Thursday, March 27, 2014

Euler Spiral

I was reading a calculus textbook when I noticed it said that cos(x2) doesn't have an elementary antiderivative. Elementary antiderivative? Clearly, they were hiding something. They didn't say it didn't have an antiderivative; they said it didn't have an elementary antiderivative. Of course, I wanted to know what the antiderivative was. If it wasn't elementary, it had to be really awesome.

I looked up the integral of sin(x2). Turns out, the integral cannot be expressed as anything other than itself. It's known as the Fresnel S integral, is written as S(x), and is defined as the integral of sin(x2). There's another Fresnel integral known as the Fresnel C integral which is written as C(x) and defined as the integral of cos(x2).

I also saw some graphs of the integrals. One really cool graph involved the parametric equations x = C(t) and y = S(t), and was called the "Euler spiral." It had a cool spirally shape, and I immediately knew that I had to graph it myself. I ended up writing an interactive JavaScript program to graph the parametric equations. Here it is; enjoy!

x = C(t)
y = S(t)

Max t:
t step size: * .001

Zoom X: %
Zoom Y: %

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