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# Conoid

A conoid is a ruled surface whose rulings are parallel to a plane (called the directrix plane) and intersect a fixed line (called the axis of the conoid) (Gellert et al. 1989, p. 202). Examples include the circular conoid, helicoid, hyperbolic paraboloid, parabolic conoid, Plücker conoid, right circular conoid, Wallis's conical edge, Whitney umbrella, and Zindler conoid. If the axis is perpendicular to the directrix plane, the conoid is called a right conoid (Gray et al. 2006, p. 436).

A conoid is a Catalan ruled surface.

A different definition was used by Archimedes in his treatise On Conoids and Spheroids, where he considered a conoid to be a solid (or surface) formed by the revolution of a conic section about one of its principal axes (Chisholm 1911, p. 964), i.e., a paraboloid, hyperboloid, or spheroid.

Catalan Ruled Surface, Right Conoid, Ruled Surface

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## References

Berger, M. and Gostiaux, B. Géométrie diffŽrentielle: variétés, courbes et surfaces. Paris: Presses Univ. France, 1987.Chisholm, H. (Ed.). "Conoid." Encyclopædia Britannica, Vol. 06 (11th Ed.). Cambridge, England: Cambridge University Press, p. 964, 1911.Coolidge, J. L. (1945). A history of the conic sections and quadric surfaces. Dover Public. Do Carmo, M. P. Differential Geometry of Curves and Surfaces, rev. upd. 2nd ed. New York: Dover, 2016.Ferréol, R. "Conoid." https://mathcurve.com/surfaces.gb/conoid/conoid.shtml.Gellert, W.; Gottwald, S.; Hellwich, M.; Kästner, H.; and Künstner, H. (Eds.). VNR Concise Encyclopedia of Mathematics, 2nd ed. New York: Van Nostrand Reinhold, 1989.Gray, A.; Abbena, E.; and Salamon, S. Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed. Boca Raton, FL: CRC Press, 2006.

## Cite this as:

Weisstein, Eric W. "Conoid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Conoid.html