Planarity

Planarity or planicity denotes the spatial arrangement (ie in three-dimensional space) of points in a plane ; the points are then flat (in mathematics: complanar ).

Two plane surfaces that are parallel are called plane-parallel .

Explanation of terms plan - planar

In German usage, it says plan , never planar , as in English, from which it is borrowed in some expressions such as screens, lenses ( Gaussian double lens or planar lens ) or other optical equipment. The term plan means “in a surface”, planar stands for “flattenable” in graph theory ( planar graph ), and for “in the (low) level (flat landscape)” in the geosciences ( lowland level ).

Planarity of graphs

Planar graph K4

A graph is planar if it can be drawn in a plane in such a way that all edges represented by Jordan curves only intersect at the end points / nodes. ${\ displaystyle G = (V, E)}$

According to Wagner and Fáry's theorem, there is a straight-line embedding for every planar graph.
The bounding edges of a graph are called borders.
Two embeddings are equivalent if there is an isomorphic map between the boundaries of their domains.
The planarity can be determined algorithmically in linear time.
If you want to color neighboring nodes of a planar graph differently, you can do this with only 4 colors according to the four-color theorem .

measuring technology

Planarity describes the surface quality; see flatness in general and surface roughness for special accuracy
A surface in construction is flat when it is sufficiently flat; see leveling
Straightening is a production process for shaping a workpiece flat.

optics

With optical lenses, planarity is a quality feature of the extent to which the cut approximates the ideal curve. The expression plane surface is used for the flat surfaces of prisms, lenses or mirrors .

A plane-parallel glass body is called a plane plate .