View source for Conoid

{{Short description|Ruled surface made of lines parallel to a plane and intersecting an axis}}
{{For|the organelle called conoid used by [[intracellular parasite]]s|myzocytosis}}
[[File:Conoid-circle.svg|300px|thumb|Right circular conoid: {{legend|#FF8080|Directrix is a circle}} {{legend|blue|Axis is perpendicular to the {{legend inline|yellow|directrix plane}}}}]]

In [[geometry]] a '''conoid''' ({{ety|el|''κωνος'' |cone||-''ειδης'' |similar}}) is a [[ruled surface]], whose rulings (lines) fulfill the additional conditions:
:'''(1)''' All rulings are parallel to a [[Plane (geometry)|plane]], the ''[[Generatrix|directrix]] plane''.
:'''(2)''' All rulings intersect a fixed line, the ''axis''.
 
The conoid is a [[right conoid]] if its axis is [[perpendicular]] to its directrix plane. Hence all rulings are perpendicular to the axis.

Because of '''(1)''' any conoid is a [[Catalan surface]] and can be represented parametrically by 
:<math>\mathbf x(u,v)= \mathbf c(u) + v\mathbf r(u)\ </math> 
Any curve {{math|'''x'''(''u''{{sub|0}},''v'')}} with fixed parameter {{math|1=''u'' = ''u''{{sub|0}}}} is a ruling, {{math|'''c'''(''u'')}} describes the ''directrix'' and the [[Euclidean vector|vector]]s {{math|'''r'''(''u'')}} are  all parallel to the directrix plane. The planarity of the vectors {{math|'''r'''(''u'')}} can be represented by 
:<math>\det(\mathbf r,\mathbf \dot r,\mathbf \ddot r)=0 </math>.
If the directrix is a circle, the conoid is called a '''circular conoid'''.

The term ''conoid'' was already used by [[Archimedes]] in his treatise ''[[On Conoids and Spheroids|On Conoids and Spheroides]]''.

== Examples ==

=== Right circular conoid ===
The parametric representation
:<math> \mathbf x(u,v)=(\cos u,\sin u,0) + v (0,-\sin u,z_0) \ ,\ 0\le u <2\pi, v\in \R</math>
:describes a right circular conoid with the unit circle of the x-y-plane as directrix and a directrix plane, which is parallel to the y--z-plane. Its axis is the line <math>(x,0,z_0) \ x\in \R \ .</math>

''Special features'': 
# The intersection with a horizontal plane is an ellipse.
# <math>(1-x^2)(z-z_0)^2-y^2z_0^2=0</math> is an implicit representation. Hence the right circular conoid is a surface of degree 4. 
# [[Simpson's rule|Kepler's rule]] gives for a right circular conoid  with radius <math>r</math> and height <math>h</math> the exact volume: <math> V=\tfrac{\pi}{2}r^2h</math>.

The implicit representation is fulfilled by the points of the line <math>(x,0,z_0)</math>, too. For these points there exist no [[tangent plane]]s. Such points are called ''singular''.

=== Parabolic conoid ===
[[File:Conoid-parabolic.svg|250px|thumb|parabolic conoid: directrix is a parabola]]
The parametric representation
:<math> \mathbf x(u,v)=\left(1,u,-u^2\right)+ v\left(-1,0,u^2\right)</math>
:::: <math>=\left(1-v,u,-(1-v)u^2\right)\ , u,v \in \R \ ,</math>
describes a ''parabolic conoid'' with the equation <math> z=-xy^2</math>. The conoid has a parabola as directrix, the y-axis as axis and a plane parallel to the x-z-plane as directrix plane. It is used by architects as roof surface (s. below).

The parabolic conoid has no singular points.

=== Further examples ===
#[[hyperbolic paraboloid]] 
#[[Plücker conoid]] 
#[[Whitney umbrella|Whitney Umbrella]]
#[[helicoid]]
<gallery widths="200" heights="200" class="float-right">
Hyp-paraboloid.svg|hyperbolic paraboloid
Pluecker-conoid.svg| Plücker conoid
Whitney-umbrella.svg| Whitney umbrella
</gallery>

== Applications ==
[[File:Pouziti konoidu1.jpg|thumb|conoid in architecture]]
[[File:Pouziti konoidu3.jpg|thumb|conoids in architecture]]

=== Mathematics ===
There are a lot of conoids with singular points, which are investigated in [[algebraic geometry]].

=== Architecture ===
Like other ruled surfaces conoids are of high interest with architects, because they can be built using beams or bars. Right conoids can be manufactured easily: one threads bars onto an axis such that they can be rotated around this axis, only. Afterwards one deflects the bars by a directrix and generates a conoid (s. parabolic conoid).

== External links ==
*[http://mathworld.wolfram.com/PlueckersConoid.html mathworld: Plücker conoid]
*{{springer|title=Conoid|id=p/c025210}}

==References==
* A. Gray, E. Abbena, S. Salamon, ''Modern differential geometry of curves and surfaces with Mathematica'', 3rd ed. Boca Raton, FL:CRC Press, 2006.  [https://www.crcpress.com/product/isbn/9781584884484] ({{ISBN|978-1-58488-448-4}})
* Vladimir Y. Rovenskii, ''Geometry of curves and surfaces with MAPLE'' [https://books.google.com/books?id=K31Nzi_xhoQC&pg=PA277&dq=conoid+maple&lr=&ei=B9hvSs_qKYzSkASR8c3XDg] ({{ISBN|978-0-8176-4074-3}})

[[Category:Surfaces]]
[[Category:Geometric shapes]]