Pascal's theorem

Pascal's theorem in the real affine plane: If two pairs of opposite sides are parallel, so also the third pair

Pascal's theorem

Pascal's theorem: Edge graph

Pascal's theorem: indices 2 and 5 swapped

The set of Pascal (after Blaise Pascal ) is a statement of a 6-corner on a non-degenerate conic section in a projective plane . It can be formulated in the real affine plane as follows:

For a hexagon on an ellipse in which two pairs of opposite sides are parallel (in the picture ), the third pair of opposite sides is also parallel (in the picture:) . ${\displaystyle P_{1}P_{2}P_{3}P_{4}P_{5}P_{6}}$ ${\displaystyle {\overline {P_{1}P_{2}}}\parallel {\overline {P_{4}P_{5}}},\;{\overline {P_{6}P_{1}}}\parallel {\overline {P_{3}P_{4}}}}$ ${\displaystyle {\overline {P_{2}P_{3}}}\parallel {\overline {P_{5}P_{6}}}}$

If one considers this theorem in the projective closure of an affine plane (one adds the "distance line" on which parallel lines intersect), then the following applies:

The points lie for any 6 points of a non-degenerate conic section in a projective plane ${\displaystyle P_{1},P_{2},P_{3},P_{4},P_{5},P_{6}}$

${\displaystyle P_{7}:={\overline {P_{1}P_{2}}}\cap {\overline {P_{4}P_{5}}},}$

${\displaystyle P_{8}:={\overline {P_{6}P_{1}}}\cap {\overline {P_{3}P_{4}}},}$

${\displaystyle P_{9}:={\overline {P_{2}P_{3}}}\cap {\overline {P_{5}P_{6}}}}$

on a straight line, the Pascal straight line (see picture).

The numbering indicates which 6 of the 15 straight lines connecting the 6 points are used and which edges are adjacent. The numbering is chosen so that the edge graph can be represented by a regular hexagon. Straight lines to opposite edges of the edge graph are therefore intersected. If other edges are to be included in the Pascal figure, the indices have to be permuted accordingly. For the 2nd Pascal configuration, the indices 2 and 5 were swapped (see picture, below).

Non-degenerate means here: no 3 points lie on a straight line. The conic section can therefore be imagined as an ellipse. (An intersecting pair of lines is a degenerate conic section.)

Conic sections are only defined in projective planes that can be coordinated via (commutative) bodies . Examples of bodies are: the real numbers , the rational numbers , the complex numbers , finite bodies . Every non-degenerate conic section of a projective plane can be described in suitable homogeneous coordinates by the equation (see projective conic section ). ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle x_{1}x_{2}=x_{0}^{2}}$

Relation to other theorems and generalizations

Theorem of Pascal: degenerations

• Pascal's Theorem is the dual version of Brianchon's Theorem .
• There are divergences with 5, 4 or 3 points (on a conic section) for Pascal's theorem . In the case of a degeneration, two points connected by an edge formally coincide and the associated secant of the Pascal figure is replaced by the tangent in the remaining point. See the figure and weblink planar circlegeometries , pp. 30–35. A suitable choice of a straight line of the Pascal figures as a distance line results in closure sentences for hyperbolas and parabolas. See hyperbola and parabola .
• If the conic section is completely contained in an affine plane, there is also the affine form of the theorem (the one described at the beginning) in which the Pascal line is the distance line. The affine form exists e.g. B. in the real and the rational affine plane, but not in the complex affine plane. In the complex projective plane, every na conic section intersects every straight line. So there is no passer-by of the conic that could be chosen as a straight line.
• The figure of the six points on the conic section is also called the Hexagrammum Mysticum .
• Pascal's theorem is also valid for a pair of straight lines (degenerate conic section) and is then identical to Pappos-Pascal's theorem .
• Pascal's theorem was generalized by August Ferdinand Möbius in 1847 :

Assume that a polygon with sides is inscribed in a conic section. Now you extend the opposite sides until they intersect in points. If these points then lie on a common line, the last point also lies on this line. ${\displaystyle 4n+2}$ ${\displaystyle 2n+1}$ ${\displaystyle 2n}$

• Another generalization is the Cayley-Bacharach theorem .

Proof of Pascal's theorem

To prove Pascal's theorem

In the real case, the proof can be carried out on the unit circle. Since a non-degenerate conic section over any body cannot always be represented as a unit circle, the representation of the conic section, which is always possible, is used here as a hyperbola.

For the proof, one coordinates the projective plane inhomogeneously so that is, that is, the distance line is (see figure). Furthermore, let a point on the x-axis, a point on the y-axis. Then and (see figure). Let the slope of the straight line be . The theorem is proven when it has been proven. ${\displaystyle P_{1}=(\infty ),P_{6}=(0)}$ ${\displaystyle g_{\infty }={\overline {P_{1}P_{6}}}}$ ${\displaystyle P_{5}=(x_{5},0)}$ ${\displaystyle P_{2}=(0,y_{2})}$ ${\displaystyle P_{9}=(x_{9},0)}$ ${\displaystyle P_{7}=(0,y_{7})}$ ${\displaystyle {\overline {P_{i}P_{k}}}}$ ${\displaystyle m_{ik}}$ ${\displaystyle m_{79}=m_{43}}$

It's easy to calculate that is. With (see picture) you get ${\displaystyle {\frac {m_{29}}{m_{25}}}={\frac {m_{79}}{m_{75}}}}$ ${\displaystyle m_{29}=m_{23},\;m_{75}=m_{45}}$

(1) . ${\displaystyle :\ m_{79}={\frac {m_{23}}{m_{25}}}\cdot m_{45}}$

The conic section is in the inhomogeneous coordinate system as a hyperbola with an equation ${\displaystyle {\mathfrak {o}}}$

${\displaystyle y={\frac {a}{x-b}}+c}$ (The asymptotes are parallel to the coordinate axes!).

The set of peripheral angles for hyperbolas applies to such a hyperbola . If one applies the theorem of peripheral angles to the base points and the hyperbolic points , the equation is obtained ${\displaystyle P_{3},P_{5}}$ ${\displaystyle P_{2},P_{4}}$

(2) . ${\displaystyle :\ {\frac {m_{23}}{m_{25}}}={\frac {m_{43}}{m_{45}}}}$

From (1) and (2) it finally follows what had to be proven. ${\displaystyle m_{79}=m_{43}}$

Meaning of Pascal's theorem and its degenerations

Since Pascal's theorem is a statement about conic sections and conic sections are only explained in Pappusian planes, the concept of the oval is introduced in any projective plane in order to be able to formulate the Pascal property in any projective plane. This is e.g. B. not necessary with Pappus' theorem , since this is a theorem about straight lines and points that exist in every projective plane. An oval is a set of points (curve) of a projective plane with the essential incidence properties of a non-degenerate conic section.

Definition of an oval

→ Main article : Oval (projective geometry)

• A set of points in a projective plane is called an oval , if ${\displaystyle {\mathfrak {o}}}$

(1) Any straight line hits a maximum of 2 points. If is, is called passer , if is, is called tangent and if is, is called secant . ${\displaystyle g}$ ${\displaystyle {\mathfrak {o}}}$
${\displaystyle |g\cap {\mathfrak {o}}|=0}$ ${\displaystyle g}$ ${\displaystyle |g\cap {\mathfrak {o}}|=1}$ ${\displaystyle g}$ ${\displaystyle |g\cap {\mathfrak {o}}|=2}$ ${\displaystyle g}$

(2) For every point there is exactly one tangent , i. H. . ${\displaystyle P\in {\mathfrak {o}}}$ ${\displaystyle t}$ ${\displaystyle t\cap {\mathfrak {o}}=\{P\}}$

Pascal property of an oval

An oval in any projective plane, which has the property given in Pascal's theorem for conic sections for any 6 points, is called 6-point Pascal or Pascal for short . Accordingly, one defines 5-point Pascal , 4-point Pascal and 3-point Pascal , if the statement of the 5-, 4- or 3-point degeneration of Pascal's theorem for the oval is fulfilled (see figure) .

Meanings

The validity of the Pascal property or the 5-point degeneration for an oval in a projective plane has the same meaning as the Pappus property (for a pair of lines):

Buekenhout's theorem

If there is a projective plane and a -point-Pascal's oval in it, then there is a papal plane and a conic section. ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathfrak {o}}}$ ${\displaystyle \color {red}6}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathfrak {o}}}$

Hofmann's Theorem,

If there is a projective plane and a -point-Pascal's oval in it, then there is a papal plane and a conic section. ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathfrak {o}}}$ ${\displaystyle \color {red}5}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathfrak {o}}}$

With the help of the 4-point degeneration and the 3-point degeneration of Pascal's theorem, conic sections can be characterized in Pappus planes:

Sentence.

(a): If there is a Pappus projective plane and a -point-Pascalian oval in it, then it is a conic section. ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathfrak {o}}}$ ${\displaystyle \color {magenta}4}$ ${\displaystyle {\mathfrak {o}}}$

(b): If there is a Pappus projective plane of the characteristic and a -point-Pascalian oval in it, then it is a conic section. ${\displaystyle {\mathcal {P}}}$ ${\displaystyle \neq 2}$ ${\displaystyle {\mathfrak {o}}}$ ${\displaystyle \color {blue}3}$ ${\displaystyle {\mathfrak {o}}}$

Comment: How far one can weaken the prerequisite pappussch in the last two cases is still unclear. The requirement in statement (a) can at least be weakened to moufangsch .

literature

• Coxeter, HSM , and SL Greitzer: Timeless Geometry , Klett Stuttgart, 1983
• Gerd Fischer: Analytical Geometry . 4th edition, Vieweg 1985, ISBN 3-528-37235-4 , p. 199
• Hanfried Lenz : Lectures on projective geometry , Akad. Verl. Leipzig, 1965, p. 60
• Roland Stark: Descriptive Geometry , Schöningh-Verlag, Paderborn, 1978, ISBN 3-506-37443-5 , p. 114

Web links

• Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes (PDF; 891 kB), Uni Darmstadt, pp. 29–35.
• Projective geometry , short script, Uni Darmstadt (PDF; 180 kB), pp. 13-16
• hexagrammum-mysticum
• Pascal's theorem on cut-the-knot (English)

Individual evidence

1. ^ Jacob Steiner's lectures on synthetic geometry , BG Teubner, Leipzig 1867 (at Google Books: [1] ), Part 2, p. 128.
2. ^ E. Hartmann: Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. Script, TH Darmstadt (PDF; 891 kB), p. 29
3. ^ F. Buekenhout: Plans Projectifs à Ovoides Pascaliens , Arch. D. Math. Vol. XVII, 1966, pp. 89-93.
4. ↑ CE Hofmann: Specelizations of Pascal's Theorem on at Oval , Journ. o. Geom., Vol. 1/2 (1971), pp. 143-153.
5. ^ E. Hartmann: Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. Script, TH Darmstadt (PDF; 891 kB), p. 32,33