User:Joshuardavis/Hemispherical projection

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WARNING: This page is obsolete. Its content has been merged into Stereographic projection and Lambert azimuthal equal-area projection.

In geometry, a hemispherical projection is a mapping from a hemisphere to a disk, used to make a flat picture of the hemisphere. Two main types are used: the equal-angle projection, which is stereographic projection, and the equal-area projection. These projections can be performed by computer, or by hand using special graph paper called a Wulff net or stereonet (for the equal-angle projection) or a Schmidt net (for the equal-area projection).

Hemispherical projections are used in structural geology, crystallography, and other disciplines to plot orientations of lines in three-dimensional space. They are also closely related to map projections used in cartography.

Contents

[edit] Motivation

As is described in later sections, various scientific disciplines require a standardized, practical way of visualizing directions in three-dimensional space \mathbb{R}^3. Directions can be thought of as lines through the origin (0, 0, 0). The space of all such lines is well-studied in mathematics as the real projective plane, but this space is difficult to visualize. The approach of this article is to first approximate the space by a hemisphere, and then to project that hemisphere onto a disk, which can be drawn easily on paper.

The unit sphere in three-dimensional space is the set of points (x, y, z) such that x2 + y2 + z2 = 1. The lower hemisphere is the set of such points with z \leq 0. Each non-horizontal line through the origin intersects unit sphere in two points, one of which is on the lower hemisphere. So these lines can be recorded as points on the lower hemisphere. A horizontal line intersects the lower hemisphere twice, at antipodal points on the equator z = 0; either of these points can be used to record the line. So the points on the lower hemisphere are in one-to-one correspondence with lines through the origin, except along the equator, where they correspond two-to-one.

Now that lines are pictured as points on a hemisphere, the remaining task is to map the hemisphere, as accurately as possible, to a flat disk. There are two primary characteristics that would ideally be preserved in such a mapping: angular relationships between curves on the hemisphere, and areas of regions on the hemisphere. It turns out that no mapping can preserve both. (If it 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.

Perhaps the most obvious map is the orthogonal projection that sends the point (x, y, z) on the lower unit hemisphere to the point (x, y) on the unit disk. However, this map preserves neither angles nor areas, so it is not used. Instead, an equal-angle projection or an equal-area projection is used, depending on which property is more valuable in the situation at hand. These projections are developed in the following sections.

The discussion thus far 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. For brevity this article remains focused on the lower hemisphere.

[edit] Equal-angle projection

Main article: Stereographic projection
Stereographic projection of the lower unit hemisphere from (0, 0, 1) onto the plane z = 0, shown here in cross section.
Stereographic projection of the lower unit hemisphere from (0, 0, 1) onto the plane z = 0, shown here in cross section.

Let H be the lower hemisphere. For any point P on H, there is a unique line through the north pole (0, 0, 1) and P, and this line intersects the plane z = 0 in exactly one point P'. The stereographic projection of P is defined to be this point P' in the plane.

The stereographic projection is a diffeomorphism (a bijection that is differentiable in both directions) between the lower unit hemisphere and the unit disk in the plane. It sends meridians on the sphere to rays emanating from the origin on the plane, and preserves angular relationships everywhere on the hemisphere. In fact, it is the only map with these properties; it is the desired equal-angle projection.

If (X, Y) are Cartesian coordinates on the disk and we parametrize the hemisphere by the inverse of stereographic projection, then the area element of the hemisphere turns out to be

dA = \frac{4}{(1 + X^2 + Y^2)^2} \; dX \; dY.

Near the edge of the disk, where X2 + Y2 approaches 1, dA approaches dX dY. This means that areas are distorted very little near the edge of the disk. In contrast, near the center of the disk, X2 + Y2 approaches 0, so dA approaches 4 dX dY, and areas are distorted by a factor of 4.

[edit] Equal-area projection

The equal-area projection of the lower unit hemisphere is defined by any one of the following formulas. In Cartesian coordinates (x,y,z) on the sphere and (X,Y) on the plane, the projection and its inverse are

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

Unlike the stereographic projection, the equal-area projection cannot be described in terms of drawing lines through points and finding where they intersect a plane; it is not a "projection" in the strictest sense of the word. Nonetheless, it is a well-defined diffeomorphism between the lower hemisphere and the disk of radius \sqrt 2.

Under the inverse of equal-area projection, the area element of the hemisphere is everywhere

dA = dX \; dY.

Thus the area of any region on the hemisphere and the area of the corresponding region on the disk are equal. In fact, this projection is the only area-preserving diffeomorphism that sends meridians to rays emanating from the origin and preserves the angles between those meridians at the origin. However, this projection does not preserve angles in general. In other words, while regions on the hemisphere are mapped to the disk with the correct area, their shapes may be greatly distorted.

For a unit hemisphere to map onto a disk in an area-preserving manner, the disk must have radius \sqrt 2. (For then it has area 2π, which is the area of the unit hemisphere.) Sometimes it is preferred to send the hemisphere to a unit disk. Formulas for that projection can be obtained by replacing X, Y, and R by \sqrt{2} X, \sqrt{2} Y, and \sqrt{2} R in the formulas above. The resulting map is not truly area-preserving, but it simply distorts the area by a constant factor of 2:

dA = 2 \; dX \;dY.

[edit] Wulff and Schmidt nets

Wulff net or stereonet, used for making plots of the equal-angle (stereographic) projection.
Wulff net or stereonet, used for making plots of the equal-angle (stereographic) projection.

Equal-angle and equal-area plots 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 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 using the desired projection. The result is called a Wulff net or stereonet (for the equal-angle projection) or a Schmidt net (for the equal-area projection).

In the Wulff net figure, the area-distorting property of the equal-angle 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 exactly the same area on the sphere. On the disk, the latter has nearly four times the area as the former; if one uses finer and finer grids on the sphere, then the ratio of the areas approaches exactly 4.

The angle-preserving property can be seen by examining the grid lines. Parallels and meridians intersect at right angles on the sphere, and so do their images on the Wulff net.

Schmidt net, used for making plots of the equal-area projection.
Schmidt net, used for making plots of the equal-area projection.

In the Schmidt net figure, the area-preserving property of the equal-area 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 either 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

Similar to Lambert azimuthal equal-area projection, Hammer projection, Aitoff projection?

!!equal-area respects area, hence integration, hence statistics

!!The orientation of a plane (through the origin) in three-dimensional space may be recorded as the orientation of the line (through the origin) that is perpendicular to that plane. Hence planes can also be plotted by stereographic projection. In crystallography, a stereographic plot of crystal lattice planes is called a pole figure.

[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.