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CONE OF REVOLUTION
Writing the
colatitude, spherical equation: ,
the axis being Oz and the halfangle at the vertex .
Cylindrical equation: . Cartesian equation: . Cartesian parametrization:
().

Another parametrization: (the coordinate lines are Viviani curves). 

The cone can be developed by mapping a point M of the cone to the point of the plane with polar coordinates ; a halfcone will then become an angular domain with angle .
Remarkable curves traced on the cone of revolution:
 give honor where honor is due: the planar sections
or conics : ellipses, parabolas
and hyperbolas. They can be developped into the polygasteroids
of index n >1.
Cutting by the plane gives an ellipse, a parabola, or a hyperbola, depending on whether is greater, equal, or less than ; if the cone rolls over a plane, it is developped into the polygasteroid with polar equation: . Opposite, the case , with the developped polygasteroid. 

 the curvature lines, which are the parallels (circles) and the meridians (lines) 

 the geodesics, which are the curves that develop into lines; there are the generatrices and the curves with spherical equation: , which project on xOy as epispirals with equation: and develop into the lines . 

 the helices, which also are the rhumb lines, which project on xOy as logarithmic spirals: see conical helix. 

 the conical spirals of Pappus, that project on xOy as Archimedean spirals. 

 the hyperbolic conical spirals. 

 the conical roses, including Viviani's curve. 

 the conical catenaries. 

See also the spherical cycloids, loci of a point on a cone of revolution rolling without slipping over another cone of revolution, as well as the Cartesian ovals, projections of the intersection between two cones of revolution with parallel axes.
Cones in nature:

and elsewhere:




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© Robert FERRÉOL, Jacques MANDONNET 2017