Definitions

# Flatness

[flat]
The intuitive idea of flatness is important in several fields.

## Flatness in mathematics

The flatness of a surface is the degree to which it approximates a mathematical plane. The term is generalized for higher-dimensional manifolds to describe the degree to which they approximate the Euclidean space of the same dimensionality. See curvature.

Flatness in homological algebra and algebraic geometry means, of an object $A$ in an abelian category, that $- otimes A$ is an exact functor. See flat module or, for more generality, flat morphism.

## Flatness in systems theory

Flatness is a property of nonlinear dynamic systems. It extends the notion of controllability from linear time-invariant systems to nonlinear systems. Flatness is closely related to Feedback linearization by dynamic state feedback.

## Flatness in cosmology

In cosmology, the concept of "curvature of space" is considered. A space without curvature is called a "flat space" or Euclidean space.

A question often asked is "is the Universe flat"? The geometry of spacetime has been measured by the WMAP probe to be nearly flat. The data are consistent with a flat geometry, with Ω = 1.02 +/- 0.02

## Flatness in mechanical engineering

Joseph Whitworth popularized the first practical method of making accurate flat surfaces during the 1830s, using engineer's blue and scraping techniques on three trial surfaces. By testing all three pairs against each other, it is ensured that the surfaces become flat. Using two surfaces would result in a concave surface and a convex surface. Eventually a point is reached when many points of contact are visible within each square inch, at which time the three surfaces are uniformly flat to a very close tolerance.

Up until his introduction of the scraping technique, the same three plate method was employed using polishing techniques, giving less accurate results. This led to an explosion of development of precision instruments using these flat surface generation techniques as a basis for further construction of precise shapes.

## Flatness in precision manufacturing

In the manufacture of precision parts and assemblies, especially where parts will be required to be connected across a surface area in an air-tight or liquid-tight manner, flatness is a critical quality of the manufactured surfaces. such surfaces are usually machined or ground to achieve the required degree of flatness. High-definition metrology, such as digital holographic interferometry, of such a surface to confirm and ensure that the required degree of flatness has been achieved is a key step in such manufacturing processes. Flatness may be defined in terms of least squares fit to a plane ("statistical flatness"), worst-case or overall flatness (the distance between the two closest parallel planes within which the surface barely will fit, or other mathematical definitions that fit the intended use of the manufactured part.

## Flatness in electrical engineering

When measuring the flatness of a particular non-time-domain response, the measure of flatness defines the difference in a maximum and minimum value. For example, in a frequency response plot for an amplifier the flatness is defined as

$text\left\{flatness\right\} = max left \left(P_text\left\{out\right\} right \right) - min left \left(P_text\left\{out\right\} right \right)$

where each output power measurement is in decibels.

## Flatness in art

In art criticism of the 1960s and 1970s, flatness described the smoothness and absence of curvature or surface detail of a two-dimensional work of art. Critic Clement Greenberg believed that flatness, or two-dimensionality, was an essential and desirable quality in painting, a criterion which implies rejection of painterliness and impasto. The valorization of flatness led to a number of art movements, including minimalism and post-painterly abstractionism.

## Flatness in liquids

A carbonated beverage becomes flat when it loses enough of its carbon dioxide that there is no more "fizz" left, although this refers to the intrinsic properties of the substance, rather than the geometric properties of the liquid.

On planet earth, the flatness of a liquid is a function of the curvature of the earth, and from trigonometry, can be found to deviate from true flatness by approximately 19.6 nanometers over an area of 1 square meter. This is using the earths mean radius at sea level, however a liquid will be slightly flatter at the poles.

## References

• Wayne R. Moore, Foundations of Mechanical Accuracy, Moore Special Tool Company, Bridgeport, CT (1970)
• Joseph Whitworth, Plane Metallic Surfaces, Longman, Brown, and Co., London (1858)