Conoid
A conoid (from the Greek κωνος cone and ειδης similar) is a ruled surface in mathematics , whose family of generators (straight lines) the two additional conditions
 (1) All generatrices of the surface are parallel to a plane, the straightening plane .
 (2) All generatrices intersect a fixed straight line, the axis .
Fulfills.
 The conoid is called straight if the axis is perpendicular to the alignment plane .
Because of (1) every conoid is a Catalan surface and can be represented by a parametric representation
to be discribed. Every surface curve with a fixed parameter is a generator, describes the guide curve and the vectors are all parallel to the straightening plane. The planarity of the vectors can be passed through with sufficient differentiability
express.
 If the trajectory is a circle, it means Konoid Kreiskonoid .
Comment:
 A conoid is unlimited (like a straight line). A graphic representation can therefore only ever show a finite part of the area.
 The term conoid was coined by Archimedes in his treatise on conoids and spheroids .
Examples
Straight circular conoid
The parametric representation
 describes a straight Kreiskonoid with the unit circle in the xy plane as a guide curve and a plane parallel to the yzplane directional plane . The axis is the straight line
Special features : 1) Every horizontal section is an ellipse, 2) The outlines of the partial area shown in the picture with regard to the main directions are a rectangle, a circle and a triangle (see 2nd picture), 3) is an implicit representation, that is , the straight circular conoid is a surface of the 4th degree. 4) The Kepler barrel typically delivers in a straight Kreiskonoid with the base circle radius and height of the exact volume: .
The implicit representation is fulfilled by the whole straight line . There are no tangent planes at the points of this straight line . Such points are called singular .
Hyperbolic paraboloid
The parametric representation

 describes the hyperbolic paraboloid with the equation It is a surface of the 2nd degree ( quadric ).
The leading curve of this conoid is the straight line (red in the picture), the straightening plane is parallel to the yz plane. If you choose the xaxis as the axis, the conoid is straight . Since in this example the further straight line runs through every point of the surface in addition to the generating line , one of these further straight lines can also be selected as the axis. However, only the firstmentioned axis is perpendicular to the plane of alignment. In this case the xaxis could be selected as both the guide curve and the axis.
The hyperbolic paraboloid has no singular points.
Plücker conoid
The parametric representation
represents a Plücker conoid with the equation
 represent.
The guide curve is a path traversed twice on the zaxis, the axis of the conoid is the zaxis and the alignment plane is parallel to the xyplane. Since the axis is perpendicular to the straightening plane, the conoid is straight .
The implicit representation is fulfilled by the entire zaxis. The points on the zaxis are singular (there are no tangent planes).
Whitney Umbrella
The parametric representation
represents a Whitney Umbrella with the equation . The surface is a conoid with twice traversed the positive zaxis as a guide curve , the zaxis as the axis and a plane parallel to the xy plane directional plane . Since the axis is perpendicular to the straightening plane, this conoid is also straight .
The implicit representation is also fulfilled by the negative zaxis, the handle of the umbrella. The points on the zaxis are singular (there are no tangent planes).
Parabolic conoid
The parametric representation
provides a parabolic conoid with the equation . The conoid having a parabola as a guide curve , the yaxis as the axis and parallel to the xz plane directional plane . Since the axis is perpendicular to the straightening plane, the conoid is straight . It is used in architecture as a roof surface (see Applications).
The parabolic conoid has no singular points.
Helical surface
The helical surface is also a straight conoid. It has no singularities.
Applications
In math
There are numerous simple examples of surfaces with singularities among the conoids .
In architecture
Like other ruled surfaces, conoids are used in architecture because they can be easily modeled from segments (beams, bars). Just Conoids can be very easily made: one threaded rods so on an axis that they can only rotate about this axis. Then you deflect the bars with the help of any guide curve and thus create a straight conoid. (See Parabolic Conoid.)
Web links
 mathworld: Plücker's Conoid
 mathcurve: Pluecker conoid
 mathcurve: parabolic conoid
 K3Dsurf: 3d surface generator
literature
 Small Encyclopedia Mathematics , Harri DeutschVerlag, 1977, p. 219.