Editing Conical surface

{{Short description|Surface drawn by a moving line passing through a fixed point}}
[[File:Elliptical Cone Quadric.Png|thumb|An elliptic cone, a special case of a conical surface, shown truncated for simplicity]]

In [[geometry]], a '''conical surface''' is an [[Unbounded set|unbounded]] three-dimensional [[surface (mathematics)|surface]] formed from the union of infinite [[line (mathematics)|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 &mdash; the ''apex'' or ''vertex'' &mdash; and any point of some fixed [[space curve]] &mdash; the ''directrix'' &mdash; 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.<ref>{{citation|title=The Theory of Engineering Drawing|first=Alphonse A.|last=Adler|publisher=D. Van Nostrand|year=1912|contribution=1003. Conical surface|contribution-url=https://archive.org/details/cu31924003943481/page/n185|page=166}}</ref>

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 [[Line (mathematics)#Ray|ray]]s that start at the apex and pass through a point of some fixed space curve.<ref name=msg>{{citation|title=Modern Solid Geometry, Graded Course, Books 6-9|first1=Webster|last1=Wells|first2=Walter Wilson|last2=Hart|publisher=D. C. Heath|year=1927|pages=400–401|url=https://books.google.com/books?id=vXENAQAAIAAJ&pg=PA400}}</ref> Sometimes the term "conical surface" is used to mean just one nappe.<ref>{{citation|title=Solid Geometry|first=George C.|last=Shutts|publisher=Atkinson, Mentzer|year=1913|contribution=640. Conical surface|page=410|contribution-url=https://books.google.com/books?id=9zAAAAAAYAAJ&pg=PA410}}</ref>

==Special cases==
If the directrix is a circle <math>C</math>, and the apex is located on the circle's ''axis'' (the line that contains the center of <math>C</math> and is perpendicular to its plane), one obtains the ''right circular conical surface'' or [[Double cone (geometry)|double cone]].<ref name=msg/> More generally, when the directrix <math>C</math> is an [[ellipse]], or any [[conic section]], and the apex is an arbitrary point not on the plane of <math>C</math>, one obtains an [[elliptic cone]]<ref name=young/> (also called a ''conical quadric'' or ''quadratic cone''),<ref name=osg>{{citation
 | last1 = Odehnal | first1 = Boris
 | last2 = Stachel | first2 = Hellmuth
 | last3 = Glaeser | first3 = Georg | author3-link = Georg Glaeser
 | contribution = Linear algebraic approach to quadrics
 | doi = 10.1007/978-3-662-61053-4_3
 | isbn = 9783662610534
 | pages = 91–118
 | publisher = Springer
 | title = The Universe of Quadrics
 | year = 2020}}</ref> which is a special case of a [[quadric|quadric surface]].<ref name=young>{{citation|title=Analytical Geometry|first=J. R.|last=Young|publisher=J. Souter|year=1838|page=227|url=https://archive.org/details/analyticalgeome00youngoog/page/n243}}</ref><ref name=osg/>

==Equations==
A conical surface <math>S</math> can be described [[Parametrization (geometry)|parametrically]] as
:<math>S(t,u) = v + u q(t)</math>,
where <math>v</math> is the apex and <math>q</math> is the directrix.<ref>{{citation|title=Modern Differential Geometry of Curves and Surfaces with Mathematica|edition=2nd|first=Alfred|last=Gray|publisher=CRC Press|year=1997|isbn=9780849371646|contribution=19.2 Flat ruled surfaces|pages=439–441|contribution-url=https://books.google.com/books?id=-LRumtTimYgC&pg=PA439}}</ref>

==Related surface==
Conical surfaces are [[ruled surface]]s, surfaces that have a straight line through each of their points.<ref>{{citation|title=Encyclopedic Dictionary of Mathematics, Vol. I: A–N|edition=2nd|publisher=MIT Press|editor-first=Kiyosi|editor-last=Ito|author=Mathematical Society of Japan|year=1993|page=419|url=https://books.google.com/books?id=WHjO9K6xEm4C&pg=PA419}}</ref> Patches of conical surfaces that avoid the apex are special cases of [[developable surface]]s, 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 <math>2\pi</math>, then each nappe of the conical surface, including the apex, is a developable surface.<ref>{{citation|title=Elasticity and Geometry: From Hair Curls to the Non-linear Response of Shells|first1=Basile|last1=Audoly|first2=Yves|last2=Pomeau|publisher=Oxford University Press|year=2010|isbn=9780198506256|pages=326–327|url=https://books.google.com/books?id=FMQRDAAAQBAJ&pg=PA326}}</ref>

A [[cylindrical surface]] can be viewed as a [[limiting case (mathematics)|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.<ref>{{citation|title=Descriptive Geometry|first1=F. E.|last1=Giesecke|first2=A.|last2=Mitchell|publisher=Von Boeckmann–Jones Company|year=1916|page=66|url=https://books.google.com/books?id=sCc7AQAAMAAJ&pg=PA66}}</ref>

==See also==
*[[Conic section]]
*[[Quadric]]

==References==
{{reflist}}

{{DEFAULTSORT:Conical Surface}}
[[Category:Euclidean solid geometry]]
[[Category:Surfaces]]