Projective Differential Geometry of Curves and Rules Surfaces
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Projective Differential Geometry of Curves and Rules Surfaces

Projective Differential Geometry of Curves and Rules Surfaces


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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. 1921 Excerpt: ...by Cayley, in 1859, who showed that by means of them it becomes possible to characterize a space-curve analytically, by means of a single equation. To Cayley, also, is due the quadratic relation between the six homogeneous line-coordinates. We repeat the definition. Let yu...yt and zl, ... zi be two points of the line. Put (1) (oik = yiZk--ykZi 0, & = 1,2,3,4). Since mn = 0, and to, -=--ra, -, we need retain only six of these quantities, say li' ral3, mU, Sf m4t2, mw We define these to be the six homogeneous coordinates of the line. The propriety of this definition has already been explained. There is a one-to-one correspondence between the lines of space and the ratios of the above six quantities. There is, of course, a relation between these six quantities, since there are not oo5, but only oo lines in space. This relation has already been found to be (cf. Chapter II, 6), (2) SI = rali a3i + a13 ra42 + rau an = 0, where 1 may be used as an abbreviation for the left member. Conversely any six quantities which satisfy (2) may be interpreted as homogeneous coordinates of a line. It is easy to see that, corresponding to any projective transformation of space, the six homogeneous line-coordinates ral undergo a homogeneous linear substitution which, of course, leaves (2) invariant. A line may be determined as the intersection of two planes, instead of being considered as joining two points. If uu... M4 and v1, ...vi are the coordinates of two planes which contain the line, the determinants ik = uivk--ukvi may also be defined as coordinates of the line. These new coordinates tik are defined in a fashion dual to the definition of the first set to.-, and line-geometry is clearly a self-dual field, its element being self-dual. As a consequence of this ...


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Product Details
  • ISBN-13: 9781130601343
  • Publisher: Rarebooksclub.com
  • Publisher Imprint: Rarebooksclub.com
  • Height: 246 mm
  • No of Pages: 80
  • Spine Width: 4 mm
  • Width: 189 mm
  • ISBN-10: 113060134X
  • Publisher Date: 01 Mar 2012
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Weight: 159 gr


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