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CONICAL SPIRAL OF PAPPUS
Curve studied by Pappus, Pascal in 1779,
and Chasles in 1843.
Pappus: mathematician from Alexandria (4th century BC). 
Spherical equation: .
Cylindrical equation: . Cartesian parametrization: . 
The conical spiral of Pappus is the trajectory of a point that moves uniformly along a line passing by a point O, this line turning uniformly around an axis Oz while maintaining an angle a with respect to Oz.
Therefore, it is the intersection between the cone of revolution (C): and the right helicoid: .
If we develop the cone (C) on a plane, the point M becoming the point with polar coordinates , then the Pappus spiral becomes the Archimedean spiral: , in other words, the Pappus spiral is a conical coiling of an Archimedean spiral.
The projection on xOy is also an Archimedean spiral, which coincides with the Pappus spiral with : the conical spiral of Pappus is a conical lift of the Archimedean spiral.
The Pappus spiral is the pedal of the cylindrical helix with respect to a point on its axis, i.e. the locus of the projections of this point on the osculating planes of the helix.
The trace on xOy of its tangent is the Galilean spiral: .
The trace on xOy of the line perpendicular to the curve and included in the tangent plane of the cone is the circle with center O and radius .
It must not be mistaken for the conical helix: the conical spiral of Pappus is to the Archimedean spiral what the conical helix is to the logarithmic spiral! Do not mistake it either for the conical spiral of Pirondini.
The planar projections of the conical spirals of Pappus are the Doppler spirals.
See also the helicoconical surfaces, that are the reunions of conical spirals.
Engraving by wentzel Jamnitzer Perspectiva corporum regularium 

In Vietnam, spirals of incense used to make wishes written on the yellow cardboard that hangs in the middle come true. 
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© Robert FERRÉOL 2018