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Quadric Surfaces - Football School?

Introduction
Quadric surfaces are a generalization of conic sections. If you remember from surfaces of revolution, then quadric surfaces are surfaces of revolution that are pushed into the third dimension about one axis which then creates a surface. Quadric surfaces come in a variety of shapes and sizes. Specifically, quadratic surfaces are the graphs of any equation that can be put into the general form: \[Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Iz+J=0\] There are six main quadric surfaces. They are the ellipsoid, the elliptic paraboloid, the hyperbolic paraboloid, the cone, and the hyperboloids of one and two sheets. All of these different quadric surfaces can be found from the general form listed above. Each quadric surface has a different equation than the others, and there are unique difference between them all. Some of the equations may need to be found by using the completing the square technique and some of the equations may need to be found by parameterizing the equation. Today, we are going to keep it rather simple by looking specifically at the ellipsoid.
Quadric Surfaces?
Ellipsoids may be the easiest of the quadric surfaces to look at, but they are great at understanding how the changing of the constants in the equation impacts the overall surface. Lets look at the equation of an ellipsoid. The equation for an ellipsoid is given by: \[\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\]Now the equation for an ellipsoid may look familiar, and that is because if \(a=b=c\) then the equation isn't actually an ellipsoid, but a sphere instead.
What makes an ellipsoid different is that it is a surface that is obtained from a sphere, but it is deformed by means of scaling or a transformation. Fun fact, Earth is actually an irregularly shaped ellipsoid.
Let's Slice the Quadric!
To relate an ellipsoid to some current events here at UK, we are going to look at an ellipsoid that makes a football like shape. Since we are a football school now I thought it would be interesting to look at an ellipsoid that makes this shape. We are going to look at an ellipsoid that has the following equation: \[\frac{x^2}{4}+\frac{y^2}{16}+\frac{z^2}{4}=1\] This ellipsoid has been stretched in the y-direction by 4 units and by 2 units in the x and z directions. This creates a football like surface.
Now, we are going to create a cross-section on our ellipsoid. I am not entirely sure why anyone would cut a football, but maybe you are just interested on what is inside a football. Mentioned before, quadric surfaces are a generalization of conics. When we take a cross-section on our ellipsoid, we will actually find a function in the second dimension. Here is our image from before, but now including a cross section with the function \(x+z=0\). This intersecting plane is a plane that is perpendicular to the vector \([1,0,1]\). This creates an almost perfect 45 degree angle that intersects our surface.
This splits our surface in half. What is revealed is hopefully not a surprise, for ellipsoids are built by ellipses! This trend is relatively similar with the other quadric surfaces. If quadric surfaces have sparked your interest, then I would recommend going and taking the cross-sections of other ones too. This will allow you to not only understand where the other quadric surfaces get their name from, but also to further understand the concept of quadric surfaces and how they behave. Below is an image that better represents the ellipsoid when we took the respective cross-section. If you were to take more cross-sections of the surface, you would easily be able to understand how the surface is built by ellipses!
Why These Functions?
This example was chosen mainly for the success that the UK football team has had this season. I thought that visualizing a football using an ellipsoid would fit the current events on campus. Also, ellipsoids and ellipses are objects that we see in our everyday lives, so understanding how they behave is interesting.

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