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Spherical Projections

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay First-time Visitors: Please visit Site Map and Disclaimer. Use "Back" to return here. 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:

A circle of radius r degrees also has a radius of 180-r degrees. Great circles divide the sphere into equal halves. The principal importance of great circles in geological applications of spherical projections is that they can represent planes. The center of a great circle is called its pole. If you know a great circle, you can find its pole, and if you know the pole, you can find the great circle. Thus it is possible to represent a plane by a single point. This fact is extensively used in advanced projection techniques.

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 PDF created with FinePrint pdfFactory trial version http://www.fineprint.com 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.

Projection How Projected Advantages Drawbacks Uses

Orthographic From sphere perpendicular to plane True visual view; all circles plot as ellipses or straight lines. Great distortion near edges Mostly in structural geology for drawing block diagrams

Gnomonic From center of sphere Great circles always plot as straight lines Extreme radial distortion, cannot plot even one hemisphere Mineralogy

Stereographic From point opposite point of tangency All circles on sphere plot as circles on plane Radial distortion Most widely used projection in mineralogy and structural geology

Equal Area Draw arc from point on sphere to plane Area conserved, moderate distortion Curves are complex Structural geology, for statistical analysis of spatial data