Gnomonic projection

 

See also: Stereographic projection, Inverse pole figure, equal area projection

 

Consider a sphere with centre O and a point P on its surface (see the figure below).  The projection plane is the plane tangential to the sphere at its north pole N.  The line OP intersects the projection plane at p, and this point is the gnomonic projection of P.   The line LM is the gnomonic projection of the plane with normal OP.  

 

For EBSD, O is the point on the sample where the diffracted electrons are generated, the projection plane is the phosphor screen and  N is the pattern centre.   Electrons are diffracted into a region defined by a pair of cones with axes normal to the diffracting plane and semi angle (90-θ), where θ is the Bragg angle.  The gnomonic projection of these cones are the edges of the Kikuchi bands and the  gnomonic projection of the crystal planes are the centre lines of the Kikuchi bands.  The edges of the bands project as hyperbolae onto the phosphor screen.  However, because the cone semi-angle is close to 90 these appear as straight lines on the diffraction pattern, although further from the pattern centre the hyperbolic shape can be seen.  Kikuchi bands further from the pattern centre appear stretched compared with those close to the pattern centre.