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Greg Stanton 13. stu
1/ When you say "Drive a block, take a right," you're giving curvature k as a function of arc length s. In 1990, curves for which k is a polynomial in s were termed "polynomial spirals." They've since been applied to self-driving cars. Great topic for 3!
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Greg Stanton 13. stu
Odgovor korisniku/ci @HigherMathNotes
2/ Definition: I made some brief notes defining polynomial spirals and explaining how to parameterize them. This will help you understand the code I wrote to produce the animations.
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Greg Stanton 13. stu
Odgovor korisniku/ci @HigherMathNotes
3/ Example: Polynomial spirals generalize the celebrated Euler spiral (a.k.a. Cornu spiral, a.k.a. clothoid), discovered independently in diverse fields of application, as far back as 1694. Here's how to derive a parameterization of the Euler spiral from its curvature.
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Greg Stanton
4/ History and applications: Has written a history of the Euler spiral: A quick overview of applications of the Euler spiral is on Wikipedia:
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Greg Stanton 13. stu
Odgovor korisniku/ci @HigherMathNotes
5/ Mathematics: The more general polynomial spirals are investigated in this 1990 paper by Dillen: {%22pages%22:[648],%22view%22:%22info%22}
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Greg Stanton 13. stu
Odgovor korisniku/ci @HigherMathNotes
6/ Application: Here are a couple papers that apply polynomial spirals to trajectory generation for autonomous vehicles.
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Greg Stanton 13. stu
Odgovor korisniku/ci @khanacademy
7/ Code: You can try out different spirals and watch the full animations in the browser by checking out my hosted :
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