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LORIGA CURVE
Curve studied by Loriga in 1910.
Juan Jacobo Duran Loriga (1854  1911): Spanish mathematician. Texeira III p. 52. 
Given a point O and identical punctual light sources placed in the plane, the Loriga curve is the locus of the points of the plane where the illumination due to the n light sources is equivalent to the illumination that would be generated by the n light sources placed at O.
The illumination is inversely proportional to the square of the distance to the source, so it is the curve with punctual equation: .
In the case where the n sources are located on the vertices of a regular polygon with radius a, we get, for n = 2, 3, 4 or 5:
n = 2  Hyperbola 

n = 3  Loriga quartic.
Polar equation:
Complex equation: . The inflection points are on the circle passing by the sources, and the tangents have the remarkable property visible on the figure: Compare to the Klein quartic. 

n = 4  Loriga Sextic.
Polar equation:


n = 5 

Remark: the curve with complex equation and polar equation, coincides with the Loriga curve only when n = 3, as it can be seen on the following figures:


Compare to the isophonic curves and, more generally, see the Goursat curves.
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© Robert FERRÉOL 2017