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!

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