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
A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimal thin.
A plane segment or planar region (or simply "plane", in lay use) is a planar surface region; it is analogous to a line segment.
[a] A face is a plane segment bounding a solid object.
A parallelepiped is a region bounded by three pairs of parallel planes.
Euclid set forth the first great landmark of mathematical thought, an axiomatic treatment of geometry.
[2] He selected a small core of undefined terms (called common notions) and postulates (or axioms) which he then used to prove various geometrical statements.
Although the plane in its modern sense is not directly given a definition anywhere in the Elements, it may be thought of as part of the common notions.
[3] Euclid never used numbers to measure length, angle, or area.
It is a geometric space in which two real numbers are required to determine the position of each point.
It is an affine space, which includes in particular the concept of parallel lines.
It has also metrical properties induced by a distance, which allows to define circles, and angle measurement.
This section is solely concerned with planes embedded in three dimensions: specifically, in R3.
In a Euclidean space of any number of dimensions, a plane is uniquely determined by any of the following: The following statements hold in three-dimensional Euclidean space but not in higher dimensions, though they have higher-dimensional analogues: In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it (the normal vector) to indicate its "inclination".
Conversely, it is easily shown that if a, b, c, and d are constants and a, b, and c are not all zero, then the graph of the equation
[6] Thus for example a regression equation of the form y = d + ax + cz (with b = −1) establishes a best-fit plane in three-dimensional space when there are two explanatory variables.
Alternatively, a plane may be described parametrically as the set of all points of the form
where s and t range over all real numbers, v and w are given linearly independent vectors defining the plane, and r0 is the vector representing the position of an arbitrary (but fixed) point on the plane.
The plane passing through p1, p2, and p3 can be described as the set of all points (x,y,z) that satisfy the following determinant equations:
This system can be solved using Cramer's rule and basic matrix manipulations.
When the intersection of a sphere and a plane is not empty or a single point, it is a circle.
Then AOE and BOE are right triangles with a common side, OE, and hypotenuses AO and BO equal.
Now consider a point D of the circle C. Since C lies in P, so does D. On the other hand, the triangles AOE and DOE are right triangles with a common side, OE, and legs EA and ED equal.
[9] The proof can be extended to show that the points on a circle are all a common angular distance from one of its poles.
The flattest surface ever manufactured is a quantum-stabilized atom mirror.
[11] In astronomy, various reference planes are used to define positions in orbit.
Anatomical planes may be lateral ("sagittal"), frontal ("coronal") or transversal.
Many features observed in geology are planes or lines, and their orientation is commonly referred to as their attitude.
The trend is the compass direction of the line, and the plunge is the downward angle it makes with a horizontal plane.
