Conical surface

In geometry, a conical surface is a three-dimensional surface formed from the union of lines that pass through a fixed point and a space curve.

Definitions

A (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the apex or vertex — and any point of some fixed space curve — the directrix — that does not contain the apex. Each of those lines is called a generatrix of the surface. The directrix is often taken as a plane curve, in a plane not containing the apex, but this is not a requirement.^[1]

In general, a conical surface consists of two congruent unbounded halves joined by the apex. Each half is called a nappe, and is the union of all the rays that start at the apex and pass through a point of some fixed space curve.^[2] Sometimes the term "conical surface" is used to mean just one nappe.^[3]

Special cases

If the directrix is a circle $C$ , and the apex is located on the circle's axis (the line that contains the center of $C$ and is perpendicular to its plane), one obtains the right circular conical surface or double cone.^[2] More generally, when the directrix $C$ is an ellipse, or any conic section, and the apex is an arbitrary point not on the plane of $C$ , one obtains an elliptic cone^[4] (also called a conical quadric or quadratic cone),^[5] which is a special case of a quadric surface.^[4]^[5]

Equations

A conical surface $S$ can be described parametrically as

S(t,u)=v+uq(t)

,

where $v$ is the apex and $q$ is the directrix.^[6]

Related surface

Conical surfaces are ruled surfaces, surfaces that have a straight line through each of their points.^[7] Patches of conical surfaces that avoid the apex are special cases of developable surfaces, surfaces that can be unfolded to a flat plane without stretching. When the directrix has the property that the angle it subtends from the apex is exactly $2\pi$ , then each nappe of the conical surface, including the apex, is a developable surface.^[8]

A cylindrical surface can be viewed as a limiting case of a conical surface whose apex is moved off to infinity in a particular direction. Indeed, in projective geometry a cylindrical surface is just a special case of a conical surface.^[9]

References

^ Adler, Alphonse A. (1912), "1003. Conical surface", The Theory of Engineering Drawing, D. Van Nostrand, p. 166
^ ^a ^b Wells, Webster; Hart, Walter Wilson (1927), Modern Solid Geometry, Graded Course, Books 6-9, D. C. Heath, pp. 400–401
^ Shutts, George C. (1913), "640. Conical surface", Solid Geometry, Atkinson, Mentzer, p. 410
^ ^a ^b Young, J. R. (1838), Analytical Geometry, J. Souter, p. 227
^ ^a ^b Odehnal, Boris; Stachel, Hellmuth; Glaeser, Georg (2020), "Linear algebraic approach to quadrics", The Universe of Quadrics, Springer, pp. 91–118, doi:10.1007/978-3-662-61053-4_3, ISBN 9783662610534
^ Gray, Alfred (1997), "19.2 Flat ruled surfaces", Modern Differential Geometry of Curves and Surfaces with Mathematica (2nd ed.), CRC Press, pp. 439–441, ISBN 9780849371646
^ Mathematical Society of Japan (1993), Ito, Kiyosi (ed.), Encyclopedic Dictionary of Mathematics, Vol. I: A–N (2nd ed.), MIT Press, p. 419
^ Audoly, Basile; Pomeau, Yves (2010), Elasticity and Geometry: From Hair Curls to the Non-linear Response of Shells, Oxford University Press, pp. 326–327, ISBN 9780198506256
^ Giesecke, F. E.; Mitchell, A. (1916), Descriptive Geometry, Von Boeckmann–Jones Company, p. 66

[1] Adler, Alphonse A. (1912), "1003. Conical surface", The Theory of Engineering Drawing, D. Van Nostrand, p. 166

[msg-2] Wells, Webster; Hart, Walter Wilson (1927), Modern Solid Geometry, Graded Course, Books 6-9, D. C. Heath, pp. 400–401

[3] Shutts, George C. (1913), "640. Conical surface", Solid Geometry, Atkinson, Mentzer, p. 410

[young-4] Young, J. R. (1838), Analytical Geometry, J. Souter, p. 227

[osg-5] Odehnal, Boris; Stachel, Hellmuth; Glaeser, Georg (2020), "Linear algebraic approach to quadrics", The Universe of Quadrics, Springer, pp. 91–118, doi:10.1007/978-3-662-61053-4_3, ISBN 9783662610534

[6] Gray, Alfred (1997), "19.2 Flat ruled surfaces", Modern Differential Geometry of Curves and Surfaces with Mathematica (2nd ed.), CRC Press, pp. 439–441, ISBN 9780849371646

[7] Mathematical Society of Japan (1993), Ito, Kiyosi (ed.), Encyclopedic Dictionary of Mathematics, Vol. I: A–N (2nd ed.), MIT Press, p. 419

[8] Audoly, Basile; Pomeau, Yves (2010), Elasticity and Geometry: From Hair Curls to the Non-linear Response of Shells, Oxford University Press, pp. 326–327, ISBN 9780198506256

[9] Giesecke, F. E.; Mitchell, A. (1916), Descriptive Geometry, Von Boeckmann–Jones Company, p. 66

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