User:Peter Mercator/Draft for scale (map)
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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. (A town plan would be an example). For maps covering larger areas, or the whole Earth, it is essential to use a map projection[1][2] from the sphere (or ellipsoid) to the plane. Such projections inevitably involve distortion and the scale can no longer be considered as constant. It is then necessary to introduce the concept of a variable point scale (or particular scale) which is defined as the ratio of the length of a small line element emanating from a point on the map to the length of the corresponding line element on the surface of the Earth. In general the point scale will vary with the position of the point and also the direction of the line element. Tissot's Indicatrix is often used to illustrate the variation of point scale. In the study of point scale it is convenient to define the projection formulae in such a way that the scale is unity, or nearly so, on some lines (or points) of the resulting map projection. Clearly such a map projection must be comparable to the size of the Earth and, in order to represent it on a small sheet of paper, it must be scaled down by a constant ratio known as the representative fraction (RF) or principal scale. Thus we have to differentiate two uses of the word scale: the variable point scale inherent in the projection and the constant scale involved in the reduction to the printed (or screen) map.