Basic structure

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In synthetic projective geometry or the geometry of the position of three-dimensional space, a distinction is made between the " incidences " of two of the three basic elements point , straight line and plane as follows:

  • A point lies in a straight line or the straight line goes through the point.
  • A point lies in a plane or the plane passes through the point.
  • A straight line lies in a plane or the plane passes through the straight line.

In simplified terms, instead of “lies in” or “goes through” one simply says “incised with”.

The set of all those basic elements (points, straight lines, planes), which each intersect with another basic element , the respective “carrier”, is called a basic structure . With the above distinction there are seven of them, namely

  • the set of all points that intersect with a straight line, called the point series ,
  • the set of all lines, the incident with a point, called line bundle ( English pencil ),
  • the set of all points that intersect with a plane, called the point field ,
  • the set of all levels, the incident with a point called Eben bundle ( English bundle of planes )
  • the set of all straight lines that intersect with a plane, called the straight line field ,
  • the set of all planes that intersect with a straight line, called plane tufts ( English sheaf of planes ),
  • and the subset of all straight lines of a straight line field which also intersect with a certain point of the support plane, it is called a straight line bundle .

If one examines the relationships between the various basic structures, it turns out that some contain something else: A basic structure I is contained in a basic structure II if I is a real subset of II. I then also means a first level basic structure and II a second level basic structure.

There are thus three basic structures at the first level:

  • Row of points, line tufts and plane tufts

and thus four basic structures of the second level:

  • Point field, line field, line bundle and plane bundle.

Special cases of the above ground structure, the parallel is finally just clump and the parallel straight bundle of infinitely far away support point and the parallel flat clumps with infinitely distant support line.

Individual evidence

  1. ^ Weisstein, Eric W. "Pencil" From MathWorld - A Wolfram Web Resource.
  2. ^ Weisstein, Eric W. "Bundle of Planes" From MathWorld - A Wolfram Web Resource.
  3. ^ Weisstein, Eric W. "Sheaf of Planes" From MathWorld - A Wolfram Web Resource.
  4. Small Encyclopedia of Mathematics ; Leipzig 1970, pp. 216-217.