Spherical Projections

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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Note: Some texts regard the projection plane for some spherical projections as passing through the center of the sphere rather than tangent to it as this page does. Formulas for projection coordinates differ by a factor of 2 between the two approaches. If your results differ from those here by a factor of 2, this is probably why.

Great and Small Circles

In any discussion of spherical projections, it is essential to understand these terms:
Great Circles have a radius of 90 degrees measured along the circumference of the sphere. The equator of the Earth and meridians of longitude are great circles. A plane passing through the center of the sphere cuts the sphere in a great circle. A great circle on the diagram is shown in blue.
Small Circles have a radius not equal to 90 degrees. Parallels of latitude are small circles. A plane not passing through the center of the sphere cuts the sphere in a small circle. A small circle on the diagram is shown in yellow (green where it overlaps the great circle).

Additional points:

The Main Types of Spherical Projections

Although every map projection projects a sphere onto a plane, in geology (mostly in mineralogy and structural geology) we make use of four main projections as shown below. All of them are azimuthal; that is, we project a sphere onto a plane tangent to the sphere. Directions on the projection are the same as directions on the sphere relative to the point of tangency. Also distances on the projection do not depend on direction; circles on the sphere centered on the point of tangency project as circles on the plane.

ProjectionHow Projected AdvantagesDrawbacksUses
OrthographicFrom sphere perpendicular to plane True visual view; all circles plot as ellipses or straight lines. Great distortion near edgesMostly in structural geology for drawing block diagrams
GnomonicFrom center of sphereGreat circles always plot as straight linesExtreme radial distortion, cannot plot even one hemisphereMineralogy
StereographicFrom point opposite point of tangencyAll circles on sphere plot as circles on planeRadial distortionMost widely used projection in mineralogy and structural geology
Equal Area Draw arc from point on sphere to plane Area conserved, moderate distortionCurves are complexStructural geology, for statistical analysis of spatial data

Some Definitions and Terms

Point of Tangency
The point of contact between the projection plane and the sphere. It is the center of the projection.
Near Hemisphere
The hemisphere closest to the projection plane (within 90 degrees of the point of tangency).
Far Hemisphere
The hemisphere opposite the projection plane (more than 90 degrees from the point of tangency).
Primitive Circle
The projected location of the great circle separating the near and far hemispheres (90 degrees from the point of tangency).
Directions and Locations
Since the most commonly used projections show the projection sphere on its side like a globe, it makes sense to refer to the projection in geographic terms:
  • The central meridian of the projection is the Prime Meridian.
  • The great circle midway between the poles is the Equator.
  • North-South means along the prime meridian, East-West means along the equator.

Orthographic Projection

Provides a true visual view of the near hemisphere but the far hemisphere is not visible. Foreshortening near the primitive circle is extreme. Great and small circles plot as ellipses or straight lines. No important properties conserved.

Once this projection saw occasional use in converting angles when drawing block diagrams, but calculators and drafting software have largely superseded this use.

Gnomonic Projection

All great circles plot as straight lines, a useful property for some aspects of crystallography. Small circles plot as other conic sections. Less than one hemisphere can be projected and radial distortion is extreme. The primitive circle plots at infinity. No important properties conserved. 

Useful in mineralogy because certain types of plot cause families of crystal faces to lie on straight lines

Stereographic Projection

All circles on the sphere plot as circles on the plane, making it easy to construct the projection. The projection is conformal, meaning that angles and small shapes on the sphere project true on the plane. Since no spherical projection allows both area and shape to be conserved, something has to give. The price we pay for conformality is areal distortion. Small regions on the sphere project true on the plane, making the stereographic a good map projection for small areas, but radial distortion increases away from the tangency point.

In principle, the entire sphere can be projected except for the projection point. In reality the areal distortion in the far hemisphere (in red) is too severe for most practical purposes (although extended stereonets have been constructed and used.) In most geological applications we use only the near hemisphere (where areal distortion is not too severe) and deal with points in the far hemisphere in one of two ways:

By the way, the stereographic is by no means the only conformal projection. The familiar wall map, the Mercator Projection, is also conformal, and there are conic and special purpose conformal projections as well. The strict definition of a conformal projection is that angles are conserved.

Equal-Area Projection

As the name implies, this projection conserves area. Shapes are not preserved but shape distortion is not too bad in the near hemisphere. In principle the entire sphere can be plotted but in practice shape distortion beyond 90 degrees (shown in red) becomes severe and beyond about 130 degrees is so extreme as to be unusable. Meridians and parallels are complex curves. Because area is conserved, this is the projection of choice for statistical comparison of spatial data.

Coordinate Data for Projections

We will use the following assumptions and notations.

The coordinates of a point at (l, w) are:

Because the sphere is a unit sphere, x2 + y2 + z2 = 1. It is also useful to note that x2 + y2 = 1 - z2. The table below lists X and Y in terms of x, y, and z. Also listed is R, the radial distance of the projected point from the center of the projection.

Because all the projections are azimuthal, the azimuth of the projected point from (X = 0, Y = 0) is always the same as the azimuth on the sphere from (l = 0, w = 0). The azimuth is most simply represented as arctan (y/x), but x/y is infinite if x = 0 and arctan A = arctan(180 + A). We can remove the ambiguity by noting that tan A/2 = sin A/(1 + cos A). Define r = sqrt(x2 + y2), the radial distance of the point on the sphere from the z axis. Then sin A = y/r and cos A = x/r, thus tan A/2 = (y/r)/(1+x/r), or tan A/2 = y/(r+x). Hence 

A = 2arctan(y/(r+x))

The only problem is that if x = -1 the formula goes to infinity, but if x = -1 then A = 180.

Coordinate Data for Spherical Projections In Terms of the Projected Sphere
ProjectionR (Sphere = 1)XY
Orthographicsqrt(x2 + y2) = sqrt(1 - z2)xy
Gnomonic sqrt(x2 + y2)/z = sqrt(1 - z2)/zx/zy/z
Equal-Areasqrt(x2 + y2 + (1 - z)2) =
sqrt(2(1 - z))
R cos A = Rx/r
= x sqrt (2/(1 + z))
R cos A = Ry/r
= y sqrt (2/(1 + z))
Stereographic 2sqrt(x2 + y2)/(1 + z) =
2sqrt((1 - z)/(1 + z))

For actual plotting of the projection, the following data might be more useful:

Coordinate Data for Spherical Projections In Terms of the Primitive Circle
ProjectionR (Primitive Circle = 1)XY
Orthographicsqrt(x2 + y2) = sqrt(1 - z2)xy
Equal-Areasqrt((x2 + y2 + (1 - z)2)/2) =
sqrt(1 - z)
R cos A = Rx/r
= x sqrt (1/(1 + z))
R cos A = Ry/r
= y sqrt (1/(1 + z))
Stereographic sqrt(x2 + y2)/(1 + z) =
sqrt((1 - z)/(1 + z))

Plotting Nets on a Computer

If you don't have software for drawing a net directly, you can still construct one using common software as follows. You will need a spreadsheet with graphing capability and a drawing program that allows you to scale, copy and flip drawings.

This method works with Excel and Windows Paint; that's how the nets on this page were created.

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Created 28 December 1998, Last Update 16 March 1999
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