The concept of scale is applicable if a system is represented proportionally by another system. This article is about proportionality the mathematical relation For example, for a scale model of an object, the ratio of corresponding lengths is a dimensionless scale, e. A scale model is a representation or copy of an object that is larger or smaller than the actual size of the object. In Dimensional analysis, a dimensionless quantity (or more precisely a quantity with the dimensions of 1) is a Quantity without any Physical units g. 1:25; this scale is larger than 1:50.

In the general case of a differentiable bijection, the concept of scale can, to some extent, still be used, but it may depend on location and direction. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property It can be described by the Jacobian matrix. In Vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its Determinant, the Jacobian determinant. 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. A map projection is any method of representing the Surface of a sphere or other shape on a plane.

In the case of an affine transformation the scale does not depend on location but it depends in general on direction. In Geometry, an affine transformation or affine map or an affinity (from the Latin affinis, "connected with" between two Vector If the affine transformation can be decomposed into isometries and a transformation given by a diagonal matrix, we have directionally differential scaling and the diagonal elements (the eigenvalues) are the scale factors in two or three perpendicular directions. In Linear algebra, a diagonal matrix is a Square matrix in which the entries outside the Main diagonal (↘ are all zero In Euclidean geometry, uniform scaling or Isotropic scaling is a Linear transformation that enlarges or diminishes objects the Scale factor In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes A scale factor is a number which scales, or multiplies some quantity 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 for one direction. A scale factor is a number which scales, or multiplies some quantity

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

In the case of an isometry the scale is 1:1. For the Mechanical engineering and Architecture usage see Isometric projection.

In the more general case of one quantity represented by another one, the scale has also a physical dimension. Dimensional analysis is a conceptual tool often applied in Physics, Chemistry, Engineering, Mathematics and Statistics to understand 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 diagram 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, torques could be represented on the same map by areas in a scale of 1 m² : 12 500 Nm, which is equal to 1 m : 12 500 N. A torque (τ in Physics, also called a moment (of force is a pseudo- vector that measures the tendency of a force to rotate an object about 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 or enlarged or reduced model indicates the ratio between the distances on the map or model and the corresponding distances in reality or the original. The scale of a Map is the ratio of a single unit of distance on the map to the equivalent distance on the ground See also Scale model A physical model is a smaller or larger physical copy of an object E. g. a map of scale 1:50,000 shows a distance of 50,000 cm (=500 m) 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. A centimetre ( American spelling: centimeter, symbol cm) is a unit of Length in the Metric system, equal to one hundredth The metre or meter is a unit of Length. It is the basic unit of Length in the Metric system and in the International 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.