conical surface

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.{{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}}

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.{{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}} Sometimes the term "conical surface" is used to mean just one nappe.{{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}}

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. 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 (also called a conical quadric or quadratic cone),{{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}} which is a special case of a quadric surface.{{citation|title=Analytical Geometry|first=J. R.|last=Young|publisher=J. Souter|year=1838|page=227|url=https://archive.org/details/analyticalgeome00youngoog/page/n243}}

A conical surface $S$ can be described parametrically as

:$S(t,u)\; =\; v\; +\; u\; q(t)$,

where $v$ is the apex and $q$ is the directrix.{{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}}

Conical surfaces are ruled surfaces, surfaces that have a straight line through each of their points.{{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}} 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 $2\backslash pi$, then each nappe of the conical surface, including the apex, is a developable surface.{{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}}

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.{{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}}

• Conic section
• Quadric

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Category:Euclidean solid geometry

Category:Surfaces