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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 Factor 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 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 s could be represented on the same map by areas in a scale of 1 m&2 : 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 (=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. SEE ALSO
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