Map Scale

Let us notice that the scale of a map is not the exact ratio of a distance on the map to the corresponding distance on the ground. It is the approximation of that ratio because maps are usually created by transforming spatial data to a spherical or ellipsoidal surface and then to a plane. A map projection is the mapping from a curved surface into a plane. Since map projection is not isometry, the linear deformations will generally change from point to point, depending also on the direction at each point. Therefore, the map scale or the principal (linear) scale is the ratio of the distance on the map (in the plane of projection) and its origin on the surface (sphere, ellipsoid) to be projected/mapped. The principal scale is usually indicated on maps because it determines the general degree of reduction of the length on the map. That is why the map scale, and the local map scale should be distinguished.

If the region of the map is small enough to ignore surface curvature, such as in a town plan, then a single value can be used as the scale without causing measurement errors. In maps covering larger areas, or the whole Earth, the map scale may be less useful or even useless in measuring distances. Tissot's indicatrix is often used to illustrate the variation of point scale across a map. If a map represents the Earth or part of it, the map scale can be explained as the ratio of the size of the generating globe to the size of the Earth. The generating globe is a spherical model to which the Earth is shrunk and from which the map is mapped/projected. The ratio of the Earth's to the generating globe's size is called the principal scale (= nominal scale = representative fraction). Many maps state the principal scale and may display a graphic scale (= bar scale = linear scale) to represent it. According to Snyder and Voxland (1989), the true scale or correct scale is the linear scale having exactly the same value as the stated or nominal scale of the map, or a linear scale factor of 1.0. This definition makes full sense if one looks at the linear scale factor in all directions, not just in one. Maps are classified as small scale, medium scale or large scale. Small scale refers to world maps or maps of large regions such as continents. They show large areas of land on a small space. They are called small scale because the principal scale is relatively small. Large-scale maps show smaller areas in more detail, such as county maps or town plans. Such maps are called large scale because the principal scale is relatively large. For instance, a town plan, which is a large-scale map, might be at a scale of 1:10,000, whereas the world map, which is a small-scale map, might be at a scale of 1:100,000,000. Mapping large areas causes noticeable distortions because it significantly flattens the curved surface. How distortion gets distributed depends on the map projection. Scale varies across the map, and the stated map scale, principal scale, is only an approximation. The linear scale changes from point to point, and usually depends on the direction. This is the local linear scale. The local linear scale factor c = ds'/ds, where ds' is a differential of arc length in the plane of projection, and ds is the corresponding differential of arc length on the original surface, usually a sphere or ellipsoid (see Map projection distortions). The local linear scale LS is the product of the principal scale PS and the local linear scale factor c:  Since the local linear scale also depends on the direction, it would be more correct to write   where we denote the observed direction as α. Analogously, the local area scale factor p = dp'/dp, where dp' is a differential of an area in the plane of projection and dp is the corresponding differential of the area on the original surface. The local area scale is independent on the direction.

Map scales may be expressed in words, as a ratio, or as a fraction. Examples are: 'one centimetre to one hundred metres' or 1:10,000 or 1/10,000. Furthermore, many maps carry one or more graphic scales (Figure 1).

Figure 1: Graphic scale on the old map

If the distance is 1 cm on the map sheet produced at the scale of 1:25,000 and the local linear scale factor is equal to 1 in the direction of the measured distance, then the corresponding distance on the ellipsoid is 1 \bullet 25,000 cm = 250 m. If the distance on the ellipsoid is 2500 m and the local linear scale factor is equal to 0.9999 in the direction of the measured distance, then the corresponding distance on the map produced at the scale of 1:100,000 will be . The scale of a map on any projection is often crucial to the map's usefulness for a given purpose. For example, extreme area distortion is present at high latitudes on a small-scale Mercator map of the World, which means that the Mercator projection is not suitable for world maps. The same holds for any cylindrical projection applied to the world map.