{{Short description|Ruled surface made of lines parallel to a plane and intersecting an axis}} {{For|the organelle called conoid used by intracellular parasites|myzocytosis}} 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, the ''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 vectors {{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 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. # 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 planes. Such points are called ''singular''.
=== Parabolic conoid === 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 #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 == thumb|conoid in architecture 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 == *[https://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