International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 44, Pages 2331-2345
doi:10.1155/S0161171204308112
Real quartic surfaces containing 16 skew lines
Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Michoaćan 58260, Mexico
Received 12 August 2003
Copyright © 2004 Isidro Nieto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
It is well known that there is an open three-dimensional subvariety Ms of the Grassmannian of lines in ℙ3 which parametrizes smooth irreducible complex surfaces of degree 4 which are Heisenberg invariant, and each quartic contains 32 lines but only 16 skew lines, being determined by its configuration of lines, are called a double 16. We consider here the problem of visualizing in a computer the real Heisenberg invariant quartic surface and the real double 16. We construct a family of points l∈Ms parametrized by a two-dimensional semialgebraic variety such that under a change of coordinates of l into its Plüecker, coordinates transform into the real coordinates for a line L in ℙ3, which is then used to construct a program in Maple 7. The program allows us to draw the quartic surface and the set of transversal lines to L. Additionally, we include a table of a group of examples. For each test example we specify a parameter, the viewing angle of the image, compilation time, and other visual properties of the real surface and its real double 16. We include at the end of the paper an example showing the surface containing the double 16.