Skip to main content

Ruled Surfaces

Introduction

So far, we've spent a lot of time navigating the world of calculus. To begin moving away from calculus, let's discuss ruled surfaces! Ruled surfaces are generated by connecting two curves with straight lines in three-dimensional space. Below, you can see how the given surface is made up of straight lines connecting the points on the left and right curves. If the number of straight lines connecting these curves is infinite, we see a solid surface. We're going to dive into ruled surfaces with an example I find interesting not just because of its shape but also its applications to biology.
(Source: http://www.grad.hr/geomteh3d/Plohe/pravcasta.png)

The curves I chose to define my surface

For an interesting example, I chose the ruled surface defined by the parameterized curve \[x(t)=rcos^5(t)\] \[y(t)=rsin^5(t)\] \[z(t)=0.4t\] and the vertical line \[z(t)=0.4t.\] Why I chose the surface and curves I did

For starters, I chose this curve because it is not self-intersecting. Additionally, it is not just changing the scale of the code given on the class page; it's taking the code and changing the curve to make it a more unique example. More importantly, though, is that I wanted to apply this math to my other major in a fun way! I study biochemistry and math, and I have spent a lot of time learning about DNA mutations and the chemistry in our bodies. I was interested in using a helicoid as my ruled surface example because DNA looks very similar to a helicoid, but rather than creating a ruled surface that resembles a double helix, I wanted to imagine a new mutation in the DNA. First, consider the structure of DNA compared to a helicoid:
(Source: www.rdworldonline.com)
As you can see, DNA base pairing is similar to having straight lines connecting two curves, and in textbooks and images like the one seen we tend to draw the chemical bonds between the DNA bases like we are creating a ruled surface between these two curves. Conversely, the given helicoid has straight lines connecting one curved line and a single straight, vertical line. I created my own DNA mutation from the given helicoid by taking the \(x\)- and \(y\)-components and changing them from just \(rcos(x)\) and \(rsin(x)\) and raising them to the fifth power. This looks like the following:
This looks like pretty messed up DNA! I mutated it like this because I like the way it looks from above even better than how weird it is from the side:
I don't know about you, but I am very ready for Christmas. This looks like a star you could find at the top of a Christmas tree. The star in two-dimensions given by the parameterization \(x(t)=rcos^5(t)\) and \(y(t)=rsin^5(t)\) has been pulled into three dimensions by \(z(t)=0.4t\), which lifts the star up along this vertical line. Because of this, I'm going to call my new mutation an X-mas mutation, just for fun. Overall, my print is going to be around two inches wide and nearly an inch and a half tall. I may scale it to be larger if printing fails or removing supports from between lines that close together proves difficult. We'll see!

Word count: 515

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

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...
Post a Comment
Read more

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...
Post a Comment
Read more