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.pic.twitter.com/tL9fFjXj7N
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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.pic.twitter.com/tL9fFjXj7N
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.pic.twitter.com/CGPhT3Zq5O
4/ History and applications:
@raphlinus Has written a history of the Euler spiral:
https://www2.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-111.pdf …
A quick overview of applications of the Euler spiral is on Wikipedia:
https://en.wikipedia.org/wiki/Euler_spiral …
5/ Mathematics: The more general polynomial spirals are investigated in this 1990 paper by Dillen: https://gdz.sub.uni-goettingen.de/id/PPN266833020_0203?tify= …{%22pages%22:[648],%22view%22:%22info%22}
6/ Application: Here are a couple papers that apply polynomial spirals to trajectory generation for autonomous vehicles. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.88.8908&rep=rep1&type=pdf … https://ri.cmu.edu/pub_files/2013/6/IV2013-Tianyu.pdf …
7/ Code: You can try out different spirals and watch the full animations in the browser by checking out my #code hosted @khanacademy:https://www.khanacademy.org/computer-programming/spiral-animator/5291927036985344 …
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