In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus any two distinct lines in a projective plane intersect at exactly one point.

Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic. The archetypical example is the real projective plane, also known as the extended Euclidean plane. This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by PG(2, R), RP², or P₂(R), among other notations. There are many other projective planes, both infinite, such as the complex projective plane, and finite, such as the Fano plane.

A projective plane is a 2-dimensional projective space. Not all projective planes can be embedded in 3-dimensional projective spaces; such embeddability is a consequence of a property known as Desargues' theorem, not shared by all projective planes.

A projective plane is a rank 2 incidence structure ${\displaystyle ({\mathcal {P}},{\mathcal {L}},I)}$ consisting of a set of points ${\displaystyle {\mathcal {P}}}$, a set of lines ${\displaystyle {\mathcal {L}}}$, and a symmetric relation ${\displaystyle I}$ on the set ${\displaystyle {\mathcal {P}}\cup {\mathcal {L}}}$ called incidence, having the following properties:

The second condition means that there are no parallel lines. The last condition excludes the so-called degenerate cases (see below). The term "incidence" is used to emphasize the symmetric nature of the relationship between points and lines. Thus the expression "point P is incident with line ℓ" is used instead of either "P is on ℓ" or "ℓ passes through P".

It follows from the definition that the number of points ${\displaystyle s+1}$ incident with any given line in a projective plane is the same as the number of lines incident with any given point. The (possibly infinite) cardinal number ${\displaystyle s}$ is called order of the plane.

To turn the ordinary Euclidean plane into a projective plane, proceed as follows:

The extended structure is a projective plane and is called the extended Euclidean plane or the real projective plane. The process outlined above, used to obtain it, is called "projective completion" or projectivization. This plane can also be constructed by starting from R³ viewed as a vector space, see § Vector space construction below.