page_title: Stereographic projection
section_title: Wulff net
hierarchical_section_title: Stereographic projection / Wulff net
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: Stereographic projection plots can be carried out by a computer using the explicit formulas given above. However, for graphing by hand these formulas are unwieldy. Instead, it is common to use graph paper designed specifically for the task. This special graph paper is called a stereonet or Wulff net, after the Russian mineralogist George (Yuri Viktorovich) Wulff.
  The Wulff net shown here is the stereographic projection of the grid of parallels and meridians of a hemisphere centred at a point on the equator (such as the Eastern or Western hemisphere of a planet).
  In the figure, the area-distorting property of the stereographic projection can be seen by comparing a grid sector near the center of the net with one at the far right or left. The two sectors have equal areas on the sphere. On the disk, the latter has nearly four times the area of the former. If the grid is made finer, this ratio approaches exactly 4.
  On the Wulff net, the images of the parallels and meridians intersect at right angles. This orthogonality property is a consequence of the angle-preserving property of the stereoscopic projection. (However, the angle-preserving property is stronger than this property. Not all projections that preserve the orthogonality of parallels and meridians are angle-preserving.)
  For an example of the use of the Wulff net, imagine two copies of it on thin paper, one atop the other, aligned and tacked at their mutual center. Let P be the point on the lower unit hemisphere whose spherical coordinates are (140°, 60°) and whose Cartesian coordinates are (0.321, 0.557, −0.766). This point lies on a line oriented 60° counterclockwise from the positive x-axis (or 30° clockwise from the positive y-axis) and 50° below the horizontal plane z = 0. Once these angles are known, there are four steps to plotting P:
  Using the grid lines, which are spaced 10° apart in the figures here, mark the point on the edge of the net that is 60° counterclockwise from the point (1, 0) (or 30° clockwise from the point (0, 1)).
  Rotate the top net until this point is aligned with (1, 0) on the bottom net.
  Using the grid lines on the bottom net, mark the point that is 50° toward the center from that point.
  Rotate the top net oppositely to how it was oriented before, to bring it back into alignment with the bottom net. The point marked in step 3 is then the projection that we wanted.
  To plot other points, whose angles are not such round numbers as 60° and 50°, one must visually interpolate between the nearest grid lines. It is helpful to have a net with finer spacing than 10°. Spacings of 2° are common.
  To find the central angle between two points on the sphere based on their stereographic plot, overlay the plot on a Wulff net and rotate the plot about the center until the two points lie on or near a meridian. Then measure the angle between them by counting grid lines along that meridian.
caption: The generation of a Wulff net (circular net within the red circle) by a stereographic projection with center C and projection plane 
    
      
        
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