Scale (ratio)

# Scale (ratio)

Discussion

Encyclopedia
The scale ratio of some sort of model
Model (physical)
A physical model is a smaller or larger physical copy of an object...

which represents an original proportional
Proportionality (mathematics)
In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...

ly is the ratio of a linear dimension of the model to the same dimension of the original. Examples include a 3-dimensional scale model
Scale model
A scale model is a physical model, a representation or copy of an object that is larger or smaller than the actual size of the object, which seeks to maintain the relative proportions of the physical size of the original object. Very often the scale model is used as a guide to making the object in...

of a building or the scale drawings of the elevations or plans of a building. In such cases the scale is dimensionless and exact throughout the model or drawing. The scale can be expressed in four ways: in words (a lexical scale), as a ratio, as a fraction and as a graphical (bar) scale
Linear scale
A linear scale, also called a bar scale, scale bar, graphic scale, or graphical scale, is a means of visually showing the scale of a map, nautical chart, engineering drawing, or architectural drawing....

. Thus on an architect's drawing we might read
'one centimetre to one metre'    or    1:100   or    1/100

and a bar scale would also normally appear on the drawing.

In general a representation may involve more than one scale at the same time. For example a drawing showing a new road in elevation might use different horizontal and vertical scales. An elevation of a bridge might be annotated with arrows with a length proportional to a force loading, as in 1 cm to 1000 newtons: this is an example of a dimensional scale. A weather map at some scale may be annotated with wind arrows at a dimensional scale of 1 cm to 20 mph.

Map scales
Scale (map)
The scale of a map is defined as the ratio of a distance on the map to the corresponding distance on the ground.If the region of the map is small enough for the curvature of the Earth to be neglected, then the scale may be taken as a constant ratio over the whole map....

require careful discussion. A town plan may be constructed as an exact scale drawing but for larger areas a map projection
Map projection
A map projection is any method of representing the surface of a sphere or other three-dimensional body on a plane. Map projections are necessary for creating maps. All map projections distort the surface in some fashion...

is necessary and no projection can represent the Earth's surface at a uniform scale: in general the scale of a projection depends on position and direction. The variation of scale may be considerable in small scale maps which may cover the globe. In large scale maps of small areas the variation of scale may be insignificant for most purposes but it is always present. The scale of a map projection must be interpreted as a nominal scale. (The usage large and small in relation to map scales relates to their expressions as fractions. The fraction 1/10,000 used for a local map is much larger than 1/100,000,000 used for a global map. There is no hard and fixed dividing line between small and large scales.)

## Mathematical note

In the general case of a differentiable bijection
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...

, 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 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 scale (map)
Scale (map)
The scale of a map is defined as the ratio of a distance on the map to the corresponding distance on the ground.If the region of the map is small enough for the curvature of the Earth to be neglected, then the scale may be taken as a constant ratio over the whole map....

.

In the case of an affine transformation
Affine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...

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 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 is a linear transformation that enlarges or shrinks objects by a scale factor that is the same in all directions. The result of uniform scaling is similar to the original...

and the diagonal elements (the eigenvalues) are the scale factor
Scale factor
A scale factor is a number which scales, or multiplies, some quantity. In the equation y=Cx, C is the scale factor for x. C is also the coefficient of x, and may be called the constant of proportionality of y to x...

s in two or three perpendicular directions.