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

Post a Comment

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...
Post a Comment
Read more

Knot 9-31

Knot 9-31 A knot is mathematics is defined as a closed, non-self-intersecting curve that is placed in three dimensions and cannot be the "unknot". The main difference between a knot in the real world and a known in mathematics is that a knot in mathematics does not contain any extra strands. The example following will help visualize this. Today, I have specifically chosen knot 9-31 from the Knot Atlas. This knot is very unique and contains some very interesting properties that we are going to look into. The Crossing Number For my knot today, I chose knot 9-31 from the Knot Atlas. This knot contains 9 crossings! The Unknotting Number The unknotting number is exactly what is sounds like. This is the minimum number of times the knot must be passed through itself to untie it. Luckily, the Knot Atlas is super useful and provides the unknotting number for us, but I still...
Post a Comment
Read more

Do Over: Integration for Over Regions in the Plane

Introduction Earlier in the semester, we visited the topic of integration for over region regions in the plane. This was our first experience with taking integration in the third dimension. To do this, we had to use double integrals to calculate the volume of a region between two surfaces. This topic seems relatively straightforward and anyone who has taken a higher calculus class knows of double integrals. Today, we are going to revisit this topic for a couple of reasons. Why Double Integrals Again? As mentioned before, I am revisiting this topic for a couple of reasons. One of the main reasons is due to how the 3D print turned out. Due to the function, I chose, the final rectangular prism was stand alone. When printing, this resulted in a sloppy prism that lacked structural integrity. The image below is a reference to my original design that includes the prism that had the issue. The main cause of ...
Post a Comment
Read more