limit thetangent at P to the given curve is parallel to the tangent atP to the derived locus. The derived locus is thus similar and similarly situated tothe given locus. Such figures are called Homothetic. § 11. Two figures may be similar without being similarlysituated : for in the case last considered either may be turnedin the plane about O through any angle. O is called the Homothetic Centre. Of two homothetic figures one is sometimes said to bederived from the other by multiplication. Two figures in the same plane, which are such that theone may be derived from the other by multiplication androtation about the homothetic centre, are said to be directlysimilar. D 50 PROJECTIVE GEOMETRY Referring to the Def. of § 6, Chapter III., we see thathomothetic figures are a particular case of figures in perspec-tive. For corresponding l
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