Lambert azimuthal equal-area projection

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A Lambert azimuthal equal-area projection of the Earth. The center is 0° N 0° E. The antipode is 0° N 180° E, near Kiribati in the Pacific Ocean. That point is represented by the entire circular boundary of the map, and the ocean around that point appears along the entire boundary.
A Lambert azimuthal equal-area projection of the Earth. The center is 0° N 0° E. The antipode is 0° N 180° E, near Kiribati in the Pacific Ocean. That point is represented by the entire circular boundary of the map, and the ocean around that point appears along the entire boundary.

The Lambert azimuthal equal-area projection is a particular mapping from a sphere to a disk. It accurately represents area in all regions of the sphere, but it does not accurately represent angles. Intuitively, it gives a planar picture of the sphere in which every region appears with the correct area but perhaps a greatly distorted shape. It is named for the Alsatian mathematician Johann Heinrich Lambert, who discovered it in 1772.[1]

The Lambert azimuthal projection is used as a map projection in cartography. It is also used, in various scientific disciplines, for plotting the orientations of lines in three-dimensional space. This plotting is aided by a special kind of graph paper called a Schmidt net.

Contents

[edit] Definition

A cross-sectional view of the sphere and a plane tangent to it at S. Each point on the sphere (except the antipode) is projected to the plane along a circular arc centered at the point of tangency between the sphere and plane.
A cross-sectional view of the sphere and a plane tangent to it at S. Each point on the sphere (except the antipode) is projected to the plane along a circular arc centered at the point of tangency between the sphere and plane.

To define the Lambert azimuthal projection, imagine a plane set tangent to the sphere at some point S on the sphere. Let P be any point on the sphere other than the antipode of S. Let d be the distance between S and P in three-dimensional space (not the distance along the sphere surface). Then the projection sends P to a point P ' on the plane that is a distance d from S.

To make this more precise, there is a unique circle centered at S, passing through P, and perpendicular to the plane. It intersects the plane in two points; let P ' be the one that is closer to P. This is the projected point. See the figure. The antipode of S is excluded from the projection because the required circle is not unique. The case of S is degenerate; S is projected to itself, along a circle of radius 0.

Explicit formulas are required for carrying out the projection on a computer. Consider the projection centered at S = (0, 0, -1) on the unit sphere, which is the set of points (x, y, z) in three-dimensional space \mathbb{R}^3 such that x2 + y2 + z2 = 1. In Cartesian coordinates (x,y,z) on the sphere and (X,Y) on the plane, the projection and its inverse are then described by

(X, Y) = \left(\sqrt{\frac{2}{1 - z}} x, \sqrt{\frac{2}{1 - z}} y\right),
(x, y, z) = \left(\sqrt{1 - \frac{X^2 + Y^2}{4}} X, \sqrt{1 - \frac{X^2 + Y^2}{4}} Y, -1 + \frac{X^2 + Y^2}{2}\right).

In spherical coordinates (φ,θ) on the sphere (with φ the zenith and θ the azimuth) and polar coordinates (R,Θ) on the disk, the map and its inverse are given by

(R, \Theta) = \left(\sqrt{2(1 + \cos \phi)}, \theta\right),
(\phi, \theta) = \left(\arccos\left(-1 + \frac{R^2}{2}\right), \Theta\right).

In cylindrical coordinates (r,θ,z) on the sphere and polar coordinates (R,Θ) on the plane, the map and its inverse are given by

(R, \Theta) = \left(\sqrt{2(1 + z)}, \theta\right),
(r, \theta, z) = \left(R \sqrt{1 - \frac{R^2}{4}}, \Theta, -1 + \frac{R^2}{2}\right).

The projection can be centered at other points, and defined on spheres of radius other than 1, using similar formulas.

[edit] Properties

The Lambert azimuthal projection of the unit sphere defined in the preceding section is undefined at (0, 0, 1). It sends the rest of the sphere to the open disk of radius 2 centered at the origin (0, 0) in the plane. It sends the point (0, 0, -1) to (0, 0), the equator z = 0 to the circle of radius \sqrt 2 centered at (0, 0), and the lower hemisphere z < 0 to the open disk contained in that circle.

The projection is a diffeomorphism (a bijection that is differentiable in both directions) between the sphere (minus (0, 0, 1)) and the open disk of radius 2. It is an area-preserving (equal-area) map, which can be seen by computing the area element of the sphere when parametrized by the inverse of the projection. In Cartesian coordinates it is

dA = dX \; dY.

This means that measuring the area of a region on the sphere is tantamount to measuring the area of the corresponding region the disk.

On the other hand, the projection does not preserve angular relationships among curves on the sphere. No mapping between a hemisphere and a flat disk can preserve both angles and areas. (If one did, then it would be a local isometry and would preserve Gaussian curvature; but the hemisphere and disk have different curvatures, so this is impossible.) This fact, that flat pictures cannot perfectly represent regions of spheres, is the fundamental problem of cartography.

As a consequence, regions on the sphere may be projected to the disk with greatly distorted shapes. This distortion is particularly dramatic far away from the center of the projection (0, 0, -1). In practice the projection is often restricted to the hemisphere centered at that point; the other hemisphere can be mapped separately, using a second projection centered at the antipode.

[edit] Schmidt net

Schmidt net, used for making plots of the Lambert azimuthal projection.
Schmidt net, used for making plots of the Lambert azimuthal projection.

The Lambert azimuthal projection can be carried out by a computer using the explicit formulas given above. However, for graphing by hand these formulas are unwieldy; instead, it is common to use graph paper, called a Schmidt net, designed specifically for the task. To make this graph paper, one places a grid of parallels and meridians on the hemisphere, and then projects these curves to the disk.

In the figure, the area-preserving property of the projection can be seen by comparing a grid sector near the center of the net with one at the far right of the net. The two sectors have the same area on the sphere and the same area on the disk. The angle-distorting property can be seen by examining the grid lines; most of them do not intersect at right angles on the Schmidt net.

Illustration of Steps 1-4 for plotting a point on a Schmidt net.
Illustration of Steps 1-4 for plotting a point on a Schmidt net.

For an example of the use of the Schmidt net, imagine that we have two copies of it on thin paper, one atop the other, aligned and tacked at their mutual center. Suppose that we want to plot the point (0.321, 0.557, -0.766) on the lower unit hemisphere. This point lies on a line oriented 60° counterclockwise from the positive x-axis (or 30° clockwise from the positive y-axis) and 50° below the horizontal plane z = 0. Once these angles are known, there are four steps:

  1. Using the grid lines, which are spaced 10° apart in the figures here, mark the point on the edge of the net that is 60° counterclockwise from the point (1, 0) (or 30° clockwise from the point (0, 1)).
  2. Rotate the top net until this point is aligned with (1, 0) on the bottom net.
  3. Using the grid lines on the bottom net, mark the point that is 50° toward the center from that point.
  4. Rotate the top net oppositely to how it was rotated before, to bring it back into alignment with the bottom net. The point just marked is then the projection that we wanted.

To plot other points, whose angles are not such round numbers as 60° and 50°, one must visually interpolate between the nearest grid lines. It is helpful to have a net with finer spacing than 10°; spacings of 2° are common.

[edit] Applications

The Lambert azimuthal projection was originally conceived as an equal-area map projection. It is now also used in disciplines such as geology to plot directional data, as follows.

A direction in three-dimensional space corresponds to a line through the origin. The set of all such lines is itself a space, called the real projective plane in mathematics. This space is difficult to visualize. However, every line through the origin intersects the unit sphere in exactly two points, one of which is on the lower hemisphere z \leq 0. (Horizontal lines intersect the equator z = 0 in two antipodal points. It is understood that antipodal points on the equator represent a single line. See quotient topology.) Hence the directions in three-dimensional space correspond (almost perfectly) to points on the lower hemisphere. The hemisphere can then be plotted as a disk of radius \sqrt 2 using the Lambert azimuthal projection.

Thus the Lambert azimuthal projection lets us plot directions as points in a disk. Due to the equal-area property of the projection, one can integrate over regions of the real projective plane (the space of directions) by integrating over the corresponding regions on the disk. This is useful for statistical analysis of directional data.

Not only lines but also planes through the origin can be plotted with the Lambert azimuthal projection. A plane intersects the hemisphere in a circular arc, called the trace of the plane, which projects down to a curve (typically non-circular) in the disk. One can plot this curve, or one can alternatively replace the plane with the line perpendicular to it, called the pole, and plot that line instead. When many planes are being plotted together, plotting poles instead of traces produces a less cluttered plot.

Researchers in structural geology use the Lambert azimuthal projection to plot crystallographic axes and faces, lineation and foliation in rocks, slickensides in faults, and other linear and planar features. In this context the projection is called the equal-area hemispherical projection. There is also an equal-angle hemispherical projection defined by stereographic projection.

The discussion here has emphasized the lower hemisphere z \leq 0, but some disciplines prefer the upper hemisphere z \geq 0. Indeed, any hemisphere can be used to record the lines through the origin in three-dimensional space.

[edit] References

Do Carmo, Manfredo P. (1976). Differential geometry of curves and surfaces. Englewood Cliffs, New Jersey: Prentice Hall. ISBN 0-13-212589-7. 

Hobbs, Bruce E., Means, Winthrop D., and Williams, Paul F. (1976). An outline of structural geology. New York: John Wiley & Sons, Inc. ISBN 0471401560. 

Spivak, Michael (1999). A comprehensive introduction to differential geometry. Houston, Texas: Publish or Perish. ISBN 0914098705. 

  1. ^ Mulcahy, Karen. "Lambert Azimuthal Equal Area". City University of New York. Retrieved on 2007-03-30.
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