In
projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, ...
, an intersection theorem or incidence theorem is a statement concerning an
incidence structure
In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean plane as the two types of objects and ignore al ...
– consisting of points, lines, and possibly higher-dimensional objects and their incidences – together with a pair of objects and (for instance, a point and a line). The "
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
" states that, whenever a set of objects satisfies the incidences (''i.e.'' can be identified with the objects of the incidence structure in such a way that incidence is preserved), then the objects and must also be incident. An intersection theorem is not necessarily true in all projective geometries; it is a property that some geometries satisfy but others don't.
For example,
Desargues' theorem
In projective geometry, Desargues's theorem, named after Girard Desargues, states:
:Two triangles are in perspective ''axially'' if and only if they are in perspective ''centrally''.
Denote the three vertices of one triangle by and , and tho ...
can be stated using the following incidence structure:
*Points:
*Lines:
*Incidences (in addition to obvious ones such as
):
The implication is then
—that point is incident with line .
Famous examples
Desargues' theorem
In projective geometry, Desargues's theorem, named after Girard Desargues, states:
:Two triangles are in perspective ''axially'' if and only if they are in perspective ''centrally''.
Denote the three vertices of one triangle by and , and tho ...
holds in a projective plane
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
is the projective plane over some
division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
(skewfield} —
. The projective plane is then called ''
desarguesian
In projective geometry, Desargues's theorem, named after Girard Desargues, states:
:Two triangles are in perspective ''axially'' if and only if they are in perspective ''centrally''.
Denote the three vertices of one triangle by and , and tho ...
''.
A theorem of
Amitsur and Bergman states that, in the context of desarguesian projective planes, for every intersection theorem there is a
rational identity such that the plane satisfies the intersection theorem if and only if the division ring satisfies the rational identity.
*
Pappus's hexagon theorem
In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that
*given one set of collinear points A, B, C, and another set of collinear points a,b,c, then the intersection points X,Y,Z of line pairs Ab and aB, Ac and ...
holds in a desarguesian projective plane
if and only if is a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
; it corresponds to the identity
.
*
Fano's axiom (which states a certain intersection does ''not'' happen) holds in
if and only if has
characteristic ; it corresponds to the identity .
References
*
*{{cite journal, doi=10.1016/0021-8693(66)90004-4, title=Rational Identities and Applications to Algebra and Geometry, journal=Journal of Algebra, volume=3, issue=3, pages=304–359, year=1966, last1=Amitsur, first1=S. A., doi-access=free
Incidence geometry
Theorems in projective geometry