Coordinate plane

The coordinate plane in two-dimensional space

As a coordinate plane is referred to in the analytical geometry one of two unit vectors spanned original plane . In two dimensions, the coordinate plane corresponds to the Euclidean plane and thus the base of a Cartesian coordinate system . There are three coordinate planes in three-dimensional space: the xy plane , the xz plane and the yz plane .

Analytical geometry

Designations

The three coordinate planes in three-dimensional space

In the following, the three coordinate axes of three-dimensional space are denoted by , and . The three coordinate planes are often identified with the letter , which is provided with two indices that indicate the two unit vectors that define the plane: ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle x_{1}}$ ${\displaystyle x_{2}}$ ${\displaystyle x_{3}}$ ${\displaystyle E}$

• the plane is of the vectors and spanned ${\displaystyle x_{1}x_{2}}$ ${\displaystyle E_{12}}$ ${\displaystyle {\vec {e}}_{1}}$ ${\displaystyle {\vec {e}}_{2}}$
• the plane is of the vectors and spanned ${\displaystyle x_{1}x_{3}}$ ${\displaystyle E_{13}}$ ${\displaystyle {\vec {e}}_{1}}$ ${\displaystyle {\vec {e}}_{3}}$
• the plane is of the vectors and spanned ${\displaystyle x_{2}x_{3}}$ ${\displaystyle E_{23}}$ ${\displaystyle {\vec {e}}_{2}}$ ${\displaystyle {\vec {e}}_{3}}$

Here are the three unit vectors , and . The three-dimensional space is divided into eight octants by the three coordinate planes . The intersection of two coordinate planes results in a coordinate axis, the intersection of all three coordinate planes the coordinate origin . ${\displaystyle {\vec {e}}_{1}=(1,0,0)}$ ${\displaystyle {\vec {e}}_{2}=(0,1,0)}$ ${\displaystyle {\vec {e}}_{3}=(0,0,1)}$

Plane equations

The three coordinate planes are characterized by the following plane equations :

Coordinate plane	Coordinate shape	Normal form	Parametric shape	Intercept shape
${\displaystyle E_{12}}$	${\displaystyle x_{3}=0}$	${\displaystyle {\vec {e}}_{3}\cdot {\vec {x}}=0}$	${\displaystyle {\vec {x}}=s\,{\vec {e}}_{1}+t\,{\vec {e}}_{2}}$	not defined
${\displaystyle E_{13}}$	${\displaystyle x_{2}=0}$	${\displaystyle {\vec {e}}_{2}\cdot {\vec {x}}=0}$	${\displaystyle {\vec {x}}=s\,{\vec {e}}_{1}+t\,{\vec {e}}_{3}}$	not defined
${\displaystyle E_{23}}$	${\displaystyle x_{1}=0}$	${\displaystyle {\vec {e}}_{1}\cdot {\vec {x}}=0}$	${\displaystyle {\vec {x}}=s\,{\vec {e}}_{2}+t\,{\vec {e}}_{3}}$	not defined

Here, a point of the respective level, the scalar product of the vectors and as well as and are real numbers. ${\displaystyle {\vec {x}}=(x_{1},x_{2},x_{3})\in \mathbb {R} ^{3}}$ ${\displaystyle {\vec {x}}\cdot {\vec {y}}}$ ${\displaystyle {\vec {x}}}$ ${\displaystyle {\vec {y}}}$ ${\displaystyle s}$ ${\displaystyle t}$

Descriptive geometry

In the representing geometry , the three coordinate planes often correspond to the floor plan, the elevation plane and the cross elevation plane.

See also : normal projection

Synthetic geometry

In the synthetic geometry is an affine or projective plane , the (a coordinate range as a set with a specific algebraic structure planar ternary ring , quasifield , Alternatively body , skew , etc.) can be assigned, as a coordinate plane on this generalized body designated.

See also : Projective Coordinate System

literature

• Wolf-Dieter Klix, Karla Nestler: Constructive geometry . Hanser, 2001, ISBN 3-446-21566-2 .
• Max Koecher, Aloys Krieg: level geometry . 3. Edition. Springer, 2007, ISBN 3-540-49328-X .

Web links

Commons : Coordinate planes  - collection of images, videos and audio files