Conoid

In geometry a conoid (from Greek κωνος  'cone', and -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

${\displaystyle \mathbf {x} (u,v)=\mathbf {c} (u)+v\mathbf {r} (u)\ }$

Any curve x(u₀,v) with fixed parameter u = u₀ is a ruling, c(u) describes the directrix and the vectors r(u) are all parallel to the directrix plane. The planarity of the vectors r(u) can be represented by

${\displaystyle \det(\mathbf {r} ,\mathbf {\dot {r}} ,\mathbf {\ddot {r}} )=0}$.

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[edit]

Right circular conoid[edit]

The parametric representation

${\displaystyle \mathbf {x} (u,v)=(\cos u,\sin u,0)+v(0,-\sin u,z_{0})\ ,\ 0\leq u<2\pi ,v\in \mathbb {R} }$

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 ${\displaystyle (x,0,z_{0})\ x\in \mathbb {R} \ .}$

Special features:

1. The intersection with a horizontal plane is an ellipse.
2. ${\displaystyle (1-x^{2})(z-z_{0})^{2}-y^{2}z_{0}^{2}=0}$ is an implicit representation. Hence the right circular conoid is a surface of degree 4.
3. Kepler's rule gives for a right circular conoid with radius ${\displaystyle r}$ and height ${\displaystyle h}$ the exact volume: ${\displaystyle V={\tfrac {\pi }{2}}r^{2}h}$.

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

Parabolic conoid[edit]

The parametric representation

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

${\displaystyle =\left(1-v,u,-(1-v)u^{2}\right)\ ,u,v\in \mathbb {R} \ ,}$

describes a parabolic conoid with the equation ${\displaystyle z=-xy^{2}}$. 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[edit]

1. hyperbolic paraboloid
2. Plücker conoid
3. Whitney Umbrella
4. helicoid

• 
  hyperbolic paraboloid
• 
  Plücker conoid
• 
  Whitney umbrella

Applications[edit]

Mathematics[edit]

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

Architecture[edit]

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[edit]

• mathworld: Plücker conoid
• "Conoid", Encyclopedia of Mathematics, EMS Press, 2001 [1994]

References[edit]

• A. Gray, E. Abbena, S. Salamon, Modern differential geometry of curves and surfaces with Mathematica, 3rd ed. Boca Raton, FL:CRC Press, 2006. [1] (ISBN 978-1-58488-448-4)
• Vladimir Y. Rovenskii, Geometry of curves and surfaces with MAPLE [2] (ISBN 978-0-8176-4074-3)