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@HigherMathNotes | |||||
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4/ History and applications:
@raphlinus Has written a history of the Euler spiral:
www2.eecs.berkeley.edu/Pubs/TechRpts/…
A quick overview of applications of the Euler spiral is on Wikipedia:
en.wikipedia.org/wiki/Euler_spi…
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Greg Stanton
@HigherMathNotes
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13. stu |
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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 #calculus 3! #MTBoS pic.twitter.com/XEaoZejNd0
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Greg Stanton
@HigherMathNotes
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13. stu |
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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
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Greg Stanton
@HigherMathNotes
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13. stu |
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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
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Greg Stanton
@HigherMathNotes
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13. stu |
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5/ Mathematics: The more general polynomial spirals are investigated in this 1990 paper by Dillen:
gdz.sub.uni-goettingen.de/id/PPN26683302…{%22pages%22:[648],%22view%22:%22info%22}
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Greg Stanton
@HigherMathNotes
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13. stu |
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6/ Application: Here are a couple papers that apply polynomial spirals to trajectory generation for autonomous vehicles.
citeseerx.ist.psu.edu/viewdoc/downlo…
ri.cmu.edu/pub_files/2013…
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Greg Stanton
@HigherMathNotes
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13. stu |
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7/ Code: You can try out different spirals and watch the full animations in the browser by checking out my #code hosted @khanacademy:
khanacademy.org/computer-progr…
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