Conoid

{{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}}

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

: $\mathbf x(u,v)= \mathbf c(u) + v\mathbf r(u)\$

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

: $\det(\mathbf r,\mathbf \dot r,\mathbf \ddot r)=0$ .

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

: $\mathbf x(u,v)=(\cos u,\sin u,0) + v (0,-\sin u,z_0) \ ,\ 0\le u <2\pi, v\in \R$

: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 $(x,0,z_0) \ x\in \R \ .$

Special features:

The intersection with a horizontal plane is an ellipse.
$(1-x^2)(z-z_0)^2-y^2z_0^2=0$ 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 $r$ and height $h$ the exact volume: $V=\tfrac{\pi}{2}r^2h$ .

The implicit representation is fulfilled by the points of the line $(x,0,z_0)$ , too. For these points there exist no tangent planes. Such points are called singular.

= Parabolic conoid =

File:Conoid-parabolic.svg

The parametric representation

: $\mathbf x(u,v)=\left(1,u,-u^2\right)+ v\left(-1,0,u^2\right)$

:::: $=\left(1-v,u,-(1-v)u^2\right)\ , u,v \in \R \ ,$

describes a parabolic conoid with the equation $z=-xy^2$ . 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 =

Hyp-paraboloid.svg|hyperbolic paraboloid

Pluecker-conoid.svg| Plücker conoid

Whitney-umbrella.svg| Whitney umbrella

Applications

File:Pouziti konoidu1.jpg

File:Pouziti konoidu3.jpg

= 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