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image of rollercoaster loop

I am trying to find the equation for the form that is in the picture. Basically it is an infinitely extending roller coaster loop. I just can not find the magic words in Google.

Any suggestions? What is it called? What is the equation?

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2 Answers 2

up vote 1 down vote accepted

$(\cos(t)+\alpha t,\sin(t))$ for some $\alpha>0$ should work.

See a graph at

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You may want to try and stitch together a bunch of clothoids, which seem to be state of the art in rollercoaster design.

They are related to the Euler spiral, see

The reference may be of particular interest.

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I actually got to both of those web pages. But surely there is a relatively simple, maybe parametric, equation for this basic form, no? – Paul Jan 13 '13 at 14:01

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