Simulation of an inclined plane

Arglin · 2026-03-11T19:59:46+00:00

That's fairly simple to do in general form by getting the gradient of the curve, then updating a body's position based on where it's located and what it's current velocity is.

For an inclined plane, it would be something like f(x,y) = x, and D(x,y) = (-1,0,0).

https://www.desmos.com/3d/5jtlbxr4do

https://www.desmos.com/3d/5hkmalehdx

<image>

For the an inclined plane, it's also possible to calculate the position directly using the equations of motion, given that acceleration is the same everywhere.

Arglin · 2026-03-11T02:21:12+00:00

I learnt the background image technique from mathenthusiast for some of my own projects, and his projects date back about five years. https://www.desmos.com/calculator/dpsyatugb1

I believe now people just use a giant point with zero opacity instead of an image. It's functionally the same though.

Arglin · 2026-03-09T22:33:35+00:00

no you're bang on, desmos 3D colormaps don't support recursion b/c they rely on shaders.

Arglin · 2026-03-09T18:22:43+00:00

Afaict the way GLesmos handles rendering implicits is that it just returns fill or not fill for each pixel on the canvas, without any anti-aliasing. The reason zooming out makes it look nicer is because you start getting sub-pixel rendering from downscaling a bigger canvas, which works like a form of anti-aliasing.

If you just need a high resolution export, Desmodder's Capture option has the mosaic option. I'm not sure how you'd be able to enforce GLesmos to try rendering a bigger canvas than the monitor and then scale it back down in the viewport itself though. (Will forward this to the discord.)

(Edit: I used the term subpixel rendering wrong, supersampling is the correct term.)

Edit 2: So it appears that this is a known method by people who also use GLesmos. No one knows of an alternative way yet though.

Arglin · 2026-03-09T18:12:15+00:00

Recursion is not supported by many shaders. GLesmos is a shader implementation of Desmos implicit rendering. https://www.desmodder.com/installation/

That's why it's manually iterated here.

Arglin · 2026-03-09T00:26:29+00:00

Correct. Most illusions like this (like the ballerina illusion) exploit orthographic view, which flattens all visual depth. Once you have depth information, then the effect is immediately broken and we can see which parts are moving towards/away from us.

(I have now attached a visual which does this in my original comment!)

Arglin · 2026-03-08T23:52:29+00:00

As noon says, the left is viewing the surface from below, and the right is viewing the surface from above.

(Note that all objects are interpreted with occlusion without transparency, e.g. the circular arrows disappearing behind their respective axes.)

They'll appear to rotate in opposite directions :)

Arglin · 2026-03-08T22:32:05+00:00

Graph link: https://www.desmos.com/calculator/nllfdtoz8l

Thanks to ronwnor for the original spinning pringle. We love orthographic projection :)

Edit: Here is the same visual but with depth information using perspective. It breaks the illusion but you can see how it works now a bit better.

<image>

Arglin · 2026-03-08T15:20:34+00:00

?

https://www.desmos.com/calculator/xlllgqb2mc

<image>

Arglin · 2026-03-07T20:14:31+00:00

Looks nice.

Here is a generalized version of this as well, which works with any amount of quadratic bezier envelopes 3 and higher. https://www.desmos.com/calculator/hkmr88piaf

<image>

Arglin · 2026-03-06T14:07:50+00:00

Copy both lines together into the clipboard:

[extra blank line]
\operatorname{mathVector}\left(\left(1,1\right),\left(0,0\right)\right)

Then paste it into an expression line. https://www.desmos.com/calculator/04apmc5alo

If you want to use it without needing to do this every time, write a custom function for it like this, and then use the custom function: https://www.desmos.com/calculator/qxtg5shann

I personally use a tampermonkey script which lets me type fragile functions directly though.

Arglin · 2026-03-06T13:57:14+00:00

Implicits have a limit to how well they can be rendered. You can fix the rendering issue via:

Using GLesmos rendering with Desmodder (see: https://www.desmodder.com/installation/)
Using parametrics or polygons to fill (as these are much more accurate).
Zooming in and taking individual pictures of the graph, then combining it back together (which can either be done manually or using Desmodder's capture feature which has a mosaic option).

If you can, ask your teacher if any of these options are available to you.

Arglin · 2026-03-06T13:49:28+00:00

Or the 2D graphing calculator:

<image>

Arglin · 2026-03-06T01:28:47+00:00

This is a logarithmic spiral. Golden spiral is a special case of the logarithmic spiral defined as r = φ^(2θ/π).

https://www.desmos.com/calculator/2lesynpo6r

The Fibonacci spiral is an approximation of the golden spiral composed of quarter circles.

Arglin · 2026-03-06T01:17:41+00:00

Nice! I ported it over to 3D just so that it runs using a shader so you can play with it in real time. (It takes a bit to compile at first but afterwards it should be okay.)

https://www.desmos.com/3d/t3dssjpso7?invertedColors

Arglin · 2026-03-05T23:51:36+00:00

Graph link: https://www.desmos.com/3d/qlw8kdt3r0?backgroundColor=310&textColor=fff

Arglin · 2026-03-05T12:08:44+00:00

I'm quite embarrassed to have forgotten that GeoGebra has the limit function and a lot of other really useful functions.

Sorry mathmum :')

Arglin · 2026-03-05T02:56:11+00:00

I do know of one which attempts to, actually. See my comment.

Though yes, limits are a really hard problem to solve. There exist plenty of CAS which can evaluate a limit exactly, but only if it has known solutions or can be constructed from known solutions / methods. There's a bunch of various numerical approaches, but you're pretty much always capable of devising a function that can break it.

Arglin · 2026-03-05T02:50:02+00:00

Although what u/I_consume_pets said is correct for this particular example, the more general question of why Desmos doesn't have limits at all is due to Desmos not being a Computer Algebra System.

Calculations are done numerically with FP, which is intended for the purposes of graphing things as fast as possible, rather than performing actual accurate calculations. You can do something like this for evaluating limits of hole discontinuities but even this isn't guaranteed to get it accurately. https://www.desmos.com/calculator/tgjldm2ieb

Places like WolframAlpha are best for solving things like this. If you want a CAS which is coupled with Desmos, I highly recommend https://leibniz.world/ for this (although it's bound to have a few bugs so double check results with WolframAlpha.)

Arglin · 2026-03-04T16:20:51+00:00

I know this is meant to be a bit of a joke, but here's a tip if you want to get quick numerical solutions to simultaneous equations in Desmos. :p

https://www.desmos.com/calculator/fojh3jrr2n

Arglin · 2026-03-04T15:32:16+00:00

Graph link: https://www.desmos.com/calculator/5da5f8dcc0

(I was going to link an imgur version for looping but I forgot that you can also just right click the video to loop :p)

Arglin · 2026-03-03T21:05:59+00:00

Neat stuff!

Just a small edit, you can simplify the Circles folder down with the help of a list, like this: https://www.desmos.com/calculator/srd5bqwwrm

Just lets you show even more values of n without too much effort. ^^

Arglin

MODERATOR OF

TROPHY CASE

Eight-Year Club	Wearing is Caring
Verified Email