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STEREOGRAPHIC PROJECTION OF A SPHERICAL CURVE


Stereo, prefix coming from the Greek stereos "solid, hard".

 
If the curve (G) is given by its spherical equation: , then the polar equation of the stereographic projection from the South pole (i.e. with pole the point ) is .

The stereographic projection of a curve (G), traced on a sphere (S) with centre O, from the pole S, a point on (S), is the locus of the intersection points between the line (SM) and a plane perpendicular to (OS); if the projection plane is modified, then the stereographic projection is transformed into its homothetic image.

Examples:
    - the nodal curves are the stereographic projections of the clelias (and, in particular, the right strophoid is the stereographic projection of the Viviani curve)
    - the logarithmic spiral is the stereographic projection of the rhumb line of the sphere.
 
 
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© Robert FERRÉOL 2017