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Ruled Surfaces

Ruled surfaces are defined by the University of Oxford Mathematical Institute as a surface where every point in the surface has at least one straight line, which also lies in the surface.

To break this down, step by step, we first start with a line. In order to form a line, we must have two points that the line will go through. Next, we choose two curves to create a surface between. On each curve there will be multiple points and we will choose two points. Then, we connect one point on the one curve with another point on the other curve creating a line between the two curves. Finally, repeating these steps for the points on the curves it creates a combination of lines that form a ruled surface.

Some good examples of ruled surfaces include cones, cylinders, and saddles.


To help demonstrate the difference, some none examples of ruled surfaces includes ellipsoids and elliptic paraboloids. These are not ruled surface because there are no straight lines that lie in the surface.

Example:

Now for an example. Let’s look at the two functions f(x) and g(x) in their parameterization forms.


The y coordinate represents the curve of the actual function, which will be a standard quadratic equation shifted one unit over. The z coordinate represents where the curve lies on the z-axis, therefore g(x) will be 15 units about f(x).

Then, as we look at points along each curve and connect them together a ruled surface forms and looks like the following.


This is an example of a saddle because the way it becomes flat in the middle but extends and curves outward.

Here is a zoomed in graphic of the model to help visualize each line being shown. With each line so close together, they combined to make a surface.


Why I Chose This Surface

My inspiration for this object comes from my hometown. Being from St. Louis, MO home of the Gateway Arch, I’ve always enjoyed working with parabolas because the arch is a perfect parabola, and it is interesting to see different ways to manipulate it. A perfect parabola means the parabola is as tall as it is wide (which is saying a lot, if you have ever been so lucky to visit the Gateway Arch!) Originally picturing The Arch on the edge of the Mississippi river, I thought it would be cool if I built a model as is there was another Arch across the river in Illinois. Then connect the curves together to make a bridge/tunnel on the river to demonstrate a ruled surface. However, that wouldn’t make a very interesting surface, so instead I wanted to see what would happen if one of the functions was flipped. By doing so, it created a saddle. Next, I shifted f(x) one unit to the left and g(x) one unit to the right to create a larger space between the functions. In doing this, I was hoping this would cause the rays to connect the curves at an angle. Instead, the rays still connected straight across, but it caused one end on each parabola to extend farther than the other. This caused the model to not be symmetric on the x or y axis. This also occurred because I had the domain set to [-5,5].


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