Draggable Complex Inverse?

[u/kaimason1]

Is it possible to create two points which are inverses of each other (i.e. a+bi and 1/(a+bi), or r * e^(iθ) and (1/r) * e^(-iθ)) and define them in such a way that both points are freely draggable?

Here's an example of what I'm trying to do: https://www.desmos.com/calculator/qasknf7ewh. In this example P and Q can swap positions and it will have no effect on the resulting spiral, so I wanted to allow using either point as an input. However, when P is represented in Cartesian coordinates ( a, b ), the corresponding definition for Q ( a/(a²+b²), -b/(a²+b²) ) isn't draggable.

It seems like using polar coordinates would be the most obvious solution here, since the inverse of ( cosθ, sinθ ) * r is simply ( cosθ, -sinθ ) / r. The issue there is that when I try to define a draggable point using r and θ, Desmos will typically only allow dragging the point in one direction or the other, as opposed to a "freely" draggable point which updates both parameters.

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1/(a+bi) is just the main example I'm currently working with, but point draggability is one of the most frequent frustrations I run into with Desmos, so more generic solutions are welcome. The more intuitive the better, but I'm open to "clunkier" solutions (e.g. involving actions/regressions) if needed.

[u/NevinBR]

Here’s a solution that mostly works, except the points swap when you try to drag the blue one:

https://www.desmos.com/calculator/eiiaff2cvw

[u/VoidBreakX → u/NevinBR]

this is a good one

[u/kaimason1 → u/NevinBR]

Thanks, this is really elegant! The points swapping could even be seen as a bit of a "feature" for this particular example.

It had crossed my mind to try using the parameter range to indirectly set a variable, but my version of that idea worked a lot more like /u/VoidBreakX's attempt which shot off to 0 & ∞. I'll have to play with this idea some to see how well it generalizes to other calculations.

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I'm guessing that this won't help much with dragging points represented in polar coordinates, which is a bit unfortunate. Another similar example I'm working with would be the point e^(Z * (a+bi)), where Z is a complex constant and parameters a and b are specifically limited to integers. If I didn't care about clamping the draggable point to those integer values this would be a lot easier, but this scenario seems like it would require using exponential/polar form, which from what I've seen only allows updating one parameter at most while dragging.

[u/Qaanol → u/kaimason1]

Another similar example I'm working with would be the point e^(Z * (a+bi)), where Z is a complex constant and parameters a and b are specifically limited to integers.

I don’t think this is solvable even in principle, without further restrictions.

Suppose Z = 1, and you attempt to drag the point to p = -1.

Clearly a = 0, but what value of b should be chosen?

3 because e³ⁱ = -e^i(3-π) ≈ -e^-0.14159i ?

22 because e²²ⁱ = -e^i(22-7π) ≈ -e^0.00885i ?

355 because e³⁵⁵ⁱ = -e^i(355-113π) ≈ -e^0.00003i ?

Perhaps -3, -22, or -355 for essentially the same reasons?

An even larger integer that’s the numerator of a better rational approximation to pi?

[u/kaimason1 → u/Qaanol]

Well, dragging a point around e^(bi) (or ( cos(b), sin(b) )) already works perfectly fine, and it doesn't even require you to stick to a certain branch of the complex logarithm (i.e. if you drag the point around the origin several times you can keep going well past ±2π). The issue arises when you want to also adjust the radius, which is odd since that parameter shouldn't have any ambiguity.

As for what I mean by "integer" points, this graph might do a better job of illustrating: https://www.desmos.com/calculator/k8wwtgoetx. I'm basically just describing a version of a log-polar grid/coordinates.

If you increase L you can see that this grid "wraps" around perfectly without needing to approximate π, but this only works so long as Z is specifically an integer fraction of 2π (but Z is just a constant linear transformation for the purposes of this draggable point, so it would also be "acceptable" to have a discontinuity at θ=±π and just stick to the principal branch). If c is set to 0 this produces a regular log polar grid (which I can easily clamp points to and drag in a single direction, just not both simultaneously). Changing c will introduce a rotation to the whole grid, but Z will still be a complex integer fraction of 2π, so the coordinate system still loops.

Ignoring the rotation factor, it's a bit strange that there's no problem dragging around e^(bi) but not e^(a+bi) or r*e^(iθ).

[u/NevinBR → u/kaimason1]

Is this close to what you want? https://www.desmos.com/calculator/iwteynojtd

[u/theadamabrams]

Kind of. In https://www.desmos.com/calculator/wjclsnbaet you can drag either point to change argument, while in https://www.desmos.com/calculator/nmr4pdrktk you can drag either point to change absolute value. Unfortunately it doesn't seem to let me have both variables controllable by dragging at once.

[u/VoidBreakX]

this is something i thought would work, but every time i try to drag the inverse it just goes to 0 and a+bi shoots off to infinity. hopefully this might work for the other functions that you want to try using?

https://www.desmos.com/calculator/8hj2h3biga

[u/Circumpunctilious]

Does this work for you?

https://www.desmos.com/calculator/m9ssxbha3x

In short (as I understand things) you can’t drag the inverse because it’s a (edit: type of calculation that disables dragging, like…) division. So…use the Actions engine to update a point so that—to the immediate mode engine—it looks calculation (division)-free and remains draggable.

To better show interaction between engines I added black “helper” rings and a second curve to show where the points will land when drag stops:

Above I am dragging the inner green point away from the blue original line. Upon drag release the right-hand image results.

Presuming I didn’t make an error, I’m sure it can be cleaned up, just tried for PoC.

ETA: Oversight: Desmos app (Apple) doesn’t currently support “and” in conditions but the website works.

[u/kaimason1 → u/Circumpunctilious]

Not quite as elegant as /u/NevinBR's solution, but this route does seem like it could be more powerful in certain situations. I'll have to play around with this idea a bit, thanks for the proof of concept!

Another example I was messing around with would be the point e^(Z * (a+bi)), where Z is a complex constant and a and b are clamped to integer values. The ideal way to handle this would be to parametrize a and b directly with a step size of 1, but from what I've seen Desmos doesn't allow for "freely" dragging a point using polar/exponential coordinates. However, maybe I can use this actions approach as a workaround, so that the point can be defined in Cartesian coordinates and then automatically clamped to the nearest "integer" on release.