Vertical direction
Encyclopedia
In astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

, geography
Geography
Geography is the science that studies the lands, features, inhabitants, and phenomena of Earth. A literal translation would be "to describe or write about the Earth". The first person to use the word "geography" was Eratosthenes...

, geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

and related sciences and contexts, a direction
Direction (geometry, geography)
Direction is the information contained in the relative position of one point with respect to another point without the distance information. Directions may be either relative to some indicated reference , or absolute according to some previously agreed upon frame of reference Direction is the...

passing by a given point is said to be vertical if it is locally aligned with the gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

of the gravity
Gravitation
Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...

field
Field (physics)
In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...

, i.e., with the direction of the gravitational force (per unit mass, i.e. gravitational acceleration 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
Cartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

.

Discussion

Although the word horizontal is common in daily life and language (see below), it is subject to many misconceptions. The concept of horizontality only makes sense in the context of a clearly measurable gravity field, i.e., in the 'neighborhood' of a planet, star, etc. When the gravity field becomes very weak (the masses are too small or too distant from the point of interest), the notion of being horizontal loses its meaning.
• A plane is horizontal only at the chosen point. Horizontal planes at two separate points are not parallel, they intersect.

• In general, a horizontal plane will only be perpendicular to a vertical direction
Vertical direction
In astronomy, geography, geometry and related sciences and contexts, a direction passing by a given point is said to be vertical if it is locally aligned with the gradient of the gravity field, i.e., with the direction of the gravitational force at that point...

if both are specifically defined with respect to the same point: a direction is only vertical at the point of reference. Thus both horizontality and verticality are strictly speaking local concepts, and it is always necessary to state to which location the direction or the plane refers to. Note that (1) the same restriction applies to the straight lines contained within the plane: they are horizontal only at the point of reference, and (2) those straight lines contained in the plane but not passing by the reference point are not horizontal anywhere.

• In reality, the gravity field of a heterogeneous planet such as Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

is deformed due to the inhomogeneous spatial distribution of materials with different densities
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

. Actual horizontal planes are thus not even parallel even if their reference points are along the same vertical line, since a vertical line is slightly curved.

• At any given location, the total gravitational force is not quite constant over time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

, because the objects that generate the gravity are moving. For instance, on Earth the horizontal plane at a given point (as determined by a pair of spirit level
Spirit level
A spirit level or bubble level is an instrument designed to indicate whether a surface ishorizontal or vertical . Different types of spirit levels may be used by carpenters, stonemasons, bricklayers, other building trades workers, surveyors, millwrights and other metalworkers, and in some...

s) changes with the position of the Moon (air, sea and land tide
Tide
Tides are the rise and fall of sea levels caused by the combined effects of the gravitational forces exerted by the moon and the sun and the rotation of the Earth....

s).

• On a rotating planet such as Earth, the strictly gravitational pull of the planet (and other celestial objects such as the Moon, the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

, etc.) is different from the apparent net force (e.g., on a free-falling object) that can be measured in the laboratory or in the field. This difference is the centrifugal force associated with the planet's rotation. This is a fictitious force
Fictitious force
A fictitious force, also called a pseudo force, d'Alembert force or inertial force, is an apparent force that acts on all masses in a non-inertial frame of reference, such as a rotating reference frame....

: it only arises when calculations or experiments are conducted in non-inertial frames of reference
Inertial frame of reference
In physics, an inertial frame of reference is a frame of reference that describes time homogeneously and space homogeneously, isotropically, and in a time-independent manner.All inertial frames are in a state of constant, rectilinear motion with respect to one another; they are not...

, such as the surface of the Earth.

Practical use in daily life

The concept of a vertical line is thus anything but simple, although, in practice, most of these effects and variations are rather small: they are measurable and can be predicted with great accuracy, but they may not greatly affect our daily life.

This dichotomy between the apparent simplicity of a usual concept and an actual complexity of defining (and measuring) it in scientific terms is because the typical linear scales and dimensions of relevance in daily life are 3 orders of magnitude (or more) smaller than the size of the Earth. Hence, the latter appears to be flat locally, and vertical directions in nearby locations appear to be parallel. Such statements are nevertheless approximations; whether they are acceptable in any particular context or application depends on the applicable requirements, in particular in terms of accuracy.

In graphical contexts, such as drawing and drafting on rectangular paper, it is very common to associate one of the dimensions of the paper with a vertical, even though the entire sheet of paper is standing on a flat horizontal (or slanted) table. In this case, the vertical direction is typically from the side of the paper closest to the user to the opposite side (farthest away). This is purely conventional (although it is somehow 'natural' when drawing a natural scene as it is seen in reality), and may lead to misunderstandings or misconceptions, especially in an educational context.

Vertically can literally be translated as up-and-down.

• Geodesy
Geodesy
Geodesy , also named geodetics, a branch of earth sciences, is the scientific discipline that deals with the measurement and representation of the Earth, including its gravitational field, in a three-dimensional time-varying space. Geodesists also study geodynamical phenomena such as crustal...

• History of geodesy
History of geodesy
Geodesy ,[1] also named geodetics, is the scientific discipline that deals with the measurement and representation of the Earth.Humanity has always been interested in the Earth...

• Vertical deflection
Vertical deflection
The vertical deflection at a point on the earth is a measure of how far the direction of the local gravity field has been shifted by local anomalies such as nearby mountains....

• Vertical tangent
Vertical tangent
In mathematics, a vertical tangent is tangent line that is vertical. Because a vertical line has infinite slope, a function whose graph has a vertical tangent is not differentiable at the point of tangency.- Limit definition :...