What is the inverse of a circle? | Hacker News Reader

What is the inverse of a circle?

mattferraro.dev

224 pointsmferraro8945 comments

I 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

https://en.wikipedia.org/wiki/Not_Knot

https://en.wikipedia.org/wiki/Knot_complement

That knot of Saturn looks like art to me. Anyway hats off for being to play with geometry like that.

Is it just me, or does it look like black holes hovering over the planets? I guess that is to be expected, if a planet with black background is used. Pseudo-scientific question: Could black holes have anything to do with something being inverted?

A black hole is just an object with so much dense mass that its Schwarzschild radius[1^] extends outside of its body. Everything has a Schwarzschild radius.

[1^]: https://en.wikipedia.org/wiki/Schwarzschild_radius

To me, inside out would mean the core on the outside with the crust in the middle. ;-)

I made this experiment I d3 quite some time ago.

http://bl.ocks.org/KKostya/6075142 http://bl.ocks.org/KKostya/6066548

Draw rectangle/circle on the right pane

For 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.

This article is interesting but not rigorous I think.

* The inverse of a geometric shape makes no sense. We only inverse operations.

* aa^-1 = 1 only if you consider the multiplication over reals.

* 1/0 is not equal to infinity.

Because the article is interesting but some people might be put off by the first few sentences, I suggest to had a disclaimer that this article lean on edutainment to the detriment of rigorous mathematics.

It depends on your definitions and which mathematical objects you are working with. The notions in the blog post are not something the author invented themselves.

For example you can find Riemann Sphere in wikipedia - https://en.wikipedia.org/wiki/Riemann_sphere

Your assertions are simply not true in the context of complex analysis. It is common to use "inverse" to refer to the multiplicative inverse as shorthand (though potentially confusing). a a^-1 = 1 is absolutely and uncontroversially applicable to any complex number. It is common and natural to extend to complex plane to include a single point at infinity (known as the extended complex plane, see e.g. https://mathworld.wolfram.com/ExtendedComplexPlane.html ). When you are working in the extended complex plane, 1/0 does equal infinity.

The answer is “it depends on how you defined the operation that acts on the circle”

This is it. Inverse is a property of functions or other relational operators, not "static" individual objects. You need a direction in order to invert it.

The multiplicative inverse, or reciprocal, or the inverse of a number, is a function acting on a number, 1/x, or x^-1.

Isn't this a case of multiplication being such a "well-known" function that the "inverse" defaults to "multiplicative inverse"?

No, inverse means reciprocal only when the context is a number or algebraic variable.

If we're talking about trig functions, inverse sine or arcsine is very different than just the reciprocal of the sine.

If we're talking about images, inverse usually means "rotate 180 degrees" or "color inversion".

I think we’re on the same page. The point I was trying to make is that reciprocal/multiplicative inverse in every day use seems like it’s a property of a number and not of a function only because most people just assume that the function we’re inverting is multiplication. The comment I was replying to missed that.

The multiplicative inverse of a number n can be seen as the inverse of the function that multiplies its input by n. That is, the inverse of multiplication by n is multiplication by 1/n.

Similarly, the additive inverse of a number m is the inverse of the function that adds m to its input.

As long as, C*Cinv = I, where C is the circle, Cinv is the inverse of circle, and I the identity. You're right. C, I, and * are entirely up in the air.

In general for inversion, we have object A (argument) and object I (identity) and a function F of two arguments, so we have equations: `F(A, X) == I, F(A, I) == I, F(X, I) == I, A != I, A != X`, where A, I, and X are objects in the same category, i.e. they must be circles `(x² + y² == r²)`.

If F is defined as `ra•rx`, then `ri == 1`, and inverse will be `rx = 1/ra`.

If F is defined as `ra + rx`, then `ri == 0`, and inverse will be `rx = 0 - ra`, where negative radius means hole.

If F is defined as `ra²•rx²`, then `ri == 1²`, and inverse will be `rx = sqrt(1/ra²)`.

If F is defined as `ra² + rx²`, then `ri == 0²`, and inverse will be `rx = sqrt(0 - ra²)`.

And so on.

Correction: `F(A, I) == A, F(X, I) == X`.

An inverse is pretty easy to generalize, and it need not rely on multiplication or the identity.

Basically: Given a function `f`, then the function `g` is the inverse of `f` if `g(f(x)) = x`.

It is easy to generalize, but it is not being generalized here.

Mountain: inverse of a circle.

Molehill: reciprocal of all points in a circle on the complex plane.

The dymaxion projection of the globe is one of my favorites and is essentially what an unbroken singular peel/shell of an orange would look like, centered on the North Pole

Take Away: 'Is a "smoke ring" also possible as an accustic phenomenon ?' (-;

beautiful and profound question

thinking of elephants stomping waves many miles... experience points to yes

In 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.

Yep, this would have the same effect. Both of these define all of the points that do not describe the circle.

However, as someone said above, f() is the inverse of g() if g(f(x)) = x. When put into practice, this means that the inverse is the reflection of the original function over y = x.

However, there's one problem with looking at the problem this way: A circle is NOT a function. Therefore, it does not have an inverse as we are thinking of it. A circle can be described by two functions, and both of these inverses combine to form the same circle. So, the inverse of a circle is (sort of) itself.

OP 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.

Great post! Here's one from Keith Peters with some interactive code examples: https://www.bit-101.com/blog/2022/01/circle-inversion/

Sorry: meaning this one copied from that earlier one, without attribution? yuck.

Otherwise know as a Möbius transformation...how is this terminology missing from the article?

Great article! The diagrams look nice. What did you use to generate those diagrams?

Maybe this article is way over my head, but is it not really simple?

If my function to generate a circle is a simple for loop 0 to 2 PI.

Then the inverse of that maps each point on the circle back to a line with points between 0 and 2 PI.

You're thinking of the inverse of the function. There's also a lot hidden in your function as generating a circle from an angle requires sin and cos functions. These are repeating functions so asin and acos don't result necessarily represent a single angle (e.g. if 1.5pi is returned, does it mean 1.5pi, 3.5pi or 5.5pi). Similarly, if you invert the formula for the unit circle, taking the square-root of the terms results in both positive and negative values.

This article is instead talking about the inverse per the identity a * 1/a = 1.

Title should read: What is the complex inverse of a circle?

IMO the opposite of a curve is an angle. One is smooth, the other disjoint such that zooming in on a curve flattens it, but zooming in on an angle makes no change. A circle is a curve that comes back around to the starting point, closing itself. The opposite of closed is open.

Therefore I propose that the inverse of the unit circle is something like the (open) region around two intersecting line segments at the origin.

I was really hoping the answer was some input/output dad joke along the lines of: I/O...

imo it would be helpful if the interactive visualizations indicated that they are interactive—I skipped right over the first one!

A triangle?

on 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

It is convenient to posit, or "define" a (unique) point at infinity which is the inverse of 0. That way, a lot of propositions work out without extra special cases. It was misleading of the author to just say "pretend infinity is a real number".

Depends on "which numbers you use". See for example https://en.wikipedia.org/wiki/Riemann_sphere where the extended complex numbers are used.

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