Conical surface
In geometry, a conical surface is an unbounded surface in three-dimensional space formed from the union of infinite lines that pass through a fixed point and a space curve.
Definitions
[edit ]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
[edit ]If the directrix is a circle {\displaystyle C}, and the apex is located on the circle's axis (the line that contains the center of {\displaystyle C} and is perpendicular to its plane), one obtains the right circular conical surface or double cone.[2] More generally, when the directrix {\displaystyle C} is an ellipse, or any conic section, and the apex is an arbitrary point not on the plane of {\displaystyle C}, one obtains an elliptic cone.[4]
Equations
[edit ]A conical surface {\displaystyle S} can be described parametrically as
- {\displaystyle S(t,u)=v+uq(t)},
where {\displaystyle v} is the apex and {\displaystyle q} is the directrix.[5]
Related surface
[edit ]Conical surfaces are ruled surfaces, surfaces that have a straight line through each of their points.[6] 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 {\displaystyle 2\pi }, then each nappe of the conical surface, including the apex, is a developable surface.[7]
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.[8]
See also
[edit ]References
[edit ]- ^ 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
- ^ Young, J. R. (1838), Analytical Geometry, J. Souter, p. 227
- ^ 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