__PROJECTION, EQUAL ANGLE__

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
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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