Lambert's Map

The Lambert conformal conic projection and how it illustrates the properties of analytic functions


The Lambert Projection

Conformal maps preserve the shapes of small areas exactly, although the scale of the map may vary from point to point. Conformality is an extremely valuable property for maps that are to be used critically, and not just for general orientation or decoration. The ancients knew the conformal property of the stereographic projection, while Mercator devised a conformal projection very useful for navigation in 1568. For general information on conformal maps and their uses, please refer to Mercator's Map.

The stereographic projection is projection onto a plane, while Mercator's projection is onto a cylinder surrounding the sphere, which, when unwrapped, becomes a plane map. A projection can also be made on a cone, which is well-suited to mapping middle latitudes, just as the Mercator is good for equatorial regions, and the stereographic is good for polar regions. A conformal conic projection was published by Johann Heinrich Lambert (1728-1777) in 1772, and is called the Lambert Conformal Conic Projection. Lambert was the inventor of the hyperbolic functions, and the first to study map projections scientifically.

French military maps had been based on the Bonne projection, which may have been adequate for France, but were not bonnes when extended eastwards, so new maps were made on the Lambert projection for the projected invasion of Germany in 1914. The invasion did not get far, as it turned out, but the excellence of the maps was obvious, and soon the Lambert projection was widely adopted. The delay in acceptance of the Lambert projection is remarkable, considering its excellent properties.

The general nature of a conic projection is shown at the right. The cone is tangent to the unit sphere at some latitude φ, and here the scale of the map is unity. It is often more convenient to work with the colatitude ψ = 90° - φ than with the latitude φ. When the cone is unwrapped, it forms a sector of a circle of angle less than 2π radians. This defines the constant k, which is this angle divided by 2π. It is easy to see in this case that k = sin ψ = cos φ. The projection of the sphere on the cone is not a simple orthogonal projection by any means, but must be mathematically determined to preserve conformality. Here we assume that we are mapping a perfect sphere. Although this is a good approximation for the earth, it is not a good enough approximation for accurate mapping.

The general appearance of a Lambert map is shown at the left. Meridians are represented by straight lines converging to a point O, which is usually more or less distant. Parallels of latitude are represented by concentric circles. The angle between meridians differing in longitude by Δλ is kΔλ. The map is located by a central meridian and two standard parallels. The simple map mentioned in the preceding paragraph has only one standard parallel, on which the scale is unity, but the usual Lambert map has two, as if the cone cut the sphere at the two standard latitudes. We will show later how to choose k so that this occurs. There is no essential change in the map, except that now the map scale is very close to unity over the majority of the map. The standard parallels are usually chosen at 1/6 and 5/6 of the latitude range of the map.

Conformality is achieved by making the scale of the map the same in orthogonal directions. For convenience, we choose the direction of the meridian and its perpendicular, which is along a parallel of latitude. The diagram at the right shows small areas of the surface of the unit sphere, and the corresponding area of the map. Corresponding points are given the same number. These points are infinitesimally close to one another. Geographic coordinates of Point 1 are (ψ,λ), point 2 (ψ + dψ,λ), and point 3 (ψ, λ + dλ). The distances are as indicated, and equality of scale in the two directions gives the differential equation for R(ψ). Once this function is known, we can plot any point on the map. The differential equation is easy to integrate, and the result is R(ψ) = C tank(ψ/2), where C is a constant depending on the scale of the map. Since the scale is kR/sin ψ, setting this equal to unity for ψ equal to the standard latitude determines C. If we use two standard latitudes, then the condition that the scale be unity on both determines k. Then we can use C to give the map any desired scale. The formula for k is:

k = ln [sin(ψ1/2)/sin(ψ2/2)] / ln [tan(ψ1/2)/tan(ψ2/2)]

These formulas look imposing, but are actually easy to calculate using a pocket calculator or a computer. They were also well-adapted to logarithmic calculations when this method had to be used. Calculations are generally carried out to as many figures as the calculator has available to avoid problems with round-off errors. All you need to know is R(ψ). From it you can plot points and determine the scale. By adjusting a multiplicative constant, the map can be made any actual size. These formulas give exact conformality, and are much simpler than those for most map projections.

The Relation to Analytic Functions

The above analysis illustrates very well some of the remarkable properties of what are called analytic functions of a complex variable. A point in the (x,y)-plane represents the complex number z = x + iy, which has all the properties of the usual numbers, and more. A complex function w = f(z), which associates a complex number w with a complex number z over some region D in the (x,y)-plane, has the geometric interpretation of mapping the point z into the point w, and vice-versa. We can use two separate complex planes for this mapping, the (x,y)-plane and the (u,v)-plane, where w = u + iv, and u,v may be considered as functions of x,y. The geometric interpretation of these coordinates is arbitrary. Above, we considered the mapping of a point (φ,λ) of the sphere to a point on the plane (x,y), which is completely analogous.

What is meant by continuity of a complex function is illustrated at the right. The point zo is mapped into the point wo by the function (subscripts not shown in the diagram). Suppose we draw a small circle around wo of radius δ. Then, however small δ may be, if we can find an ε such that when z is within this distance of zo, w is within a distance δ of wo, then we say that f(z) is continuous at zo. Continuity is an obvious requirement for a practical map. If w = u + iv, then w is continuous if u and v are continuous functions of their arguments x,y. In other words, a function is continuous if it approaches its value at any point (instead of having the limit different from the value at the point).

Now we come to what is perhaps the most important property of all. The derivative is a limit fundamental to the differential calculus. For a function of one variable, f(x), the derivative is lim [f(x + dx) - f(x)]/dx as dx approaches zero. If it exists, it defines the derivative f'(x), and does not imply much more about the function. For a complex function, the analogous expression is lim [f(z + dz) - f(z)]/dz, but now dz can vary in any direction. For example, dz = dx, or dz = i dy. In either case the limits may exist, and may be different. If the limits are the same, we have a very special case because then the limit is the same no matter how z is approached, and we have a unique derivative f'(z) = ae, where we have expressed this complex number in terms of a magnitude and an angle. The important thing is that we get the same number for any direction of approach to z. The existence of the derivative says a great deal about a complex function, usually implying that it has continuous derivatives of all orders. It is a very restrictive condition.

If the derivative f'(z) exists everywhere in a region D, then f(z) has some very important properties. Such a function is called analytic. To illustrate just how peculiar such a function is, it has the property that if the function is known over a small area, or along a curve of small length, then the function is determined everywhere in the region D. Polynomials or convergent power series are analytic functions, as are ratios of polynomials, except at the zeros of the denominator (poles). The exponential function is analytic, as well as all the functions derived from it, such as logarithmic, trigonometric and hyperbolic. Any power series defines an analytic function, and Weierstrass made this the basis for his investigations.

The mapping of a small region by an analytic function multiplies all distances by a =|f'(z)|, and rotates the area by an angle φ = arg f'(z), if the derivative is f'(z) = ae. We recognize such a mapping as conformal, from what we have observed with the Lambert projection. Any analytic function defines a conformal mapping, yet another remarkable property of these functions.

If we define a complex function by w = u(x,y) + iv(x,y), the functions u and v must satisfy special requirements if w is to be analytic. The mapping by this function of a typical small area dydx is shown in the diagram at the right, where the change of scale and the rotation are clearly shown. Since lines 1-2 and 1-3 are of equal length and at right angles, for a conformal map lines 1'-2' and 1'-3' must also be of equal length and right angles. This requirement, expressed in terms of the partial derivatives of u and v, is known as the Cauchy-Riemann conditions. This provides another way to recognize an analytic function when it is expressed in this way. It is easy to show that both u and v are solutions of Laplace's equation in two dimensions, or harmonic functions, yet another amazing property of analytic functions.

The Spheroidal Earth

The earth, to the detriment of simplicity, insists on being very close to an oblate spheroid, the figure obtained by rotating an ellipse about its minor axis, which in this case is the polar axis of the earth. The reason for this, of course, is that the earth rotates about its polar axis. The centrifugal force competes with gravitation to give the earth the shape it would have if it were a fluid. Now, the surface of equal gravitational potential corresponding to sea level is not an exact spheroid, and the earth acts as if it had some rigidity, but these effects are small. The spheroid, therefore, is a useful reference surface for the earth. The force of gravity is normal to the spheroid, the direction a plumb bob would assume, and called vertical. Any point on the actual surface of the earth can be located by longitude, latitude and height above (or below) the spheroid, where the latitude is the angle between vertical and the equatorial plane.

The geometry of the spheroidal earth is shown at the left. This meridional cross section is an ellipse, of semiaxes a and b. The distance from the center to a focus is c = &sqrt;a2 - b2, and the eccentricity is c/a. For the earth, c = 521 828 m, and e = 0.0818. The relation between e and the flattening f is e2 = 2f -f2. The vertical at point P, extended downwards, meets the axis at Q, and the length PQ = N. φ is the geographical latitude, and &psi is 90° - φ, the geographic colatitude, convenient in many formulas. The angle between a line OP (not shown) and the equatorial plane is the geocentric latitude. Relative to the center of the spheroid, any point (φ,λ) has the coordinates x = N cos φ sin λ, y = N cos φ cos λ, z = (1 - e2) N sin φ.

A conformal conic map is made of the spheroid just as for the sphere. Only the results are stated here. The radius of a parallel of latitude is N sin &psi instead of simply a sin ψ, so where we had sin ψ we now have sin ψ / (1 - e2cos2ψ)1/2. The meridional distance corresponding to dφ is no longer simply this angle, but dφ (1 - e2)/(1 - e2sin2φ)3/2. When the resulting differential equation for R(φ) is integrated, the result is as before, except that sin ψ is replaced by the value above, and tan(ψ/2) is multiplied by {(1 + e cos ψ)/(1 - e cos ψ)] e/2. These are small corrections, but necessary for accurate work.

The spheroid whose constants were given above is the International Astronomical Union (IAU) spheroid of 1976. Table 15.5 of the Explanatory Supplement to the Nautical Almanac lists 14 reference spheroids that have been used. In the U.S., the Clarke spheroid of 1866 with a = 6 378 206.4 m, 1/f = 294.978698 was the basis for official maps, while in the U.K. the Airy spheroid of 1830 with a = 6 377 563.396 m, 1/f = 299.324964 played a similar role. For more recent spheroids, a varies from 6 378 388 m (Hayford, 1924) to 6 378 137 m (MERIT, 1983). The corresponding values of 1/f are 297 and 298.257. Any of these spheroids is satisfactory, since it is not a question of increasing accuracy--there is no exact spheroid. The proliferation of spheroids may be fun, but it is also confusing when comparing figures from different sources. Measurements on earth satellites give a great deal of excellent information on the earth's shape, and an inspiration for new spheroids.

The use of many decimal places in geodetic calculations may give an impression of great accuracy, but unless everything is correct and perfectly known, this is only an illusion. The large number of places is used to minimize the effects of roundoff error.

Plane Coordinates

For ordinary surveying, geographical latitude and longitude are not a convenient way to specify points, because of the difficulty of correlating them with linear distances on the earth as measured by ordinary surveying, classically with a transit and tape. It is much more convenient to locate points by rectangular coordinates, but this, too, has its problems because of the curvature of the earth. The two can be brought together by the use of a conformal map, of which the Lambert is an excellent example. Choosing an origin, we superimpose a set of rectangular coordinates on the map, with y usually along a meridian, and x perpendicular to it. For any small area, this map can be made to coincide almost exactly with a plane survey (one that does not take the curvature of the earth into consideration). Relative coordinates can then be determined easily.

The conversion between geographical and plane coordinates is very easy for the Lambert projection. The principle is shown at the right. The easting is x = R(φ) cos [k(λ - λo)] + xo, where xo is a constant added to x so that the easting is greater than zero everywhere on the map, since O is on the standard meridian. The northing is R(φo) - R(φ) cos [k(λ - λo)]. Nothing need be added, usually, since O is chosen at or below the lowest point on the map. These calculations are as easy as they appear, and make the Lambert projection an excellent basis for plane coordinates.

In the U.S., each state is divided into plane coordinate zones. Those using the Lambert projection are generally wider than high. Colorado has three of these Lambert plane coordinate zones. The transverse Mercator projection is also used for plane coordinates, although the calculations are much more difficult than for the Lambert. Transverse Mercator zones are generally higher than wide.

A great circle route on a Lambert map does not deviate much from a straight line. Deetz and Adams show a Lambert map of the North Atlantic with great circle routes drawn on it, and they differ little from straight lines. A Lambert map of the coterminous United States with standard parallels at 33°N and 45°N, and a central meridian at 95°W represents the whole country excellently. The maximum scale error in this map is only 2.5%.

The first director of the U.S. Coast and Geodetic Survey, Ferdinand R. Hassler, invented a peculiar projection, the polyconic, which was used for U.S. maps ever since. This projection is not conformal, but is approximately so along its central meridian. The meridians are laid out at true length, and the parallels are drawn with their true curvatures. Hassler observed that the U.S. of his time was predominantly north-south, and of little breadth, so this would serve well. It happened, however, that the U.S. spread way to the west, so that many additional standard meridians were necessary. Although maps on the same meridian fit together well, this is not the case for those on different meridians. Errors of 7% on polyconic maps drop to a few percent on Lambert maps. Nevertheless, government inertia is great, and perhaps the polyconic still breathes in U.S.G.S. topographic maps. Its French cousin, the Bonne projection, has already been mentioned above. The approximately conformal Transverse Mercator Projection has been adopted for U.S. military maps, the military mind preferring complication and obscurity to simplicity.

Stereographic Projection

The principle of stereographic projection is shown at the right. It is a true geometric projection, and is conformal. The map plane is shown tangent at the north pole, but it could be at any height. Often it is taken as the equatorial plane. The only difference is the scale. The scale varies by a factor of 2 from the center to the periphery when mapping an entire hemisphere, which is really not too large for many purposes. It is a conformal map, with meridians represented by radial lines and parallels by circles. It maps circles on the sphere into circles on the map, a valuable property. A stereographic net is used in crystallograpy to work with directions. In this case, the equatorial plane is the usual plane of the map, and both hemispheres are represented on it.

A stereographic map may be drawn on a plane tangent to any point of the sphere, but then the meridians and parallels are no longer lines and circles, but complicated curves. Radial lines from the center of the map are great circles. A projection from the center O instead of the south pole S results in a gnomonic map, which has much more variation of scale over a hemisphere--in fact, a full hemisphere cannot be represented. The gnomonic map is not conformal, stretching areas radially. The meridional scale is sec2&psi, while the transverse scale is secψ. However, a gnomonic map has the advantage that any straight line on it is a great circle route, not just those from the center. Along such a route, the radius moves in a plane, and the intersection of two planes is a line, the projection of the great circle. A great circle on the sphere is probably a close enough approximation for most purposes. Finding shortest distances, or geodesics, on the spheroid is more difficult.

To make a sterographic map of the spheroid, "a" is replaced by a/(1 - e2)1/2, and the tangent is multiplied by the familiar factor [(1 + e cosψ)/(1 - e cosψ)]e/2. The scale of the map is proportional to (R/sinψ)(1 - e2cosψ2) 1/2. Once the scale is chosen to be unity at some standard latitude, the scale at any latitude can be found.

The stereographic projection also has an application in pure mathematics, as the Riemann Sphere of complex numbers, as shown in the diagram. Instead of the usual problem of mapping the sphere on a plane, here we do the reverse, and map the plane on the sphere, to help us understand the properties of the point at infinity. The sphere has diameter 1, and its south pole is at the origin, the point z = 0. Its north pole is the point z = ∞, and also the point of projection. The equator is the map of the unit circle |z| = 1. Points on the lower hemisphere are within the unit circle, and points in the upper hemisphere are outside the unit circle. Points of the sphere symmetric by reflection in the equator represent points of the plane inverse to the unit circle. If φ is the latitude of a point, then r = tan(π/4 - φ/2), while the symmetric point at -φ has r' = tan(π/4 + φ/2) = cot(π/4 - φ/2), so we easily see that rr' = 1. A straight line in the plane maps into a circle through the north pole.

References

C. H. Deetz and O. Adams, Elements of Map Projection (Washington, DC: U.S.C.&G.S. Special Publication No. 68, 1931).

For complex variables, an excellent and classic text is R. V. Churchill, Complex Variables and Applications (New York: McGraw-Hill, 1960). There are, of course, many texts on this important subject, but none any better than this one for the scientist and engineer. The subject is essential in applied mathematics, besides being beautiful. We have treated only the differential calculus here, but the integral calculus has the most useful and amazing results.

E. K. Schuman, Plane Coordinates (Rolla, MO: mimeographed, 1953).

P. K. Seidelmann, ed., Explanatory Supplement to the Astronomical Almanac (Mill Valley, CA: University Science Books, 1992).


Return to Mathematics Index

Composed by J. B. Calvert
Created 27 August 2002
Last revised 7 September 2002