This tutorial explains how to create complex 3D shapes inside volumetric shaders. **Signed Distance Functions** (often referred as **Fields**) are mathematical tools used to describe geometrical shapes such as sphere, boxes and tori. Compared to traditional 3D models made out of triangles, signed distance functions provide virtually **infinite resolution**, and are amenable to geometric manipulation. The following animation, from formulanimation tutorial :: making a snail, shows how a snail can be created using simpler shapes:

- Introduction
- Part 1. SDF Sphere
- Part 2. Union and Intersection
- Part 3. SDF Box
- Part 4. Shape Blending
- Part 5. Smooth Union
- Part 6. SDF Algebra
- Conclusion

You can find here all the other posts in this series:

- Part 1: Volumetric Rendering
- Part 2: Raymarching
- Part 3: Surface Shading
**Part 4: Signed Distance Fields**- Part 5: Ambient Occlusion
- 🚧 Part 6: Hard and Soft Shadows
- 🚧 Part 7: Volume Raycasting

### Introduction

The way most modern 3D engines – such as Unity – handle geometries is by using triangles. Every objects, no matter how complex, must be composed of those primitive triangles. Despite being the de-facto standard in computer graphics, there are objects which cannot be represented with triangles. Spheres, and all other curved geometries, are impossible to tessellate with flat entities. It is indeed true that we can approximate a sphere by covering its surface with lot of small triangles, but this comes at the cost of adding more primitives to draw.

Alternative ways to represent geometries exist. One of this uses **signed distance functions**, which are mathematical descriptions of the objects we want to represent. When you replace the geometry of a sphere with its very equation, you have suddenly removed any approximation error from your 3D engine. You can think of signed distance fields as the SVG equivalent of triangles. You can scale up and zoom SDF geometries without ever losing detail. A sphere will always be smooth, regardless how close you are to its edges.

Signed distance functions are based on the idea that every primitive object must be represented with a function. It takes a 3D point as a parameter, and returns a value that indicates how distant that point is to the object surface.

### SDF Sphere

In the first post of this series, Volumetric Rendering, we’ve seen a hit function that indicates if we are inside a sphere or not:

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bool sphereHit (float3 p) { return distance(p,_Centre) < _Radius; } |

We can change this function so that it returns the distance from the sphere surface instead:

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float sdf_sphere (float3 p, float3 c, float r) { return distance(p,c) - r; } |

If sdf_sphere returns a positive distance, we’re not hitting the sphere. A negative distance indicates that we are inside the sphere, while zero is reserved for the points of the space which actually make up the surface.

### Union and Intersection

The concept of signed distance function was briefly introduced in Raymarching tutorial, where it guided the advancement of the camera rays into the material. There is another reason why SDFs are used. And it is because they are amenable to composition. Given the SDFs of two different spheres, how can we merge them into a single SDF?

We can think about this from the perspective of a camera rays, advancing into the material. At each step, the ray must find its closest obstacle. If there are two spheres, we should evaluate the distance from both and get the smallest. We don’t want to overshoot the sphere, so we must advance by the most conservative estimation.

This toy example can be extended to any two SDFs. Taking the minimum value between them returns another SDF which corresponds to their union:

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float map (float3 p) { return min ( sdf_sphere(p, - float3 (1.5, 0, 0), 2), // Left sphere sdf_sphere(p, + float3 (1.5, 0, 0), 2) // Right sphere ); } |

The result can be seen in the following picture (which also features few other visual enhancements that will be discussed in the next post on Ambient Occlusion):

With the same reasoning, it’s easy to see that taking the maximum value between two SDFs returns their intersection:

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float map (float3 p) { return max ( sdf_sphere(p, - float3 (1.5, 0, 0), 2), // Left sphere sdf_sphere(p, + float3 (1.5, 0, 0), 2) // Right sphere ); } |

### SDF Box

Many geometries can be constructed with what we already know. If we want to push out knowledge further, we need to introduce a new SDF primitive: the half-space. As the name suggests, it is nothing more than just a primitive that occupies half of the 3D space.

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// X Axis d = + p.x - c.x; // Left half-space full d = - p.x + c.x; // Right half-space full // Y Axis d = + p.y - c.y; // Left half-space full d = - p.y + c.y; // Right half-space full // Z Axis d = + p.z - c.z; // Left half-space full d = - p.z + c.z; // Right half-space full |

The trick is to intersect six planes in order to create a box with the given size s, like shown in the animation below:

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float sdf_box (float3 p, float3 c, float3 s) { float x = max ( p.x - c.x - float3(s.x / 2., 0, 0), c.x - p.x - float3(s.x / 2., 0, 0) ); float y = max ( p.y - c.y - float3(s.y / 2., 0, 0), c.y - p.y - float3(s.y / 2., 0, 0) ); float z = max ( p.z - c.z - float3(s.z / 2., 0, 0), c.z - p.z - float3(s.z / 2., 0, 0) ); float d = x; d = max(d,y); d = max(d,z); return d; } |

There are more compact (yet less precise) ways to create a box, which take advantage of the symmetries around the centre:

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float vmax(float3 v) { return max(max(v.x, v.y), v.z); } float sdf_boxcheap(float3 p, float3 c, float3 s) { return vmax(abs(p-c) - s); } |

### Shape Blending

If you are familiar with the concept of **alpha blending**, you will probably recognise the following piece of code:

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float sdf_blend(float d1, float d2, float a) { return a * d1 + (1 - a) * d2; } |

It’s purpose is to create a blending between two values, d1 and d2 , controller by a value a (from zero to one). The exact same code use to blend colours can also be used to blend shapes. For instance, the following code blends a sphere into a cube:

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d = sdf_blend ( sdf_sphere(p, 0, r), sdf_box(p, 0, r), (_SinTime[3] + 1.) / 2. ); |

### Smooth Union

In a previous section we’ve seen how two SDFs can be merged together using
min. If it is true that SDF union is indeed effective, it is also true that its results is rather unnatural. Working with SDFs allows for many ways in which primitives can be blended together. One of this technique, **exponential smoothing** (link: Smooth Minimum), has been used extensively in the original animations of this tutorial.

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float sdf_smin(float a, float b, float k = 32) { float res = exp(-k*a) + exp(-k*b); return -log(max(0.0001,res)) / k; } |

When two shapes are joined using this new operator, they merge softly, creating a gentle step that removes any sharp edge. In the following animation, you can see how the spheres merges together:

### SDF Algebra

As you can anticipate, all those SDF primitives and operators are part of a signed distance function algebra. Rotations, scaling, bending, twisting… all those operations can be performed with signed distance functions.

In his article title Modeling With Distance Functions, Íñigo Quílez has worked on a vast collection of SDFs that can be used as primitive for the construction of more complex geometries. You can see some of them by clicking in the interactive **ShaderToy** below:

An even larger collection of primitives and operators is available in the library **hg_sdf** (link here) curated by the MERCURY group. Despite being written in GLSL, the functions are easily portable to Unity’s Cg/HLSL.

### Conclusion

The number of transformations that can be performed with SDFs is virtually endless. This post provided just a quick introduction to the topic. If you really want to master volumetric rendering, improving your knowledge of SDFs is a good starting point.

#### Other Resources

- Part 1: Volumetric Rendering
- Part 2: Raymarching
- Part 3: Surface Shading
**Part 4: Signed Distance Fields**- Part 5: Ambient Occlusion
- 🚧 Part 6: Hard and Soft Shadows
- 🚧 Part 7: Volume Raycasting

⚠ Part 6 and 7 of this series are available, but yet to be completed!

If you are interested in volumetric rendering for non-solid materials (clouds, smoke, …) or transparent ones (water, glass, …) the topic is discussed in the Atmospheric Volumetric Scattering series!

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You are free to use, adapt and build upon this tutorial for your own projects (even commercially) as long as you credit me.

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Is there a way to go about building larger scenes comprised of many SDF’s without endless nested unions?

Wouldn’t it be more correct to call the sdf_blend function a linear interpolation also commonly referred to as lerp? Wikipedia has an example like this and I guessing it would give the same result?:

float lerp(float v0, float v1, float t) {

return (1 – t) * v0 + t * v1;

}

Hey!

Yes, indeed it would. But I wanted to use names that were resonating more with 3D modelling, than Maths. :p

Seems like there is a mistake in the sdf_box function. It should use the input variable c and not _Centre as the center position?

Indeed! Well spotted, thank you! <3

There's a problem with my website and I can't change the page right now unfortunately.

Hopefully I'll remember to do it when is all done!

In the sdf_box function, is the following statement correct?

float x = max

( p.x – c.x – float3(s.x / 2., 0, 0),

c.x – p.x – float3(s.x / 2., 0, 0)

);

How can a vector (float3) can be subtracted from scalars (p.x and c.x)?

I’d like to point out a fairly important thing about SDFs that is not covered by any introduction article I found online.

An SDF describes the distance between a point and a shape. Among the functions you introduced above, only the sphere and half plane are correct SDFs.

Let’s consider the box: intersecting 6 half planes is NOT equivalent to the point-box distance. If the point is outside of a corner of the box, its distance will be the diagonal line connecting it to the vertex. However, intersection will only get the maximum of the three distances along the three axes.

Why do we care? If we just need to test a point against an SDF, we don’t care. We just care if a point is inside or outside, we don’t care about the correct distance. Following this, we could just have the function return a boolean, and maybe we could squeeze in some optimizations (for example, compare the squared length in the Sphere function, instead of the more expensive length which requires a square root).

So, when do we care about an SDF being correct? We do if the SDF is then used for further processing. For example: blending!

If you blend two SDF where at least one is wrong (i.e. sphere and box described in the article), the blended result will not be exactly right! Maybe it will look ok, but not quite right! If you were to spend time writing a mathematically correct box SDFs and lerp it with a sphere, you would see how better the transition looks!

That being said, I also want to list which functions are conservative (i.e. if two input SDF are correct, the result is correct) and which arent:

Union: Conservative!

Intersection: Not Conservative! (that’s why intersecting 6 correct half planes does NOT give you a correct Box!)

Blend: Not conservative!

Be careful to use intersection and blending if you plan to use other SDF operations afterwards!

One final note: if you have a correct SDF, you can also check the intersection with a sphere! If the SDF is not correct, the distance distribution will be wrong and that wont work.

I hope this was useful to read, I might wrap it up in an article more nicely, as I discovered most of these things the hard way while working on a project, and it could save other people pain and effort!

Thank you, this is a very interesting topic!