In Euclidean geometry, a plane is a flat two-dimensional surface that extends indefinitely. Euclidean planes often arise as subspaces of three-dimensional space .A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimally thin.While a pair of real numbers suffices to describe points on a plane, the relationship with out-of-plane points requires special consideration for their embedding in the ambient space .
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Euclidean planes in three-dimensional space in the context of Hyperplane
In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is one less than that of the ambient space. Two lower-dimensional examples of hyperplanes are one-dimensional lines in a plane and zero-dimensional points on a line.
Most commonly, the ambient space is n-dimensional Euclidean space, in which case the hyperplanes are the (n − 1)-dimensional "flats", each of which separates the space into two half spaces. A reflection across a hyperplane is a kind of motion (geometric transformation preserving distance between points), and the group of all motions is generated by the reflections. A convex polytope is the intersection of half-spaces.
Euclidean planes in three-dimensional space in the context of Bed (geology)
In geology, a bed is a layer of sediment, sedimentary rock, or volcanic rock "bounded above and below by more or less well-defined bedding surfaces". A bedding surface or bedding plane is respectively a curved surface or plane that visibly separates each successive bed (of the same or different lithology) from the preceding or following bed. In cross sections, bedding surfaces or planes are often called bedding contacts. Within conformable successions, each bedding surface acted as the depositional surface for the accumulation of younger sediment.
Euclidean planes in three-dimensional space in the context of Plane of reference
In celestial mechanics, the orbital plane of reference (or orbital reference plane) is the plane used to define orbital elements (positions). The two main orbital elements that are measured with respect to the plane of reference are the inclination and the longitude of the ascending node.
Depending on the type of body being described, there are four different kinds of reference planes that are typically used: