Pascal's theorem

The set of Pascal (after Blaise Pascal ) is a statement in projective planes and states:

For arbitrary 6 points of a nondegenerate conic in a projective plane, the points

On a straight line, the Pascal line ( see picture). ( For the proof see weblink planar circle geometries, pp. 29-30 and R. Stärk: Descriptive Geometry, p 114 )

The numbering indicates that 6 of the 15 straight line connecting the points 6 are used, and which edges are adjacent. The numbering is chosen so that the line graph can be represented by a regular six - corner. Straight to opposite edges of the graph 's edges are so cut. If other edges in the Pascal figure enter, you have to permute the indexes accordingly. For the second Pascal configuration, the indices 2 and 5 were reversed (see image below).

Non- degeneracy means here: no 3 points lie on a straight line. The conic section can be thought of as an ellipse, therefore. ( A pair of lines to be cut is a degenerate conic. )

Conic sections are only defined in such a projective planes, which can be koordinatisieren over ( commutative ) bodies. Examples of solids include the real numbers, the rational numbers, complex numbers, finite body. Each nondegenerate conic in a projective plane can be suitable homogeneous coordinates by the equation describe (see projective conic ).

Remarks:

  • The set of Pascal is the dual version of the set of Brianchon.
  • There are set of Pascal degeneracies with 5 or 4 or 3 points ( on a conic ). In a degenerate two points connected by an edge coincide formally and the associated secant of the Pascal figure is replaced by the tangent in the remaining point. See the figure below weblink and planar circle geometries, p.30 - 35th By a suitable choice of a straight line of Pascal figures as line at infinity is closing rates correspond to hyperbolas and parabolas. See hyperbola and parabola.
  • If the conic is completely contained in an affine plane, there is also an affine form of the sentence:
  • The figure of six points on the conic is called Hexagrammum Mysticum.
  • The set of Pascal is also valid for a pair of straight lines ( degenerate conic ) and is then identical with the set of Pappus -Pascal.
  • The set of Pascal was generalized by August Ferdinand Möbius in 1847:
Brianchon's theorem Hanfried Lenz
635251
de