Conical surface

Ingeometry,aconical surfaceis a three-dimensionalsurfaceformed from the union oflinesthat pass through a fixed point and aspace 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 — theapexorvertex— and any point of some fixedspace curve— thedirectrix— that does not contain the apex. Each of those lines is called ageneratrixof the surface. The directrix is often taken as aplane 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 anappe,and is the union of all theraysthat 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'saxis(the line that contains the center of $C$ and is perpendicular to its plane), one obtains theright circular conical surfaceordouble cone.^[2]More generally, when the directrix $C$ is anellipse,or anyconic section,and the apex is an arbitrary point not on the plane of $C$ ,one obtains anelliptic cone^[4](also called aconical quadricorquadratic cone),^[5]which is a special case of aquadric surface.^[4]^[5]

Equations

A conical surface $S$ can be describedparametricallyas

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

,

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

Related surface

Conical surfaces areruled surfaces,surfaces that have a straight line through each of their points.^[7]Patches of conical surfaces that avoid the apex are special cases ofdevelopable 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]

Acylindrical surfacecan be viewed as alimiting caseof a conical surface whose apex is moved off to infinity in a particular direction. Indeed, inprojective geometrya 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 ^bWells, 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 ^bYoung, J. R. (1838),Analytical Geometry,J. Souter, p. 227
^^a ^bOdehnal, 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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