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Spongebob's Pringle

So, a lot of you probably read the title of my blog post and got really confused because you either a.) didn't watch Spongebob as a child or b.) have no idea where I'm going with this or c.) both. Well, as you all have probably figured out by now, I like to choose objects with some relation to the real world in order to allow everyday people (non-math people) to either understand the math we're talking about or at least be somewhat entertained/interested in what we're talking about. Quadric surfaces have been the most "mathy" topic we've discussed so far, so I had to be even more creative when trying to relate it to something non-math people would understand.

I thought about creating a hyperboloid of one sheet and comparing it to an hourglass, but I decided against it because I couldn't think of a good reason why someone would want to slice an hourglass. The only other quadric surface that has a shape that can be applied to the real world, in my opinion, is a hyperbolic paraboloid. In my head, a hyperbolic paraboloid, with the right equation, resembles that of a pringle. The only problem that I saw with using a pringle was that it's slices are parabolas. This is a positive thing because even adults who still watch Spongebob have heard of what a parabola is. The only downfall is that parabolas give boring equations. So, in order to spruce it up, I decided to create the pringle from the Spongebob episode where Spongebob's only friends are a coin, a tissue, and a pringle. I decided to split the pringle up into 3 equal pieces so he could share it with his other two friends when they got hungry. Below you can find a picture of Spongebob and his friends.
Now that you all know what goes through my brain when picking an example, let's talk about the math. A quadric surface is a surface in three-dimensions with the general equation: \[Ax^{2}+By^{2}+Cz^{2}+Dxy+Exz+Fyz+Gx+Hy+Iz+J=0\] Since that equation is a bit overwhelming, it may help to think about these quadric surfaces after they are rotated and translated to get certain shapes. The surface I am focusing on today is the hyperbolic paraboloid. It's name is very helpful in my opinion because it decribes what two-dimensional equations its made up of, hyperbolas and parabolas. The general equation for a hyperbolic paraboloid is: \[z= \dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}\] The specific equation that I thought resembled a pringle most closely is: \[z= \dfrac{x^2}{8}-\dfrac{y^2}{10}\] When this equation is restricted to both x and y staying between \(-3\) and \(3\). The following shape results:
Now, we must slice our surface. If this surface is sliced horizontally, the equation of the intersection would be a hyperbola, but since my goal is to split the pringle into 3 pieces, this will be best done by slicing vertically. These vertical slices will be perpendicular to the xy-plane and will create intersections of parabolas. In order to get 3 pieces, we must do 2 slices and since our range of y-values is from \(-3\) to \(3\), the planes I have chosen to slice with are \(y=-1\) and \(y=1\). The surface sliced with those 2 planes looks like:
In order to calculate the equation of the parabolas we need to find exactly where the plane intersects with the surface. We can do this by taking the equation of our hyperbolic paraboloid and plugging in 1 for y and then -1 for y. Once you do this, you can see that both planes give us the same two-dimensional intersections of a parabola. The following is the equation for both of our parabolas: \[z= \dfrac{x^2}{8}-\dfrac{1}{10}\] The parabolic equation graphed in two-dimensions will show us what our slice will look like once we print the object. Below is what the graph of both slices will look like:
From this graph you can see that the minimum of both our slices occurs at \(z=-\dfrac{1}{10}\) and our parabola is shrunken by one-eighth. Looks to me to be about the same curvature as a pringle, but I'm no expert on pringles.

Our final object will be about 46mm by 46mm by 17mm and will look like a pringle split up into 3 pieces. The final object will resemble the picture below:

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