Conoid

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right circular conoid: directrix (red) is a circle, the axis (blue) is perpendicular to the directrix plane (yellow)

In geometry a conoid (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

Any curve with fixed parameter is a ruling, describes the directrix and the vectors are all parallel to the directrix plane. The planarity of the vectors can be represented by

.
  • If the directrix is a circle the conoid is called 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

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

Special features:

  1. The intersection with a horizontal plane is an ellipse.
  2. 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 and height the exact volume: .

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

Parabolic conoid[edit]

parabolic conoid: directrix is a parabola

The parametric representation

describes a parabolic conoid with the equation . 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

Applications[edit]

conoid in architecture
conoids in architecture

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]

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)