What is the inverse of a circle?
mattferraro.dev
What is the inverse of a circle?
1–10 of 51 posts
Re: What is the inverse of a circle?
#2OP gives a complicated but interesting answer.
OP's equation 1.7 suggests something to me that wasn't highlighted.
Centered at the origin in R2, I expected inverse of r times e^iTheta to be be 1/r times e^-iTheta. Their product is then 1. I believe that is in equation 1.7 .
For that apparently special case, points on a larger-than-unit circle map to points on a smaller-than-unit circle.
Re: What is the inverse of a circle?
#3on 1.2) is 1 divided by zero not undefined instead of infinite?
Re: What is the inverse of a circle?
#4In polar coordinates a circle can be defined by all points where r = a. The inverse would be all points where r doesn't equal a.
Re: What is the inverse of a circle?
#5on 1.2) is 1 divided by zero not undefined instead of infinite?
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Re: What is the inverse of a circle?
#6In polar coordinates a circle can be defined by all points where r = a. The inverse would be all points where r doesn't equal a.
I interpreted the question myself from another angle: a circle is a function where every f(x) is an equal linear distance to an arbitrary fixed point z. So, the "inverse" to this function could be a function where every f(x) must have a different linear distance to z.
Re: What is the inverse of a circle?
#7on 1.2) is 1 divided by zero not undefined instead of infinite?
It’s just some sleight of hand. You can switch all of the parts to be lim x to zero of 1/x m.m. if you wish to satisfy yourself
Re: What is the inverse of a circle?
#8Great article! The diagrams look nice. What did you use to generate those diagrams?
Re: What is the inverse of a circle?
#9For anyone else who's mind went blank trying to work out just what sort of "inverse" can be applied to a circle. It's interesting math and the diagrams are worth checking out, some very nice use of interactivity to assist with understanding the nature of the function.
But definitely not the kind of inverse I was expecting.
I went to "A function that's equal everywhere not the circle you define in the function", some sort of f(x,y) = !f(circle) by way some sort of algebraic geometry math. Then I was trying to work out if it meant something else... then I loaded it and was genuinely surprised to find its a much more specific kind of inverse that never even occurred to me.
Re: What is the inverse of a circle?
#10I did some wacky inside-out pics of planets years ago, inspired by the inside-out Mandelbrot set looking like a leaf/teardrop.
https://www.adamponting.com/inside-out/
Related: Not Knot, 1991 Thurston-ish short film about knots and knot complements - "the space where the knot isn't".
https://www.youtube.com/watch?v=zd_HGjH7QZo