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Ray Marching Distance Functions · Acedra
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Ray Marching Distance Functions
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0:00
Hello and welcome to another coding adventure. While I was messing around with ray-tracing last episode, I came across a technique called ray marching which lets one do some pretty cool stuff. But before we can get to the cool stuff, we need to be able to calculate distances, starting with the distance between two points. Fortunately, Pythagoras has this one figured out already. Next the distance from a point to a circle.
0:27
This is just the distance between the point and the circle Center minus the radius of that circle. By the way, this is called a signed distance function, because if the point was inside the radius of the circle the distance would actually be negative and you'll see why this is useful a little later on. But for now, I'm just going to add one more distance function, which is the distance from a point to the edge of a box.
0:48
Wherever the point travels in this world of boxes and circles we can figure out how far away it is from the nearest surface by just looping over all of the shapes in the scene, calculating their distances and returning the smallest of those values. Imagine now that this point is actually an eye, and we want to see from its point of view. So we could cast out a bunch of rays and see which surfaces they collide with.
1:10
To tell where this collision occurs, we could write some kind of intersection function, but that would limit us to using only shapes whose intersection we know how to calculate. To get around this we'll start by calculating the distance to the scene. Which gives us a radius in which we can safely move our point along the ray without it going through anything. We then simply repeat this process until
1:30
the distance becomes really, really small, which means we've collided with something. So ray marching really just means: moving a ray along in small steps, until you find an intersection. The example I'm showing here is a type of ray marching called sphere tracing, which speeds things up by using the radius of this sphere or circle that we're calculating as the step size. If you extend this idea to three dimensions and
1:53
manage not screw it up in the process, like I did, you can then render cubes, spheres, and really any other shape that you can define a distance function for. With this representation, we can do some interesting stuff. For example, take this function, which calculates the distance to a cube and a sphere and returns whichever is smallest. In other words the min of the two values. If I change it to the max of the two values, suddenly both objects disappear.
2:19
This is because the surface of the shape is where the distance is zero and the only way the maximum of two values is going to be zero or less is if both values are zero or less. In other words, we've created a new shape that only exists at the intersection of the cube and the sphere. If I then multiply the distance to the sphere by negative 1 this will essentially invert the sphere, making
2:44
the outside of it solid and the inside hollow. So the shape now only exists at the intersection of the cube and the outside of the sphere. Meaning, wherever the inside of the sphere touches will get cut away. This ability to invert a shape is one of the reasons why it's useful for our distance function to be negative on the inside. Another cool thing we can do is replace the min function with a smooth version. For example, if we have two equations this would be the min of them.
3:14
But a smooth min would look like this. The result of this is that objects that are close together will be blended together, which I find tremendously satisfying for some reason. The smoothing can of course be dialed down to more reasonable levels, but I prefer it like this. Yet another thing is: notice how dramatically the number of ray march steps increases as we get close to the edge of a shape. We can use this data to generate
3:39
a nice glowing outline around our objects entirely free of charge. Once I had all this working, I thought it'd be fun to just mess around a bit, so I made up a distance function that takes the sign of the current position and blends that with some other nonsense. It resulted in this repetitive little world that I could fly around in, and tweaking the parameters made it all glitchy and weird. It was kind of fun to explore for a little
4:00
bit and for whatever reason this got me thinking about fractals. Since I've previously made a very underwhelming visualization of the Mandelbrot Set, I searched for some kind of 3D variation on it, and came across a distance function for the Mandel Bulb. I have no idea how it works but I plugged the code in and let it run. And I think the results is somewhat mesmerizing.
4:40
[Music] Well that was quite an adventure but I've of course only barely scratched the surface of what can be done with ray marching. If you'd like to learn more, have a look at the links in the description. You can also find my project files there, if you'd like to play with the stuff I showed in this video. Anyway, I hope you'll join me on more coding adventures in the future, but until then: Cheers
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