Horizontal and vertical

The usage of the inter-related terms horizontal and vertical varies with context. There are important symmetries and asymmetries between the two terms which change as one goes from two to three dimensions, from a flat earth scenario to the spherical earth one.

In astronomy, geography, and related sciences and contexts, a direction passing by a given point is said to be vertical if it is locally aligned with the local gravity vector at that point. In general, something that is vertical can be drawn from up to down (or down to up), such as the y-axis in the Cartesian coordinate system.

Girard Desargues defined the vertical to be perpendicular to the horizon in his Perspective of 1636.

In the context of a two-dimensional orthogonal Cartesian coordinate system on a Euclidean plane, to say that a line is horizontal or vertical, an initial designation has to be made. One can start off by designating the vertical direction, usually labelled the Y direction. The horizontal direction, usually labelled the X direction, is then automatically determined. Or, one can do it the other way around, i.e., nominate the x-axis, in which case the y-axis is then automatically determined. There is no special reason to choose the horizontal over the vertical as the initial designation: the two directions are on par in this respect.

The following hold in the two-dimensional case:

a) Through any point P in the plane, there is one and only one vertical line within the plane and one and only one horizontal line within the plane. This symmetry breaks down as one moves to the three-dimensional case.

b) A vertical line is any line parallel to the vertical direction. A horizontal line is any line normal to a vertical line.

c) Horizontal lines do not cross each other.

d) Vertical lines do not cross each other.

Not all of these elementary geometric facts are true in the 3-D context.

In the three-dimensional case, the situation is more complicated as now one has horizontal and vertical planes in addition to horizontal and vertical lines. Consider a point P and designate a direction through P as vertical. A plane which contains P and is normal to the designated direction is the horizontal plane at P. Any plane going through P, normal to the horizontal plane is a vertical plane at P. Through any point P, there is one and only one horizontal plane but a multiplicity of vertical planes. This is a new feature that emerges in three dimensions. The symmetry that exists in the two-dimensional case no longer holds.

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Wikipedia