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The concept of scale is applicable if a system is represented proportional
Proportionality (mathematics)

In mathematics, two quantity are called proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio....
ly by another system. For example, for a scale model
Scale model

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 of an object, the ratio of corresponding lengths is a dimensionless scale, e.g. 1:25; this scale is larger than 1:50. In other words the lower the number on the right side of the ration the bigger the scale of a map.

In the general case of a differentiable bijection
Bijection

In mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that f = y....
, the concept of scale can, to some extent, still be used, but it may depend on location and direction.






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The concept of scale is applicable if a system is represented proportional
Proportionality (mathematics)

In mathematics, two quantity are called proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio....
ly by another system. For example, for a scale model
Scale model

Sorry, no overview for this topic
 of an object, the ratio of corresponding lengths is a dimensionless scale, e.g. 1:25; this scale is larger than 1:50. In other words the lower the number on the right side of the ration the bigger the scale of a map.

In the general case of a differentiable bijection
Bijection

In mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that f = y....
, the concept of scale can, to some extent, still be used, but it may depend on location and direction. It can be described by the Jacobian
Jacobian

In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant.In algebraic geometry the Jacobian of a algebraic curve means the Jacobian variety: a group variety associated to the curve, in which the curve can be embedded....
 matrix. The modulus of the matrix times a unit vector is the scale in that direction. The non-linear case applies for example if a curved surface like part of the Earth's surface is mapped to a plane, see map projection
Map projection

A map projection is any method of representing the surface of a sphere or other shape on a Plane . Map projections are necessary for creating maps....
.

In the case of an affine transformation
Affine transformation

In geometry, an affine transformation or affine map or an affinity between two vector spaces consists of a linear transformation followed by a translation :...
 the scale does not depend on location but it depends in general on direction. If the affine transformation can be decomposed into isometries and a transformation given by a diagonal matrix
Diagonal matrix

In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero....
, we have directionally differential scaling
Scaling (geometry)

In Euclidean geometry, uniform scaling or isotropic scaling is a linear transformation that enlarges or increases or diminishes objects; the scale factor is the same in all directions; it is also called a homothety....
 and the diagonal elements (the eigenvalues) are the scale factor
Scale factor

A scale factor is a number which scaling, or multiplies, some quantity. In the equation, is the scale factor for . is also the coefficient of , and may be called the constant of proportionality of to ....
s in two or three perpendicular directions. For example, on some profile maps horizontal and vertical scale are different; in particular elevation may be shown in a larger scale than horizontal distance.

In the case of directional scaling (in one direction only) there is just one scale factor
Scale factor

A scale factor is a number which scaling, or multiplies, some quantity. In the equation, is the scale factor for . is also the coefficient of , and may be called the constant of proportionality of to ....
 for one direction.

The case of uniform scaling corresponds to a geometric similarity. There is just one scale throughout.

In the case of an isometry
Isometry

In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces....
 the scale is 1:1.

In the more general case of one quantity represented by another one, the scale has also a physical dimension
Dimensional analysis

Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving certain physical quantities....
. E.g., if an arrow is drawn to represent a physical vector, the "scale" has a physical dimension equal to that of the vector, divided by length. For example, if a force of 1 newton is represented by an arrow of 2 cm, the scale is 1 m : 50 N. There is typically consistency in scale among quantities of the same dimension, but otherwise scales within the same horse may vary; e.g. "5 m" may also be represented by an arrow of 2 cm; in that case the scale for vectors which represent length is 1:250. Correspondingly, torque
Torque

Torque is the tendency of a force to rotate an object about an axis . Just as a force is a push or a pull, a torque can be thought of as a twist....
s could be represented on the same map by areas in a scale of 1 mē : 12 500 Nm, which is equal to 1 m : 12 500 N. Torques in the plane of the map could be represented by arrows with an independent scale of e.g. 1 m : 300 Nm.

The scale of a map
Scale (map)

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 or enlarged or reduced model
Model (physical)

A physical model is a smaller or larger physical copy of an object. The object being modelled may be small or large .The geometry of the model and the object it represents are often similar in the sense that one is a rescaling of the other; in such cases the Scale is an important characteristic....
 indicates the ratio between the distances on the map or model and the corresponding distances in reality or the original. E.g. a map of scale 1:50,000 shows a distance of 50,000 cm
Centimetre

A centimetre is a Units of measurement of length in the metric system, equal to one hundredth of a metre, which is the current International System of Units SI base unit of length....
 (=500 m
Metre

The metre or meter is a Unit of measurement of length. It is the SI base unit of length in the metric system and in the International System of Units , used around the world for general and scientific purposes....
) as 1 cm on a map, and a model on a scale 1:25 of a building with a height of 30 m has a model height of 1.20 m. An alternative method of indicating the scale is by a scale bar. This can also be applied on a computer screen etc., where the ratio may vary, and also remains valid when enlarging or reducing a paper map.

See also

  • Scale (map)
    Scale (map)

    Sorry, no overview for this topic
  • Scale
    Scale

    Scale can refer to:...
     (disambiguation)
  • List of scale model sizes
    List of scale model sizes

    MODEL SCALES This is a list of scale model sizes, listing a variety of size ratios for scale model....
  • Scale factor
    Scale factor

    A scale factor is a number which scaling, or multiplies, some quantity. In the equation, is the scale factor for . is also the coefficient of , and may be called the constant of proportionality of to ....