Scale distortion

Scale distortion refers to the difference in scale on one map, depending on the map's projection. Though all maps have a universal scale, that scale can only be used with perfect accuracy where there is no distortion on the map. Thus, it is important to understand how the scale varies on the map.

Point Scale
The point scale can be thought of as the scale of a map at a specific point.

Suppose P is a point at latitude $$\phi$$ and longitude $$\lambda$$ on the sphere (or ellipsoid). Let Q be a neighbouring point and let $$\alpha$$ 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 points on the projection corresponding to P and Q.

Definition: the point scale k at P is  the ratio of the two distances P'Q' and PQ in the limit that Q approaches P. We write this as
 * $$k(\lambda,\,\phi,\,\alpha)=\lim_{Q\to P}\frac{P'Q'}{PQ},$$

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 the point scale depends only on position and not on direction we say that it is isotropic and denote the scale by $$k(\lambda,\phi)$$— this is the case for all conformal projections. In the projection is also axi-symmetric then the scale will be independent of $$\lambda$$ and it is denoted by $$k(\phi)$$— this is the case for the (normal) Mercator projection.

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.)

Visualisation of point scale: the Tissot indicatrix


Consider a small circle on the the surface of the Earth centred at a point P at latitude $$\phi$$ and longitude $$\lambda$$. Since the point scale varies with position and magnitude the corresponding small circle on the projection will be distorted. Tissot proved that, as long as the distortion is not too great, the circle will become an ellipse on the projection. In general the dimension, shape and orientation of the ellipse will change over the projection and by superimposing these distortion ellipses on the map projection we can convey the way in which the point scale is changing over the map. The distortion ellipse is known as Tissot's Indicatrix. The example shown here is the Winkel tripel projection, the standard projection for world maps made by the National Geographic Society. The minimum distortion is on the central meridian at latitudes of 30 degrees (North and South). (Other examples ).

Point scale for normal cylindrical projections of the sphere


The key to a quantitative understanding of scale is to consider an infinitesimal element on the sphere. The figure shows a point P at latitude $$\phi$$ and longitude $$\lambda$$ on the sphere. The point Q is at latitude $$\phi+\delta\phi$$ and longitude $$\lambda+\delta\lambda$$. The lines PK and MQ are arcs of meridians which must converge at the pole: the length of PK is $$a\delta\phi$$ where $$a$$ is the radius of the sphere. The lines PM and KQ are arcs of parallel circles: the length of PM is $$(a\cos\phi)\delta\lambda$$ since the radius of a parallel circle at latitude $$\phi$$ is $$(a\cos\phi)$$. In deriving a property of the projection at P it suffices to take an infinitesimal element PMQK of the surface. In the limit of Q approaching P such an element tends to an infinitesimally small planar rectangle.

Consider the infinitesimal element on the sphere and the corresponding infinitesimal in the plane of the projection. For all the normal cylindrical projections of the sphere we know that $$x=a\lambda$$ and $$y$$ is some function of latitude only. Therefore on the projection the meridians are vertical (without approximation) and the parallels horizontal so that the element P'M'Q'K' is also a rectangle with a base $$\delta x=a\delta\lambda$$ and height $$\delta y$$. We defer the treatment of the scale in a general direction to a mathematical addendum to this page. Here we simply note that
 * horizontal scale factor  $$\quad k\;=\;\frac{\delta x}{a\cos\phi\,\delta\lambda\,}=\,\sec\phi\qquad\qquad{}$$
 * vertical scale factor     $$\quad h\;=\;\frac{\delta y}{a\delta\phi\,}=\frac{y'(\phi)}{a}$$

Note that the scale in the horizontal direction is independent of the definition of $$y(\phi)$$ so it is the same for all normal cylindrical projections. The following examples illustrate three such normal cylindrical projections and in each case the variation of scale with position and direction is illustrated by the use of Tissot's Indicatrix.

The equidistant projection
The equidistant projection, or the Plate Carrée (french for "flat square"), is the simplest and most ancient projection, having been in use since before the time of Ptolemy. The sphere is mapped into a rectangle by the equations.
 * $$x = a\lambda \qquad\qquad y = a\phi,$$

where $$a$$ is the radius of the sphere, $$\lambda$$ is the longitude from the central meridian of the projection (here taken as the Greenwich meridian at $$\lambda =0$$) and $$\phi$$ is the latitude. Note that $$\lambda$$ and $$\phi$$ are in radians (obtained by multiplying the degree measure by a factor of $$\pi/180$$). The value of  $$\lambda$$ is in the range $$[{-}\pi,\pi]$$ and the value of $$\phi$$ is in the range $$[{-}\pi/2,\pi/2]$$.

The results for the horizontal and vertical  scale factors follow immediately from the previous section. Since $$y'(\phi)=1$$ we have
 * horizontal scale, $$\quad k\;=\;\frac{\delta x}{a\cos\phi\,\delta\lambda\,}=\,\sec\phi\qquad\qquad{}$$      vertical scale  $$\quad h\;=\;\frac{\delta y}{a\delta\phi\,}=\,1$$

The calculation of the point scale in an arbitrary direction is given below.



Thus for the equidistant projection there is no scaling in the $$y$$-direction. In the $$x$$-direction the scaling increases with latitude by the factor $$\sec\phi$$. This is true for all normal cylindrical projections since the parallels, of true length $$2\pi a\cos\phi$$ at latitude $$\phi$$ must be stretched by the factor $$\sec\phi$$ to give the width of the rectangle, namely $$2\pi a$$. This stretching in the $$x$$ direction alone is illustrated by using Tissot's Indicatrix. The semi-minor axis of each ellipse is equal to that of the undistorted circles on the equator where the scale is unity. The mafor axis of each is stretched by the factor of $$\sec\phi$$. The area of any ellipse is greater than that of the circles showing that the projection does not preserve area.

Mercator projection


The Mercator projection maps the sphere to a rectangle (of infinite extent in the $$y$$-direction) by the equations
 * $$x = a\lambda \qquad\qquad y = a\ln \left(\tan \left(\frac{\pi}{4} + \frac{\phi}{2} \right) \right)$$

where a, $$\lambda\,$$ and $$\phi \,$$ are as in the previous example. Since $$y'(\phi)=a\sec\phi$$ the scales in the horizontal and vertical directions are:
 * horizontal scale, $$\quad k\;=\;\frac{\delta x}{a\cos\phi\,\delta\lambda\,}=\,\sec\phi\qquad\qquad{}$$
 * vertical scale    $$\quad h\;=\;\frac{\delta y}{a\delta\phi\,}=\,\sec\phi$$

In the mathematical addendum below we prove that the point scale in an arbitrary direction is also equal to $$\sec\phi$$ so the scale is isotropic, its magnitude increasing with latitude as $$\sec\phi$$. In the Tissot diagram each infinitesimal circular element preserves its shape but is enlarged more and more as the latitude increases.

Lambert's equal area projection


Lambert's equal area projection maps the sphere to a finite rectangle by the equations
 * $$x = a\lambda \qquad\qquad y = a\sin\phi$$

where a, $$\lambda$$ and $$\phi$$ are as in the previous example. Since $$y'(\phi)=\cos\phi$$ the scales in the horizontal and vertical directions are


 * horizontal scale, $$\quad k\;=\;\frac{\delta x}{a\cos\phi\,\delta\lambda\,}=\,\sec\phi\qquad\qquad{}$$
 * vertical scale    $$\quad h\;=\;\frac{\delta y}{a\delta\phi\,}=\,\cos\phi$$

The calculation of the point scale in an arbitrary direction is given below.

The vertical and horizontal scales now compensate each other and in the Tissot diagram each infinitesimal circular element is distorted into an ellipse of the same area as the undistorted cirles on the equator.

Scale variation on the Mercator projection
The Mercator point scale is unity on the equator but varies with latitude as $$k=\sec\phi$$. Since $$\sec\phi$$ tends to infinity as we approach the poles the Mercator map is grossly distorted at high lines. For this reason the projection is totally inappropriate for world maps (unless we are discussing navigation and rhumb lines). However, at a latitude of about 25 degrees the value of $$\sec\phi$$ is about 1.1 so Mercator is accurate  to within 10% in a strip of width 50 degrees centred on the equator. Narrower strips are better: a strip of width 16 degrees is accurate to within 1% or 1 part in 100.

A standard criterion for good large scale maps is that the accuracy should be within 4 parts in 10,000, or 0.04%, corresponding to $$k=1.0004$$. Since $$\sec\phi$$ attains this value at $$\phi=1.62$$ degrees and the  Mercator projection is highly accurate  within  a strip of width 3.24 degrees centred on the equator. This corresponds to north-south distance of about 360 km or 200 miles.

Therefore Mercator is very good in a narrow strip near the equator and is well suited to accurate mapping of territory there. This is the motivation for the universal application of the Transverse Mercator projection in which the formulae of the normal Mercator projection are applied treating some meridian as an equator: the result is a highly accurate map in a narrow strip near that meridian. By repeating this for many meridians we can cover the whole globe with highly accurate maps. For example the Universal Transverse Mercator (UTM) system employs 60 separate transverse projections to cover the ellipsoidal globe (at least between latitudes 80S and 85N). Although the transverse Mercator projection is conformal with an istropic scale the scale factor is a complicated function of latitude and longitude in these transverse projections.

Modified projections
The demand that the scale satisfies $$1<k<1.0004$$ may be relaxed a little to $$0.9996<k<1.0004$$. In this case we still have a scale variation that is within 0.04% of true scale so the mapping is still highly accurate. As an example suppose that the Mercator projection given above is modified to
 * $$x = 0.9996a\lambda \qquad\qquad y = 0.9996a\ln \left(\tan \left(\frac{\pi}{4} + \frac{\phi}{2} \right) \right).$$

This alteration does not alter the shape of the projection but it does mean that the scale factors given above are reduced. In fact the scale factor remains isotropic but we now have
 * modified Mercator scale,  $$\quad k\;=0.9996\sec\phi.$$

Thus (a) the scale on the equator is 0.9996, (b) the scale is $$k=1$$ at a latitude given by $$\phi_1$$ where   $$\sec\phi_1=1/0.9996=1.00004$$ so that $$\phi_1=1.62$$ degrees and (c)  $$k=1.0004$$ at a latitude $$\phi_2$$ given by $$sec\phi_2=1.0004/0.9996=1.0008$$ for which $$\phi_2=2.29$$ degrees. This modification of the transformation equations means that the projection is just as accurate in a wider strip (4.58 degrees).

Clearly the line of unit scale at latitude $$\phi_1$$ is where the cylindrical projection surface intersects the sphere: the radius of the parallel at $$\phi_1$$ must equal the radius of the projection cylinder. This is an example of a secant (in the sense of cutting) projection as opposed to a tangent (touching) projection. The same ideas may be used to extend the zone of accuracy of all cylindrical and conical projections.

The limits $$0.9996<k<1.0004$$ are not adopted slavishly but many projections are close. For the important UTM (conformal) projections the maximum scale factor can almost reach $$k=1.001$$ (over the mapped region); for the British OSGB projection (again conformal transverse Mercator) the maximum scale value is $$k=1.0007$$. In both these cases the scale on the central meridian is constant at $$k_0=0.9996$$ and the isoscales lines with $$k=1$$ are slightly curved lines approximately 180 km east and west of the central meridian.