Maths question about circles

nm0 · September 25, 2020, 11:20pm

Hello,

I have what I hope is a trivial maths question… I am writing a spatialisation process with spat in max. I simply would like to have a source move smoothly between a randomly selected loudspeaker e.g. 1-6 at a certain speed.

The patch design simply outputs one of the 6 loudspeaker positions at random every time there is a bang, generated by a metro object. The speaker positions form a hexagon, at -30 30 90 150 -150 -90 degrees. I use a line object to make the source glide between speaker positions with a time interval related to the speed of the bang interval. Patch copied below.

The problem is that the line object and list of positions are only 1D and so the line moves only back and forth along the list of speaker positions i.e. it doesn’t always take the shortest path on the circle e.g. between speakers 4 and 5 at 150 and -150 degrees it goes anticlockwise 300 degrees rather than clockwise 60 degrees.

Is there an alternative paradigm to work in a more circular manner that would enable the system to always glide the shortest path between speakers? I’m hoping it’s a trivial problem and I’m just a little ignorant with my bad teenage maths knowledge.

It would be great to know an alternative paradigm that might work for larger speaker arrays, and eventually in 3D too.

Thanks!
Nick


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tcarpent · September 26, 2020, 9:54am

Hi Nick,

Just some hints (I haven’t tried to actually implement it) :
you want to use the shortest path (i.e. shortest arc length). So, at some point, you need a “shortest path”-aware line.
For that, you need to keep track of the last output value from line. Compare that last value to the next (target) azimuth.
If difference is greater than 180, you need to modulo (%360) the start/end azimuths of the line.
(NB : incidentally, you could also adapt the duration of the line so that the trajectory has a constant angular speed)

I believe you can keep your p speaker-strobe unchanged; just modify the line part of your patcher.

Even though the underlying math is rather simple, you might want to opt for a js implementation, for the sake of readibility/maintenance ?

Extension to 3D is a little more complex, as you need to use the great-circle distance.

Hope this helps,
T.

nm0 · September 26, 2020, 1:18pm

Thanks T,

I think I understand the principle but not the next step. If I modulo 360 the start/end points when the difference between the last and next value is greater than 180 then I’ll still just get the same values from before, -150 to 150, as they’re all less than 360. Did I misunderstand the hint?

Are you saying that whenever the difference is greater than 180 the points should be transformed e.g. -150 to 150 becomes 210 to 150, 150 to -90 becomes 150 to 270 etc? This seems to be a solution but I’m not sure how to implement it in max.

I know I could rewrite the speaker points as positive integers, 30 90 150 210 270 330 but then I’d have the same problem with the source never moving over 0 degrees.

Or, is there a way to move a source by instruction e.g. move 210 degrees clockwise? This could bypass the problem because I wouldn’t need to reconcile absolute positions within a circle.

I made a workaround that makes the previous patch take the shortest route from speakers 4 to 5, but then it’s v inelegant to apply this to the other affected paths (3-5, 5-3, 5-4, 4-6, 6-4), and besides I’d like to work it out so that it can be used more flexibly for different speaker layouts, not just this hexagon.

I don’t know js yet…

Thanks again!
Nick

nm0 · September 26, 2020, 9:13pm

I have found a workaround for the specific hexagon layout where I just substitute different values whenever the difference between the last two positions exceeds 180, as as hinted at in T’s suggestion, patch below (a little messy). Now the source takes the shortest path between positions.

It would be cool to know if there is a more flexible or smarter way to implement this so that it would work with any number of speakers and not rely on the workaround hand-written for the hexagon layout. I don’t know which bit of circle geometry I’d have to study to work it out myself, I’m no engineer

Cheers
Nick


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tcarpent · September 26, 2020, 9:50pm

Here is a quick and dirty example using javascript.
Could definitely be improved.

(should work with arbitrary number of speakers…)

Hope this helps,
T.

strobe-example.maxpat (11.2 KB) strobe.js (2.1 KB)

nm0 · September 27, 2020, 8:41am

Thanks T,

When I try to download the js file I get taken to the following error page:

The change you wanted was rejected.
Maybe you tried to change something you didn’t have access to.

It doesn’t work if I try to save the linked address too.

Some kind of problem with sharing js files over the forum?

smalllotus · September 27, 2020, 4:05pm

Hi Thibaut,

there’s really a trouble for downloading the script…

I’m thrilling, waiting for fix !

@both : Really interesting topic !

tcarpent · September 27, 2020, 5:00pm

Let’s give it another try :

smalllotus · September 27, 2020, 7:01pm

Thank you very much !!!

Nice work

nm0 · September 27, 2020, 9:31pm

thanks T. I can see that js is easier to use for these kinds of problems.

Am I correct in thinking that your script can only work with the lists always read in sequence (normal or reverse) / only in one direction, it can’t change direction itself?

I’m trying to work out just with Max a more universal approach for a list of n positions in any sequence/ at random, but I don’t feel like I have the chops to conceptualise the problem and solution properly. So far, I think if I still want to use a Line as the core object like above then it comes down to making a patch that can switch between giving the next value as either positive or negative to get the shortest path depending the last value and the difference with the next. I think this means a patch has to incorporate the below 8 possible operations to fully cover all the possibilities:

next value is positive when last value was positive to move clockwise (0-360)
nv is + when lv was + to move anticlockwise
nv is + when lv was - to move clockwise
nv is + when lv was - to move anticlockwise
nv is - when lv was + to move clockwise
nv is - when lv was + to move anticlockwise
nv is - when lv was - to move clockwise
nv is - when lv was - to move anticlockwise

There is some redundancy to account for all the possible outcomes. Maybe this is overthinking it, or not the right way, but I’ll try more this week to make something that works.

I’d be grateful if anyone has some tips or can just magic up the solution, if it’s a trivial maths problem.

tcarpent · September 28, 2020, 9:59am

Am I correct in thinking that your script can only work with the lists always read in sequence (normal or reverse) / only in one direction, it can’t change direction itself?

Yes. The js script was not intended to be a generic solution; it just implements clockwise/anticlockwise rotations.
But feel free to use it as a starting point, and adapt it to your needs and more complex scenarios.

(I’m not a big fan of js, but usually a few lines of code can convey your intent / logic more expressively than a patcher)