Stereographic projection
Definition
A stereographic projection is a specific type of conformal (shape-preserving) azimuthal map projection created by projecting points from the surface of a sphere onto a plane from a single perspective point located on the sphere's surface opposite (antipodal to) the point of tangency, with the critical property that all circles on the sphere�both great and small circles�are represented as circles or straight lines on the map (where straight lines are considered circles of infinite radius), preserving local angles and shapes at infinitesimal scale everywhere while severely distorting areas, making it particularly valuable for mapping polar regions and for applications in complex analysis, crystallography, and navigation where the preservation of angular relationships is more critical than the preservation of area.
Application
A fundamental practical application of the stereographic projection is in structural geology and crystallography for analyzing three-dimensional angular relationships on a two-dimensional plane. Geologists use a stereographic net (or Wulff net) to plot the orientation of planar features like bedding, foliation, and fault planes, as well as linear features like fold axes and mineral lineations. Because the projection is conformal, it preserves the true angles between these lines and planes, allowing geologists to perform accurate graphical analyses�such as determining the axis of folding, calculating true dip from apparent dips, or identifying dominant joint sets�directly from field measurements, which is essential for understanding tectonic stress fields, rock mechanics, and mineral exploration.
FAQ
What is the most unique geometric property of the stereographic projection?
Its defining property is that it maps all circles on the sphere (both great and small circles) to circles or straight lines on the plane. A straight line is considered a circle of infinite radius. This elegant geometric transformation, combined with its conformal nature (preserving angles), makes it mathematically powerful and visually useful for representing spherical information in a planar format.
If it distorts areas so much, why is it valuable?
Its value lies in its conformality�it preserves local angles and shapes at every point, which is often more critical than preserving area for specific technical applications. This makes it the perfect tool for fields where analyzing directional data, angular relationships, and orientations is the primary goal, such as mapping complex crystalline structures or geological features, where the precise angle between planes is the key piece of information.
Where is it most commonly used outside of traditional geography?
Its most prominent non-cartographic uses are in specialized scientific and technical fields:
Structural Geology & Crystallography: For plotting and analyzing the 3D orientation of rock layers, faults, and crystal faces using a stereographic net (Wulff net).
Complex Analysis: In mathematics, it provides a one-to-one mapping between the complex plane and the Riemann sphere.
Astronomy: For creating star charts and planning telescope observations, as it can represent large areas of the celestial sphere with manageable shape distortion.
How does it compare to other azimuthal projections for mapping the poles?
For polar mapping, the Stereographic is often preferred over other azimuthal types when shape preservation is paramount. While the Gnomonic shows great circles as straight lines (for navigation) and the Orthographic gives a perspective view, their shape distortion is more severe. The Stereographic projection offers a balanced, conformal polar map where shapes of landmasses like Antarctica or the Arctic archipelago are recognizable and angles are true, making it the standard choice for official polar maps from organizations like the United States Geological Survey (USGS).
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