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The concept of scale is applicable if a system is represented proportionally by another system. For example, for a scale model of an object, the ratio of corresponding lengths is a dimensionless scale, e.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. 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.

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Keith Reynolds



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