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Right Circular Conoid: An Example of Ruled Surfaces

What is ruled surface

A ruled surface is a curved surface which can be generated by the continuous motion of a straight line in space along a space curve. Any point on a ruled surface can be expressed in a parametric form: \[r(u,v)=B(u)+vD(u),\] where \(B(u)\) is a directrix or base curve of the ruled surface, and \(D(u)\) is a generatrix or a unit vector which gives the direction of the ruling at each point on the directrix, i.e. generatrix is a generator which moves along a path to generate a surface, and directrix is just the path for it to follow to generate the surface.
Alternatively, the surface can be described by connecting corresponding points on two space curves: \[r(u,v)=(1-v)S(u)+vT(u), \quad 0\le v\le1\] where \(S(u)\) and \(T(u)\) are two directrices.
The two representations are identical if \[B(u)= S(u)\] and \[D(u)= T(u) - S(u)\]. A conoid is a ruled surface, if the generatrices of which stay parallel to a plane \(P\), called directrix plane of the conoid, while intersecting a line \(D\), called axis of the conoid. When \(P\) and \(D\) are perpendicular, the conoid is said to be right. If the directrix is a circle the conoid is called circular conoid. A right circular conoid is shown in the following figure, where the directrix (red) is a circle, the axis (blue) is perpendicular to the directrix plane (yellow)
The parametric representation of a right circular conoid is as following: \[r(\theta,v)=(k*cos \theta, k*sin \theta, 0) + v(cos \theta , 0, z_0 ), 0 \le \theta \le 2\pi, k,v \in R, k,v \ne 0 \] The two curves or directrices are the circle on the \(X-Y\) plane and its axis.

An example of right circular conoid

Let us consider an example or the right circular conoid which is defined as The parametric representation \[r(\theta,v)=(1-v)(2*cos \theta, 2*sin \theta, 0) +v* (2*cos \theta, 0, 4), 0 \le \theta \le 2\pi, 0 \le v \le 1\] Cartesian coordinates for this right circular conoid can be defined as: \[\left\{\begin{split} \begin{aligned} x&=2\cos \theta \\ y&=2(1- v)\sin \theta \\ z&=4 v\\ \end{aligned} \end{split}\right.\] where \(0 \le \theta \le 2\pi, 0 \le v \le 1\).
The two curves are the circle on the \(X-Y\) plane and the axis. The equation of the circle in Cartesian coordinate system is： \[\left\{\begin{split} \begin{aligned} x^2+y^2&=4\\ z&=0\\ \end{aligned} \end{split}\right.\] The axis is a line segment. The coordinates of the points on the line segment are: \[(x,0,4), -2 \le x \le 2\] We can design a 3D model of the right circular conoid by using OpenSCAD.

Why to use this example

This example of the right circular conoid was chosen for the following reasons:
  1. In geometry, conoid is a Catalan surface, which is named after the Belgian mathematician Eugène Charles Catalan. The Catalan surface is a ruled surface all of whose rulings are parallel to a fixed plane. The right circular conoid has some special features, e.g. the intersection with a horizontal plane is an ellipse.
  2. Ruled surfaces are of interest to architects, especially with free-form architecture and complicated shapes. There are numerous examples of ruled surface structures in contemporary architecture. Like other ruled surfaces, conoids are of high interest with architects, because they can be built using beams or bars. The basic principle is that one edge of the shell is curved while the opposite edge is kept straight. Right conoids can be manufactured easily. We can put bars onto an axis such that they can be only rotated around this axis. Afterwards we limit the positions of the bars by a directrix, then we get a conoid. Glass panes can be set at the curved front of a conoid for illumination. If the conoid faces north, it is suitable to get the best natural light, which can be important for space such as factories.
In the above example, we investigate the right circular conoid. This helps us to better understand the concept of ruled surfaces.

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