Scale (map) - Point Scale (or Particular Scale)

Point Scale (or Particular Scale)

As proved by Gauss’s Theorema Egregium, a sphere (or ellipsoid) cannot be projected onto the plane without distortion. This is commonly illustrated by the impossibility of smoothing an orange peel onto a flat surface without tearing and deforming it. The only true representation of a sphere at constant scale is another sphere such as the schoolroom globe.

Given the limited practical size of globes, we must use maps for detailed mapping. Maps require projections. A projection implies distortion: A constant separation on the map does not correspond to a constant separation on the ground. While a map may display a graphical bar scale, the scale must be used with the understanding that it will be accurate on only some lines of the map. (This is discussed further in the examples in the following sections.)

Let P be a point at latitude and longitude on the sphere (or ellipsoid). Let Q be a neighbouring point and let be the angle between the element PQ and the meridian at P: this angle is the azimuth angle of the element PQ. Let P' and Q' be corresponding points on the projection. The angle between the direction P'Q' and the projection of the meridian is the bearing . In general . Comment: this precise distinction between azimuth (on the Earth's surface) and bearing (on the map) is not universally observed, many writers using the terms almost interchangeably.

Definition: the point scale at P is the ratio of the two distances P'Q' and PQ in the limit that Q approaches P. We write this as

where the notation indicates that the point scale is a function of the position of P and also the direction of the element PQ.

Definition: if P and Q lie on the same meridian, the meridian scale is denoted by .

Definition: if P and Q lie on the same parallel, the parallel scale is denoted by .

Definition: if the point scale depends only on position, not on direction, we say that it is isotropic and conventionally denote its value in any direction by the parallel scale factor .

Definition: A map projection is said to be conformal if the angle between a pair of lines intersecting at a point P is the same as the angle between the projected lines at the projected point P', for all pairs of lines intersecting at point P. A conformal map has an isotropic scale factor. Conversely isotropic scale factors across the map imply a conformal projection.

Isotropy of scale implies that small elements are stretched equally in all directions, that is the shape of a small element is preserved. This is the property of orthomorphism (from Greek 'right shape'). The qualification 'small' means that at some given accuracy of measurement no change can be detected in the scale factor over the element. Since conformal projections have an isotropic scale factor they have also been called orthomorphic projections. For example the Mercator projection is conformal since it is constructed to preserve angles and its scale factor is isotopic, a function of latitude only: Mercator does preserve shape in small regions.

Definition: on a conformal projection with an isotropic scale, points which have the same scale value may be joined to form the isoscale lines. These are not plotted on maps for end users but they feature in many of the standard texts. (See Snyder pages 203—206.)