In projective geometry, an intersection theorem or incidence theorem is an incidence structure consisting of points, lines, and possibly higher-dimensional objects and their incidences, together with a pair of nonincident objects A and B (for instance, a point and a line). The "theorem" states that, whenever a set of objects satisfies the incidences (i.e. can be matched up with the objects of the incidence structure in a way that preserves incidence), then the objects corresponding to A and B must also be incident. An intersection theorem is not necessarily true in all projective geometries; it is rather a property which some geometries satisfy but not others.
For example, Desargues' theorem can be stated using the following incidence structure:
- Points:
- Lines:
- Incidences (in addition to obvious ones such as :
The implication is then —that point R is incident with line PQ.
Read more about Intersection Theorem: Famous Examples
Famous quotes containing the words intersection and/or theorem:
“If we are a metaphor of the universe, the human couple is the metaphor par excellence, the point of intersection of all forces and the seed of all forms. The couple is time recaptured, the return to the time before time.”
—Octavio Paz (b. 1914)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)