Monday, May 14, 2012

Cat Ear Parabolas

So, I promise this isn't going to turn into a cat blog.  I mean, I am obsessed with my cat, I ain't gonna lie, but I have a modicum of control.

However, I was thinking recently about how Copernicus' ears look like 3-D conic sections.



And then I thought about the parabola that is the front outline of his ear, and I decided, since I am done with school and can do random fun things like this now, that it had to be done.

So I took a picture of his ear.
And then I traced it in Adobe Illustrator (badly), rotated the drawing so that the vertex was at the top, and overlaid a graph.  The points that I found were based on the 8-by grid that was visible for editing on Illustrator.  I placed it so that my graph could have 2 x-intercepts.  For the graph I'm going for, it doesn't matter where on the coordinate system it lies.  It's about the shape.

Then, I looked up how to find the equation of a parabola from points on the graph because I couldn't remember it.  Turns out you need the vertex and one other point for the simplest process.

For a vertically symmetrical parabola, you start with y=a(x-h)^2 + k, where (h, k) is the vertex.

My vertex was (0, 25), so I started with that:  y=a(x-0)^2 + 25, i.e., y=ax^2 + 25.  Then input another point and solve for a.  I chose (-8, 16), because I was fairly sure of my sketch at that point on the graph.

16=a[(-8)^2] + 25 . . . 16= 64a + 25 . . . -9 = 64a . . . a = -0.138461538

So my equation for that view of Buddy's ear is


 and the graph of that function is ...



 Which, if you imagine some angling and perspective isn't bad.

I really want to try one with a better head-on picture.  Getting Bud to sit still for that, however, is the hard part.













1 comment:

  1. Hey! Theoretically, just using that equation wouldn't be enough, would it? You'd need to calculate for more than that. Use y=a(x)+bx+c and method of elimination/substitution of system of equations, yes, but what about putting the image in an online graphing calculator like Desmos and putting in the equation to see if the parabola fits with he picture for certainty. Plus finding the regression of r.

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