Skip to main content

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.

Comments

Popular posts from this blog

Do Over: Integration Over a Region in a Plane

Throughout the semester we have covered a variety of topics and how their mathematical orientation applies to real world scenarios. One topic we discussed, and I would like to revisit, is integration over a region in a plane which involves calculating a double integral. Integrating functions of two variables allows us to calculate the volume under the function in a 3D space. You can see a more in depth description and my previous example in my blog post, https://ukyma391.blogspot.com/2021/09/integration-for-over-regions-in-plane_27.html . I want to revisit this topic because in my previous attempt my volume calculations were incorrect, and my print lacked structural stability. I believed this print and calculation was the topic I could most improve on and wanted to give it another chance. What needed Improvement? The function used previously was f(x) = cos(xy) bounded on [-3,3] x [-1,3]. After solving for the estimated and actual volume, it was difficult to represent in a print...

Minimal Surfaces

Minimum surfaces can be described in many equivalent ways. Today, we are going to focus on minimum surfaces by defining it using curvature. A surface is a minimum surface if and only if the mean curvature at every point is zero. This means that every point on the surface is a saddle point with equal and opposite curvature allowing the smallest surface area possible to form. Curvature helps define a minimal surface by looking at the normal vector. For a surface in R 3 , there is a tangent plane at each point. At each point in the surface, there is a normal vector perpendicular to the tangent plane. Then, we can intersect any plane that contains the normal vector with the surface to get a curve. Therefore, the mean curvature of a surface is defined by the following equation. Where theta is an angle from a starting plane that contains the normal vector. For this week’s project, we will be demonstrating minimum surfaces with a frame and soap bubbles! How It Works Minimum surfac...

Ruled Surfaces : Trefoil

Ruled Surfaces : Trefoil A ruled surface is a surface that consists straight lines, called rulings, which lie upon the surface. These surfaces are formed of a set of points that are "swept" by a straight line. This is relatively intuitive once you see a good visual, but can be a bit abstract without that concrete example. A very basic example of a ruled surface is a cylinder; if we have a straight line and move it in a circle we create a cylinder made entirely of straight line. Note that the surface will only be a cylinder if all the lines are parallel. If the lines are not parallel we can create hyperboloids and cones depending on how much we have rotated. The rotation we are describing here is not a simple turning action, but more of a twisting motion—less like rotating a can by turning it and more like wringing out a washcloth by twisting it. Specifically, a cylinder is essentially two circles connected by rulings, if we keep one of the circles...