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:
R.
3
{\ displaystyle \ mathbb {R} ^ {3}}
x
1
{\ displaystyle x_ {1}}
x
2
{\ displaystyle x_ {2}}
x
3
{\ displaystyle x_ {3}}
E.
{\ displaystyle E}
the plane is of the vectors and spanned
x
1
x
2
{\ displaystyle x_ {1} x_ {2}}
E.
12
{\ displaystyle E_ {12}}
e
→
1
{\ displaystyle {\ vec {e}} _ {1}}
e
→
2
{\ displaystyle {\ vec {e}} _ {2}}
the plane is of the vectors and spanned
x
1
x
3
{\ displaystyle x_ {1} x_ {3}}
E.
13
{\ displaystyle E_ {13}}
e
→
1
{\ displaystyle {\ vec {e}} _ {1}}
e
→
3
{\ displaystyle {\ vec {e}} _ {3}}
the plane is of the vectors and spanned
x
2
x
3
{\ displaystyle x_ {2} x_ {3}}
E.
23
{\ displaystyle E_ {23}}
e
→
2
{\ displaystyle {\ vec {e}} _ {2}}
e
→
3
{\ 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 .
e
→
1
=
(
1
,
0
,
0
)
{\ displaystyle {\ vec {e}} _ {1} = (1,0,0)}
e
→
2
=
(
0
,
1
,
0
)
{\ displaystyle {\ vec {e}} _ {2} = (0,1,0)}
e
→
3
=
(
0
,
0
,
1
)
{\ 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
E.
12
{\ displaystyle E_ {12}}
x
3
=
0
{\ displaystyle x_ {3} = 0}
e
→
3
⋅
x
→
=
0
{\ displaystyle {\ vec {e}} _ {3} \ cdot {\ vec {x}} = 0}
x
→
=
s
e
→
1
+
t
e
→
2
{\ displaystyle {\ vec {x}} = s \, {\ vec {e}} _ {1} + t \, {\ vec {e}} _ {2}}
not defined
E.
13
{\ displaystyle E_ {13}}
x
2
=
0
{\ displaystyle x_ {2} = 0}
e
→
2
⋅
x
→
=
0
{\ displaystyle {\ vec {e}} _ {2} \ cdot {\ vec {x}} = 0}
x
→
=
s
e
→
1
+
t
e
→
3
{\ displaystyle {\ vec {x}} = s \, {\ vec {e}} _ {1} + t \, {\ vec {e}} _ {3}}
not defined
E.
23
{\ displaystyle E_ {23}}
x
1
=
0
{\ displaystyle x_ {1} = 0}
e
→
1
⋅
x
→
=
0
{\ displaystyle {\ vec {e}} _ {1} \ cdot {\ vec {x}} = 0}
x
→
=
s
e
→
2
+
t
e
→
3
{\ 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.
x
→
=
(
x
1
,
x
2
,
x
3
)
∈
R.
3
{\ displaystyle {\ vec {x}} = (x_ {1}, x_ {2}, x_ {3}) \ in \ mathbb {R} ^ {3}}
x
→
⋅
y
→
{\ displaystyle {\ vec {x}} \ cdot {\ vec {y}}}
x
→
{\ displaystyle {\ vec {x}}}
y
→
{\ displaystyle {\ vec {y}}}
s
{\ displaystyle s}
t
{\ 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.
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.
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
<img src="https://de.wikipedia.org//de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">