# The wrong way to use a signed distance function (sdf)

Disclaimer: there’s nothing wrong with using a sdf this way.

Recently, my good friend Mike Brondbjerg posted this on Twitter:

You can follow along and see how this idea is evolving into the beautiful, elegant line drawings that are the hallmark of his style.

## Spheres

Mike’s post gave me an idea, a way to introduce a concept I like to use in some of my work: signed distance functions. Sdfs are more commonly associated with raytracing and shaders, and by far the best source to learn about them in that context is Inigo Quilez, of Shadertoy fame: https://iquilezles.org/.

But there is a delightfully wrong way to use a sdf. Their primary use in raytracing and shaders is to define meshless geometry. For example, a way to draw smooth spheres without generating a ton of triangles. The use I’m showing here is the exact opposite: to generate geometry, point clouds in fact. Geometry that then needs to be processed in a conventional way before it can be rendered on the screen.

Let’s imagine we want to tackle something similar to the tweet: particles colliding with a sphere. One way to do this is by calculating the distance of the particle to the center of the sphere, let’s keep it at the origin. For a particle $\vec{p}$ at position $(x,y,z)$ , this distance is given by: ${d( \vec{p} )=\sqrt{x.x+y.y+z.z}}$. If that distance $d$ is larger than the radius $r$ of the sphere, the particle is outside the sphere, if it’s smaller then it is inside, if it’s exactly the same then it is on the surface.

$\begin{cases} d( \vec{p} )r, & \text{if } \vec{p}\text{ is outside} \end{cases}$

Using this, we can check the particles every step. As long as their distance to the sphere is larger than the radius, they’re fine. If a particle takes a step and its distance becomes smaller or equal, it has hit the sphere. A grid of particles flying into a sphere. Some have collided with the sphere, the rest is zooming off into infinity.

If the sphere is not in the origin but centered in a point $\vec{c} (cx,cy,cz)$ the distance function becomes ${d( \vec{p} , \vec{c} )=\sqrt{(x-cx).(x-cx)+(y-cy).(y-cy)+(z-cz).(z-cz)}}$. Typically, this is explained as the length of the vector going from $\vec{p}$ to $\vec{c}$. Another way to see it, that will serve us well further on, is to imagine that we shift the sphere to the origin, $\vec{c}\to\vec{0}$, and the entire space moves with it . Our particle is then moved $\vec{p}\to\vec{p}-\vec{c}$. Checking a particle at $\vec{p}$ against a sphere in center $\vec{c}$ is the same as doing it for a particle at $\vec{p}-\vec{c}$ against a sphere in the origin.

Since we can put the sphere wherever we want, we can add multiple spheres. Each particle is then tested against the different spheres one by one.

## Distance Fields

In creative coding, it is often useful to have several perspectives to look at things. The equations above can be refactored:

$\begin{cases} d( \vec{p} )-r<0, & \text{if } \vec{p}\text{ is inside} \\ d( \vec{p} )-r=0 , & \text{if } \vec{p}\text{ is on sphere}\\ d( \vec{p} )-r>0, & \text{if } \vec{p}\text{ is outside} \end{cases}$

Nothing has really changed, the equations are still the same. But what they describe is a function ${d( \vec{p} )-r}$ that separates space in three regions: one region outside the sphere, with positive values; one region inside the sphere, with negative values; and the surface of the sphere itself, where the function equals zero. This is the signed distance function of a sphere in the origin, with radius $r$ .

Testing the particles at each step is essentially the same evaluation: calculate the function at $\vec{p}$ and see how it classifies. The advantage of seeing it this way is that the signed distance function can be easily replaced, and suddenly we’re checking collisions with a box, a torus, or any of the many shapes that have a well-defined sdf, like the list Inigo keeps.

In itself, there’s nothing stopping us from using the first approach to calculate the distance of a point to some specific geometry. But I find sdfs make it more intuitive, especially when we start modifying and combining them.

Not every function is a signed distance function, there are certain requirements. I’m not going into the rigorous math details, mainly because I’m not qualified enough to pull that off. In essence, its rate of change has to correspond to what you expect for a distance. If we follow the slope of the function downwards, we expect to end up at the surface. If we take small steps towards it, the distance should decrease with small steps, not suddenly change slope or start increasing.

As a creative coder, one of the first things we think of is “add noise”, which is not a mathematically valid distance function. Adding noise to the sdf of a sphere doesn’t give us the sdf of a noisy sphere. Fortunately, as creative coders, we can choose to forego the rigor and let the mayhem surprise us. Noise might not be a valid distance function, but it’s still fun to use.

## Tracer classes

Although a lot of information and functions are available online, it might not be immediately clear how to use any of it in Processing. Typically, code is given in OpenGL Shading Language (GLSL) and the used functions aren’t always obvious. GLSL is created to deal with numbers and vectors in a unified way. A function like max(v,0.0) looks familiar but in GLSL can also work component-wise on vectors, something that Processing doesn’t handle. Transcribing sdf functions can be confusing when unfamiliar with GLSL.

In this section, we’ll be creating the code used for the images above. In the end, we will have a rudimentary framework to build on for more complex pieces. Let’s start with some convenience classes, Point and Vector.

Both are just containers for coordinates. There is no real reason why we can’t use Processing PVector for this, or why we have both Point and Vector. But for this tutorial, it is easier to talk about points and vectors with as little abstraction as possible.

Another class that will come in handy is Ray, a half-line starting at a Point origin, the direction is given by the Vector direction. On creation, direction is normalized, its length is rescaled to 1.0. The function get(t) will return a new Point on the ray, a distance t from its origin.

For our mini-framework, we need signed distance functions. We could hardwire the functions into a sdf(Point p) function and change that code every time. But, a bit of structure goes a long way to help exploration. First, we need to tell Processing/JAVA what a signed distance function is:

Don’t worry if the details on what an interface is aren’t clear. We can consider it a promise to Processing: everything we identify as a SDF will have this function signedDistance(Point p) that returns a float. It doesn’t matter what class it is precisely, how we create objects of that class, how the object calculates the distance, as long as we tell Processing it’s an SDF, it will be able to call that function. An interface can be used to define variables, to pass objects to functions, pretty much everywhere we can use a class. What we can’t do, is create a new object with new SDF().

We already encountered one signed distance function, that of a sphere at the origin with radius r, ${d( \vec{p} )-r}$. We can now implement a class that encapsulates this.

We tell Processing that this class implements SDF. In return, we need to fulfill our promise and implement signedDistance(Point p) . Everywhere Processing expects a SDF we can now pass a SphereSDF and it will work.

Similarly, we can define the sdf of a box at the origin of size X, Y, and Z.

One piece missing, the particles. We’re going to shoot particles, Tracers, into the scene along straight paths. When they hit something, they stop. Otherwise, they come to a stop after a certain distance. If our collision geometry would be defined by meshes, we could try to intersect the ray of the particles with the faces of the geometry. However, in this case, the whole point of this tutorial after all, we define the geometry by signed distance functions. So, we will be using another technique of finding collisions: sphere tracing.

Imagine we have an arbitrary collision geometry defined by a sdf and assume we have some particles starting outside this geometry. We know that in every point in space we can calculate the signed distance ${sdf( \vec{p} )}$. That distance, let’s call it $d$ , tells us how close the geometry is to the point, but it doesn’t give us a direction. The only thing we can say is that the particle can safely move in any direction over a distance $d$. In other words, we know that at that point we can put a sphere of radius $d$ and know for sure that the geometry isn’t in that sphere. In the extreme case, if the particle is moving straight towards the geometry, it might end up directly on the surface, but it will never cross it or go inside.

Take the figure below. We start at $P_{0}$ and the chosen direction is along the blue line. The black triangle and rectangle are our collision geometry. $sdf( P_{0})$ tells us it that it is safe to move anywhere in the green circle centered on $P_{0}$.

If we take a big step, the maximum safe distance, along the blue line, we end up in $P_{1}$. $sdf( P_{1})$ isn’t zero. Our direction of movement wasn’t the shortest path to the surface. We move on, this time a step of size $sdf( P_{1})$ and the particle ends up in $P_{2}$. We can repeat this until the returned distance is close to zero, or if we never hit the surface, after we reach a cutoff distance. In the figure, the fourth step takes us to $P_{4}$, on the surface.

The nice thing about this technique is that we don’t have to guess step sizes. There is no risk of taking too many small steps and wasting time, or of taking too large steps and overshoot a collision. The sdf automatically takes care of this by dynamically adapting the step size.

Our Tracer particle class looks like this:

Each Tracer starts in a Point origin, along a Vector direction. Since our particles only move in a straight line, we store them as a Ray. We also need to define when we stop tracing the particles. Numerical roundoff makes it unlikely we’ll get exactly 0.0 distance, so instead, we check if the distance becomes smaller than some value precison. If the tracer doesn’t hit, we want to stop after a certain distance, called cutoff. Just to be safe, we limit the maximum number of steps our tracer can take, MAXSTEPS.

The current state of a Tracer is held in 4 variables:

• t: the current distance traveled along ray
• closestDistance: the closest the particle has come to the surface so far
• steps: the number of steps taken so far
• p: the current position of the particle along the ray

At every step, the signed distance function is checked, and the particle is moved that distance forward along the ray. The tracing is stopped once one of three conditions is met:

• closestDistance < precision: the particle has come closer to the surface than our precision threshold: collision.
• t>=cutoff: The distance traveled along the ray exceeds the cutoff distance: no collision.
• steps>=MAXSTEPS: The number of steps taken exceeds the maximum number allowed. This should only occur when we’ve made a mistake in the code.

In any case, at the end of the trace, the particle is either on the surface or beyond our region of interest.