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. 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 map projection.
In the case of an affine transformation 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, we have directionally differential scaling and the diagonal elements (the eigenvalues) are the scale factors 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 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 the scale is 1:1.
In the more general case of one quantity represented by another one, the scale has also a physical dimension. 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, 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. 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. 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 1.20 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.