page_title: Stereographic projection
section_title: Crystallography
hierarchical_section_title: Stereographic projection / Applications to other disciplines / Crystallography
context_page_description: In geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles at which curves meet. It is neither isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures.
  Intuitively, then, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. Because the sphere and the plane appear in many areas of mathematics and its applications, so does the stereographic projection; it finds use in diverse fields including complex analysis, cartography, geology, and photography. In practice, the projection is carried out by computer or by hand using a special kind of graph paper called a stereographic net, shortened to stereonet, or Wulff net.
context_section_description: In crystallography, the orientations of crystal axes and faces in three-dimensional space are a central geometric concern, for example in the interpretation of X-ray and electron diffraction patterns. These orientations can be visualized as in the section Visualization of lines and planes above. That is, crystal axes and poles to crystal planes are intersected with the northern hemisphere and then plotted using stereographic projection. A plot of poles is called a pole figure.
  In electron diffraction, Kikuchi line pairs appear as bands decorating the intersection between lattice plane traces and the Ewald sphere thus providing experimental access to a crystal's stereographic projection.  Model Kikuchi maps in reciprocal space, and fringe visibility maps for use with bend contours in direct space, thus act as road maps for exploring orientation space with crystals in the transmission electron microscope.
caption: A crystallographic pole figure for the diamond lattice in [111] direction
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