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Saturday, May 25, 2013

Hyperbolas and ellipses with one equation

Sometimes you don't discover something new in math. Actually, it's really hard to discover something new in math as people have been kind of busy, and I have a result I like which I rediscovered a few years back.

It is a way to get hyperbolas and ellipses with a single equation.

The equation is x2 - Dy2 = 1, in rationals, so to program it you need to use doubles:

y = -2t/(D - t2)

and

x = (D + t2)/(D - t2)

and you get hyperbolas with D greater than 0, and ellipses with D less than 0, and the circle when D=-1, giving the well-known circle parameterization:

With x2 + y2 = 1: y = 2t/(1 + t2) and x = (1 - t2)/(1 + t2)

So yeah, you can just draw an ellipse or a hyperbola by incrementing t, which gives you x and y. Easy.

I like easy.

When I came up with my own solution for x and y with a parameter, I was shocked to discover that I was so far off from the time of the initial discovery that Fermat himself knew the above.

And Pierre Fermat died in 1665. So he's been dead for over 340 years.

Turns out it's really hard to find something new in math. No big deal though. I also like rediscovering things too, as it's fun!

So you can use a single equation for hyperbolas and ellipses, just by shifting that thing D, and yup, if you remember your trigonometry--or is it algebra?--that means that D has to be connected to eccentricity. And if you're really smart, figure out the equation that connects them directly to prove your intelligence.

Now, derive the equations yourself. If some dude centuries ago could do it, why not you today?

I did it. It's not all that hard actually, and might be a fun exercise to test your limits with something easy.

See! Isn't math fun?


James Harris
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