Conic Sections Treated Geometrically

Conic Sections Treated Geometrically

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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1881 edition. Excerpt: ...and of the axis BB' project into the middle points of two parallel lines pp' and bb' by Props. 1 and 2. Again PN: BC:: pn: be AN: AC:: an: ac A'N: AC:: a'n: ac Hence pn2: an. a'n: be2; ac2 Therefore apa' is an ellipse. Q.E.D. 195. Cor.--An ellipse can be projected into a circle by properly choosing the plane of projection. For describe a semicircle on AC in a plane perpendicular to the plane of the ellipse, and place in it a line CE equal to CB; then if we take a plane parallel to BEB' as plane of projection, the line AC will project into a line equal to CE or CB, while the length of CB remains unchanged. Conversely, in the same manner as in Prop. 3, the projection of a circle is an ellipse. PROP. 4. 196. The projection of a parabola is a parabola. For all the ordinates (being parallel lines) project in the same ratio, so also do all the absciss. But in the original curve the square of the ordinate varies as the abscissa, whence this relation holds also in the projected curve, which is therefore a parabola. 197. Also, by a proof similar to that given for an ellipse, any hyperbola may be projected into an hyperbola, with its axes in any given ratio, and, as a special case, into a rectangular hyperbola. 198. A diameter of a circle bisects all chords at right angles to it, and projects into a straight line in an ellipse which passes through the middle points of a system of parallel chords. Such a line is called a diameter of the ellipse. 199. Diameters at right angles in a circle project into conjugate diameters in an ellipse, for each is the locus of middle points of chords parallel to the other. Also the converse is true, that if an ellipse be projected into a circle each pair of conjugate diameters project into a pair of diameters at right...
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Product details

  • Paperback | 30 pages
  • 189 x 246 x 2mm | 73g
  • United States
  • English
  • black & white illustrations
  • 1236847652
  • 9781236847652