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 right circular conoid , helicoid ,
hyperbolic paraboloid , parabolic
conoid , Plücker's 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 .
Conoidal shells have been used in architectural structures and are also discussed among analytic surfaces with applications in machine-building products (Bock Hyeng
et al. 2025).
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 .
See also Catalan Ruled Surface ,
Right Circular Conoid ,
Right
Conoid ,
Ruled Surface
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References Berger, M. and Gostiaux, B. Géométrie différentielle: variétés, courbes et surfaces. Bock Hyeng, C. A.; Krivoshapko,
S. N.; Kouamou Nguessi, A.; Yamb Bell, E.; and Bahel, B. "Application of
Curvilinear Analytical Surfaces in Forms of Architectural Objects and Machine Building
Products." Int. J. Archit. Arts Appl. 11 , 19-35, 2025. https://doi.org/10.11648/j.ijaaa.20251101.13 . Chisholm,
H. (Ed.). "Conoid." Encyclopædia Britannica, Vol. 06 (11th
Ed.). Cambridge, England: Cambridge University Press, p. 964, 1911. Coolidge,
J. L. A
History of the Conic Sections and Quadric Surfaces. Do Carmo, M. P. Differential
Geometry of Curves and Surfaces, rev. upd. 2nd ed. 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. Gray, A.; Abbena, E.; and Salamon, S. Modern
Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed. Referenced on Wolfram|Alpha Conoid
Cite this as:
Weisstein, Eric W. "Conoid." From MathWorld https://mathworld.wolfram.com/Conoid.html
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