Problems, Theorems and Examplesin Descriptive Geometry (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1888 Excerpt: ...if the lines meet at a point, in space, they will do so in projection. Hence the tangents at the extremities of any chord will meet on the diameter conjugate to that chord. Constructions on the Parabola. 184. "1. Having given a parabola to find a diameter, bisect any two parallel chords. 2. To find the axis, draw a chord perpendicular to a diameter; and a parallel to that diameter through the middle point of the chord will be the axis. 3. To construct the tangent line at a given point of the curve, draw three equidistant parallel diameters, the middle one of which shall pass through this point. Then a line through the vertex of this diameter, and parallel to the chord connecting the vertices of the others, will be the required tangent. Elementary properties of the Hyperbola, and resulting constructions. I.---Properties corresponding to those of the Ellipse. 185. Let VATB--Y'A'B', PI. XL, Fig. 98, be a vertical cone of revolution, cut by the vertical plane CD, parallel to its axis, . in a hyperbola. This plane will thus cut all the elements of the cone, except the two contained in the parallel plane, KSy through the vertex, YV7, of the cone. The plane CD may be taken as described, since out of the infinite number of cones of revolution from which a given hyperbola can be cut (177), one will have its axis parallel to the plane of the curve. The properties of the ellipse derived from those of the circle may now be elegantly transferred to the hyperbola, as follows: 186. 1. The hyperbola, cccbd--da'd g'Vh', evidently consists of two equal, opposite and infinite branches, having two axes of symmetry, XZ and Y/0/; for, as the figure is placed, the vertical projection of the curve is identical in form with the curve itself. 2. Let PQP/ be a plane, perpendicula...