intersecting

Intersecting refers to the relationship between two or more objects that share one or more common points. In geometry, this often means two geometric figures meet at a point or set of points. For example, in the plane two distinct lines either intersect at a single point, are parallel and do not intersect, or are the same line and intersect at infinitely many points. Curves can intersect at finite or infinite sets of points, with the number and nature of intersection points depending on the shapes and equations involved; curves may cross, touch (tangent), or coincide over a segment.

In mathematics more broadly, intersecting extends to sets, events, and other collections. The intersection of sets A and B is the set of elements that are in both A and B, denoted A ∩ B. The intersection of a collection of sets contains the elements common to all members of the collection. This operation extends to higher dimensions and to algebraic structures, where intersections reflect common solutions or shared elements.

In algebraic geometry, intersection theory studies how subvarieties meet and the multiplicities of their intersections. Bezout’s theorem describes the maximum number of intersection points of projective plane curves, counting multiplicities, based on their degrees. In probability, the intersection of events A and B, written A ∩ B, represents the event that both occur; probabilities of intersections relate to concepts of independence and conditional probability.

Intersecting also appears in combinatorics and graph theory, where terms like intersecting families describe collections of sets with pairwise nonempty intersections, or where paths and subgraphs may intersect within a larger structure.