Ruled Surfaces

Introduction
Taking a step up from quadratic surfaces we are introduced to the topic of ruled surfaces. Ruled surfaces can create some very unique and interesting shapes. The gist of ruled surfaces is that ruled surfaces are a set of points that are swept by a moving straight line. To simplify things, visualize a cone. A cone is formed by simply fixing a point and having another point move along a circle. The straight lines that are formed, form a cone. Some surfaces can be doubly ruled if they are two distinct lines that lie on the surface. Examples of these include one-sheet hyperboloid and hyperbolic paraboloid. Ruled surfaces are very interesting to understand, as they allow you to further understand how quadratic surfaces are formed. Today we will be working at a slightly more unique shape.
Ruled Surfaces?
As mentioned before, ruled surfaces are a set of points that are swept by a moving straight line. Let’s revisit the cone example from earlier. Below is an image of the cone to help visualize.

From the visualize, you can see exactly how the cone behaves. It is formed from the center point being fixed and then having another point that moves along the circle. Then, following the process of ruled surfaces, the points are swept by a moving straight line. This gives the shape of the cone. This is true with a lot of different shapes. Also, since we are visualizing shapes in the xyz plane, we have to parameterize each of the curves, so we can truly understand what is occurring here. We will look into the parametrization further once we get into the example, but just for a slight foundation we can look at the cone example again. For our cone above, the parameterization is \[x(t)=rcos(t)\] \[y(t)=rsin(t)\] \[z(t)=0\] This will give us the circle or the foundation of the cone. We will then raise the z value to a positive value that will be the top circle of the cone. Now that we have a slight understanding of the process to achieve a ruled surface, we are going to get into the example.
Ruled Surfaces Example
As stated previously, the unique shapes that you can make with ruled surfaces is quite interesting. Today, we are going to look at a shape that has a foundation of a keylock like curve and a banana like curve. The purpose of these two shapes is to show just how unique you can make ruled surfaces. First, let’s look at the curve for the keylock like curve. We have parameterized this curve to make it easier to understand. The following equations give the keylock like curve: \[x(t)=4cos(t)+cos(2t)^2\] \[y(t)=2sin(t)+sin(3t)\] \[z(t)=0\]

Next, let’s look at the parametrization for the banana like curve. This one was a lot of fun to try and make it work. The following equations give the banana like curve: \[x(t)=cos(t+phase)+cos(t)\] \[y(t)=sin(t+phase)+sin(2t)\] \[z(t)=10\]

From the banana curve like equations, there is a constant that is called phase. Now, what this actually does is that it shifts the banana curve 60 degrees. This might seem odd but remember the information that was included with ruled surfaces. Ruled surfaces are swept by a moving straight line. What the shift from the phase does is very interesting. It will actually slant the lines that connect the two surfaces. The lines are still straight lines, but they now have a slant to them due to the phase shift. This can be better visualized with the addition of the straight lines that connect the two surfaces.

As you can tell from the image, there are straight lines present, but they have a slight slant from them. One problem that I may run into when printing my object is that I cannot tell if the straight lines do overlap one another, but we will find out!
Why These Functions?
These functions were chosen for quite a few different reasons. First, I really wanted to exaggerate just how unique of a curve you can make with ruled surfaces. I really liked the curves that I was able to produce. Second, I think that the parametrization of the curves is quite easy to understand once a foundation of ruled surfaces is learned. Lastly, I wanted to challenge my own knowledge to see if I truly understood how ruled surfaces behaved. The dimensions of my object came out to roughly 1.5 in x 0.70 in x 1 in.