Projective Differential Geometry of Curves and Ruled Surfaces

Projective Differential Geometry of Curves and Ruled Surfaces

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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. 1906 edition. Excerpt: ...proportional to them. We shall then find an equation of the form i, (utvk--ukv, ) = 0, also of the wth degree. By the same reasoning as above, we find that the lines of a complex of the nih degree, which are situated in a given plane, envelop a curve of the nth class, the complex curve of the plane considered. The locus of all lines which satisfy two independent equations, homogeneous in the line coordinates, consists of oo2 straight lines. It is known as a congruence. Clearly the lines common to two complexes form a congruence. But a congruence need not be the complete intersection of two complexes, just as a space curve need not be the complete intersection of two surfaces. The essential part of our definition of a congruence is that it contains oo2 straight lines. The locus of oo1 straight lines is a ruled surface, which may or may not be the complete intersection of three complexes. We shall have occasion to make use of the expression of the homogeneous line-coordinates in terms of non-homogeneous pointcoordinates. For this purpose we need merely put Vi = x, V2 = V, Va =, 2/4 = 1, t = af, zt = tf, zs = z s4-1, so that co12 = xy1--x'y, ra13 = xz'--afz, nu = x--x1, In particular the two points x, y, z and a?, y1, #' may be infinitesimally close to each other, so that x1 = x + dx, y1 = y + dy, z' = z + dz. Then the line coordinates become xdy--ydx, xdz--zdx, dx, ydz--zdy, dy, dz, in which form they have been employed principally by IAe. If x is the independent variable, and and--are denoted by y' and z', they become xy'-y, xz'-z, 1, yz'--zy', y1, z1, in which form Halphen has made use of them. 2. The linear complex. Null-system. Let us consider in greater detail the case in which the equation of the complex is of the more

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  • Paperback | 84 pages
  • 189 x 246 x 4mm | 168g
  • United States
  • English
  • black & white illustrations
  • 1236766385
  • 9781236766380