You write GLSL, which is a shader language. Specifically, in this case, you write code that determines the color of a pixel given its coordinates (and some other input variables).
On Shadertoy, they use WebGL to create a little rendering pipeline that in the simplest configuration renders two triangles to form a rectangle, using your GLSL code. Since it's WebGL (should be supported by all modern browsers, mobile included), that code is executed directly on your GPU.
In more complex Shadertoy scenarios, you can do multiple passes where you render to different buffers and then use those buffers as textures in other passes. One of the uses of that is keeping state between frames, which would not be possible in a purely shader-based system otherwise. For example, if you have some sort of player character you move around the world, you may use one pixel to store the position, another to store the rotation and so on. Or you can use the image output from the previous frame as the input for the next frame to compute IIR functions, simulate cellular automatons and so on. I think by now you can even render to cube maps and multiple viewports (for VR).
There's even a "sound" shader you can write that is called once and then renders a waveform for one minute and then plays that back while displaying the regular shader.
Watching this reminds me of trying to use the 'golden ratio' and other mathy formulas in CSS typography and column grids to get "perfect" alignments from scratch. For ex from film: arranging the body to the min(d1, d2) alignment to the head sphere.
But obviously this is on a whole other level of knowledge and complexity.
Anyone know of graphing software on the computer that can graph arbitrary inequalities and logical expressions like the HP Prime's advanced graphing can? [0] For example, if you say x^2 = y it will draw a sideways parabola, top and bottom (yes, it's not a function). Or if you say 10 = x^2 + y^2 it will draw a circle without any gaps. Or if you say
it draws something like a chain link fence. [1] It works well on a battery operated calculator, so a desktop computer should have no trouble with it, but does the software exist?
On HN a few days ago, I saw a Windows program which seemed like it would do as you describe (and much more.)
But - curses - I didn't bookmark it, at least not in the normal place. I'll look a little harder for you. If I don't update/follow up on this comment, assume I didn't find it. :(
EDIT: Aha. It's called "EvalDraw." Cheers to user 'TeMPOral for sharing.
Yup, I usually used EvalDraw in (x,y) mode to graph inequalities. You have to remember though that equations like x=y or x-y = 0 represent a line - an infinitely thin object - so EvalDraw will likely not draw you anything given such input. You have to translate your equation into something that has thickness. E.g. an f(x) = 0 into abs(f(x)) < 0.01
For instance, GP's example:
(x,y)
abs(sin((1+sin(x+y))asin(.5(sin(x)+sin(y)))+(1+sin(x-y))asin(.5(cos(x)+cos(y)))))<0.01
Draws a beautiful red chain-link fence on blue background.
If you can rewrite your formula to give you a value between 0 and 1, you'll get a nice color gradient. And then you can have more direct control over colors using some of the things I described in the linked comment.
If you have a mac the built in Grapher.app (in Applications/Utilities) can graph those things. I pasted your chain-link-fence example and it graphs fine
GrafEq [1] from Pedagoguery Software is one of best-known softwares that can handle a variety of implicit 2D plots (it's a shareware, though). The algorithm for its earlier version is based on generalized interval arithmetics and described in the author's thesis [2]. I like it so much that reimplementing it is one of my (ever-growing) longer-term projects...
Have you tried the Wolfram Alpha app? I didn't try all of the examples you posted, but graphing a circle works fine.
I use this app all the time to help me simplify expressions (I'm an engineer - I use math, I love math, but I am no mathematician, so I seek shortcuts where I can to save time).
Desmos and GeoGebra can both do this (though GeoGebra has little gaps in this particular example, and you'll need arcsin rather than asin for both tools). If you want programming-oriented, Sympy can do an okay job (asin), as can Numpy (arcsin), but the latter involves more mucking about, evaluating on a grid.
I absolutely adore this space. Adding the time dimension to graphing tools is crazy powerful. I've done something similar a while back that allows you to produce sound (shameless plug): https://noise.sh (and previous discussion: https://news.ycombinator.com/item?id=24441518)
