Stereographic Projection

This is an outline of the basic construction for stereographic projection and a statement of some basic properties.

The central projection of 3-space from a point N to a plane onto a plane F (not through N) is defined as follows:

  1. For any point P of 3-space, construct the line NE.
  2. Intersect the line NE with the plane F.  The point of intersection c(P) is the central projection of P to the plane with center N.
  3. Note:  This projection is not defined if point P is on the plane through N parallel to F.

The stereographic projection of a sphere E onto a plane F from a center N is defined as follows.

  1. Point N must be on the sphere E and the plane must be perpendicular to the diameter line of the sphere through N.
  2. For any point P of the sphere E distinct from N, construct the line NP.
  3. Intersect the line NP with the plane.  The point of intersection s(P) is the stereographic projection of P on the plane with center N.
  4. Note:  This is the same as central projection of P but with domain restricted to the sphere.  From the way the center and the plane were chosen, the line NP is never parallel to the plane for a P distinct from N, so the stereographic projection maps the sphere – {N} to the plane.
  5. Note:  Unlike the case of central projection from 3-space, there is an inverse map from the plane to the sphere.  For a point Q on the plane, construct line NQ.  This line intersects the sphere at N and also another distinct point P (since the line is not tangent to E).  This point P is the inverse stereographic projection of Q.

The figure below shows the stereographic projection from P to Q.  The plane F is through the center of the sphere (think N = north pole, plane = plane of equator).  It also shows the projection of the antipodal point P* of P to Q*.

Properties of Stereographic Projection

Additional information about Stereographic Projection from a point N on a sphere is given in the web references.

The properties of Stereographic Projection that we will use are these  (we have indicated a proof of (a), the other two will be proved in class).

(a)    Stereographic projection is a 1-1 map of the sphere onto the plane, except that projection image of the projection center N is not defined (or may be set = infinity).

(b)   Circles on the sphere not through N map to circles on the plane.  Circles through N map to lines on the plane.

(c)    Angles are preserved by stereographic projection.