Ruled Surfaces

Introduction

At some point (perhaps in an arts and crafts class) you’ve probably turned a sheet of printer paper into a cylinder by taping the ends together. Similarly, you’ve also probably created a long paper-cone by wrapping one end of the paper around your finger. Though you didn’t realize it at the time, these are both very simple examples of ruled surfaces, which this post will examine in more detail.

What are Ruled Surfaces?

Put simply, ruled surfaces are any surface that can be constructed of entirely straight lines (even if the edges may not appear particularly straight). A basic example is the paper cylinder and cone mentioned earlier. In the below image, you can see each one is basically infinitely many lines connecting the top and bottom circles:

However, ruled surfaces can get much more complicated. Imagine we take the cylinder above but twist it. We would get the hyperboloid of one sheet shown below (leftmost image). Furthermore, the lines don’t have to be bounded by a closed shape like a circle. They can also be drawn between two lines, like the helix below (rightmost image):

Now that you understand the basics of ruled surfaces, we will take a look at how to construct one ourselves.

Parametric Equations

Previously, the lines for our ruled surfaces were drawn between two different functions, such as a line or a circle. Before we can create a ruled surface, we need a way to express these functions in the 3-dimensional plane. Remember, a Cartesian function must pass the vertical line test, making it problematic to graph even a circle with a single function. This is what parametric equations come in handy for.

Say we want to graph a circle in 3-dimensional space. Instead of having one equation as a function of \( x \), we will have one equation for each coordinate of our line, and make them a function of a separate variable (we’ll use \( t \) ). The coordinates for our circle become: \[ x(t) = \cos(t) \] \[ y(t) = \sin(t) \] \[ z(t) = 1 \] The result will be a circle of radius 1 at a height of 1:

Let’s look at some more interesting parametric curves. For example, consider the following parametrization: \[ x(t) = 4 \cos(t) + \cos(4t) \] \[ y(t) = 4 \sin(t) - sin(4t) \] \[ z(t) = 0 \] The resulting graph is star-shaped:

So far these shapes have had constant \( z \) values. By adjusting this we can graph even crazier lines. Take a look at the following equations and graph: \[ x(t) = 4 \cos(t) \] \[ y(t) = 4 \sin(t) \] \[ z(t) = \sin(4t) + 10 \]

Now that you have a better understanding of parametric equations, let’s turn these into a ruled surface.

An Example

Imagine that you work for a successful pasta company. You’ve noticed that your customers are getting bored of the same-old cylindrical penne and rotini shaped noodles. You are determined to remedy this by inventing a creative new shape and realize that ruled surfaces are the perfect tool to do so! Thus, you decided to use the surface created by drawing lines connecting a star on the bottom and an oscillating circle on the top (just like our parametrizations from earlier!). To make the result even more interesting, you decide to offset the point at which the lines are connected to the oscillating circle by 50 degrees. As more lines are added, the resulting surface will look like the below:

Why this example?

This example was chosen for a few reasons:

First of all, the parametrizations used to create the ruled surface are intriguing all on their own. The star-shaped line is a good example of how parametrizations can be used to graph lines that would be impossible with a regular function. Since the star is constant in the \( z \) dimension, the oscillating circle is a good partner since it combines the simplicity of the circle shape with the novelty of a non-constant \( z \). Therefore, on their own these two functions are a good demonstration of the capabilities of parametrization.

The points of the star shape and the waves of circle make it very difficult to imagine that these curves could be connected through straight lines. As a result, this was an informative example compared to, say, a cylinder, since it shows the reader how a surface with a lot of curves can still be deconstructed into straight lines.

Author: Sarah Bombrys