Euclidean planes in three-dimensional space

In Euclidean geometry, a plane is a flat two-dimensional surface that extends indefinitely. Euclidean planes often arise as subspaces of three-dimensional space ${\displaystyle \mathbb {R} ^{3}}$. A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimal thin. While a pair of real numbers ${\displaystyle \mathbb {R} ^{2}}$ suffices to describe points on a plane, the relationship with out-of-plane points requires special consideration for their embedding in the ambient space ${\displaystyle \mathbb {R} ^{3}}$.