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STEREOGRAPHIC PROJECTION OF A SPHERICAL CURVE

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

If the curve 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 ,
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