**Updates**:

**2023/02/09**: finally published**2023/02/08**: worked on text and illustrations a lot, many sample prints, multiple visualization approaches, details on f1 + f2 vs f1 * f2 and cylindrical and spherical transformation of TMPS**2023/01/05**: adding mesh/voxel renderings, slicing geometry to generate G-code**2022/12/11**: first FDM G-code generated using 2D / contour approach**2022/12/07**: included many suitable periodic minimal surfaces**2022/12/02**: start with implicit surface focus

As I progress I will update this blog-post.

Table of Contents

## Introduction

Infill geometries are geometries which are continuous, repetitive or periodic; they fill a boundary defined geometry aka outer form often defined via meshs. Let’s dive into some of the simple geometries and then looking at some more complex structures:

## The Implicit Geometries

Implicit geometries are geometries defined via *f(x,y,z) = 0* defining their surface, the boundary between inside and outside and they are ideal to define repetitive or periodic 3D infill geometries.

### Sphere

Sphere: x^{2} + y^{2} + z^{2} – r^{2} = 0

When you ever tried to compose a sphere as a mesh, you know there are many ways to do so, and all are more complex than this simple description, and as you realize, the formula is perfect, it’s not an approximation – this is the nature of implicit formula. When you try to visualize an implicit formula, **then you need to discretize** and there the approximation takes place, as a mesh or as voxels.

Another nifty property of the sphere, it is the minimal surface to circumvent a volume, and through this blog-post, the minimal surface will become a common theme.

### Cube

Cube: max(abs(x),abs(y),abs(z)) – w/2 = 0

### Plane

Plane: z = 0

As I render only -10 to 10 to each axis, it creates a small plate:

## Triply Periodic Minimal Surface (TPMS)

Let’s move to the world of minimal surfaces, so called *Triply Periodic Minimal Surfaces* (TPMS), those can be expressed in implicit form and have some properties as sought for infill geometries.

In differential geometry, a

Wikipedia: Triply Periodic Minimal Surface (TPMS), retrieved 2023/02/08triply periodic minimal surface (TPMS)is a minimal surface in ℝ^{3}that is invariant under a rank-3 lattice of translations. These surfaces have the symmetries of a crystallographic group. Numerous examples are known with cubic, tetragonal, rhombohedral, and orthorhombic symmetries. Monoclinic and triclinic examples are certain to exist, but have proven hard to parametrise.

### Schwarz P aka Primitive

One of the simplest yet powerful formula:

Schwarz P: cos(x) + cos(y) + cos(z) = 0

increasing the frequency or scale of the structure:

By extending the formula with +a, we can animate it:

### Schwarz D aka Diamond

Schwarz D: sin(x)*sin(y)*sin(z) + sin(x)*cos(y)*cos(z) +

cos(x)*sin(y)*cos(z) + cos(x)*cos(y)*sin(z) = 0

### Neovius

Neovius: 3*(cos(x)+cos(y)+cos(z)) + 4*cos(x)*cos(y)*cos(z) = 0

### C(Y) Surface

C(Y) Surface: sin(x)*sin(y)*sin(z) + sin(2x)*sin(y) + cos(x)*sin(2y) + sin(2y)*sin(z) + sin(2z)*sin(x) + cos(x)*cos(y)*cos(z) + sin(2x)*cos(z) + cos(x)*sin(2y) + cos(y)*sin(2z) = 0

### Fischer Koch

Fischer Koch: (cos(x)*cos(y)*cos(z) + cos(z)*cos(x)) –

(cos(2x)+cos(2y)+cos(2z)) = 0

### S Surface

S Surface: cos(2x)*sin(y)*cos(z) + cos(2y)*sin(z)*cos(x) +

cos(2z)*sin(y)*cos(y) – 0.4 = 0

### Gyroid

Gyroid: cos(x)*sin(y) + cos(y)*sin(z) + cos(z)*sin(x) = 0

### FRD

FRD: 8 * a*cos(x)*cos(y)*cos(z) + b*(cos(2x)*cos(2y)*cos(2z)) –

c*cos(2x)*cos(2y) – d*cos(2y)*cos(2z) – e*cos(2z)*cos(2x)

Let’s explore this form more thoroughly, we animate a, b, c, d, and e and see what it does, essentially we animate -1 to 1 in sinus, 0 eliminates of the chunk of the formula:

### Gyroid Skeletal

Gyroid Skeletal: 10*cos(x)*sin(y)+cos(y)*sin(z)+cos(z)*sin(x)) –

0.5*(cos(2x)*cos(2y)+cos(2y)*cos(2z)+cos(2z)*cos(2x)) – 14

### P Skeletal

P Skeletal: 10*(cos(x)+cos(y)+cos(z) –

5.1*(cos(x)*cos(y)+cos(y)*cos(z)+cos(z)*cos(x)) – 14.6

By changing the last substraction of 14.6 to 10 or 8, the structure get more dense – ideal to use.

The P Skeletal connects 6 arms to each other.

### IWP Skeletal

IWP Skeletal connects 8 arms to each other.

### Schwarz D Skeletal

Schwarz D Skeletal connects with 4 arms to each other.

The above “skeletal” minimal surfaces are ideal for lattice structures, likely most usable in context of voxel-based 3D printing approaches, such as SLA, SLS, SLM and so forth, but less ideal for traditional FDM where the lattice is sliced Z-planar again kind of defeating the overall purpose of lattice structures.

### D Surface

D Surface: cos(x)*cos(y)*cos(z) – sin(x)*sin(y)*sin(z)

As Juergen Meier created a variant, adding **a**, which gives these variants:

providing a structure using 4 arms to connect each other.

## Miscellaneous

## Using Implicit Geometries as Infill Structures

Slic3r and Prusa Slicer are providing **gyroid infill pattern** since early version, but beyond that it seems no to little development happened since (2022/12).

Let’s see how implicit geometry can be transformed into slices (FDM) or voxels/pixels (SLA, SLS etc)

### Algorithm A: 3D Cache

- create point cloud of surface of implicit geometry
- create surface of implicit geometry using marching cube
- (optional) determine x, y, z size where it repeats itself
- slice surface for infills at certain scale
- clip inner surface with outer perimeter of slice

**Pros**

- with caching: fast lookup of infill geometry

**Cons**

- many steps
- x, y, z repeatability must be given, hard to determine programmatically from outside
- clipping to perimeter can be computational expensive depending

### Algorithm B: 2D Cache

- create 2D point cloud of a slice of implicit geometry based on clipped 2D area / slice
- convert 2D point cloud to polylines (FDM) or pixels (SLA)

**Pros**

- reduction to 2D problem at first stage
- fast 2D point cloud creation as only one z-level is used

**Cons**

- create 2D point cloud at arbitrary resolution, loss of curves unless refitted
- caching without knowing repeatability of the geometry makes little sense

## FDM G-code

Here some early G-code for FDM 3D printer using *PyImplicit* tool tracking the implicit surface as 2D contour:

## Meshs & Voxels

The implicit surfaces only define the surface, either:

**inside vs outside**– a solid; or**certain thickness**of such surface

In order to create watertight meshs the volume needs to be limited with a boundary box, and Marching Cube is performed from outside to get proper mesh to post-process afterwards.

Now you may wonder, what’s the fuss with all those forms, why doing this complicate implicit form, why not just create a few forms as meshs right away and repeat them orderly – **well, here it comes why**:

## Frequency or Scale Gradients

Changing the frequency or scale ** s_{0}** and

**can be achieved by:**

*s*_{1}*z _{norm} = (z-z_{min}) / (z_{max}-z_{min})*

*s =***or**

*(1-z*_{norm})*s_{0}+ z_{norm}*s_{1}

*s = lerp(s*_{0}, s_{1}, z_{norm})

*f = surface(x*s, y*s, z*s)*This shows the power of generative geometries, we simply can define the scale or frequency of a geometry at any point, given we transit within reason and not too sharply to cause discontinuty.

## Thickness Gardients

Alike changing thickness:

*z _{norm} = (z-z_{min}) / (z_{max}-z_{min})*

*t = lerp(t*

f = abs(surface(x,y,z)) – t_{0}, t_{1}, z_{norm})f = abs(surface(x,y,z)) – t

## Form Gradients

What looks very complex is done quite simply with:

*z _{norm} = (z-z_{min}) / (z_{max}-z_{min})*

*f = lerp(surface1(x,y,z) , surface2(x,y,z), z*_{norm})This is quite powerful property, to be able to morph from one implicit form to another with such a simple formula.

Contineous Transitions:

- Schwarz D – Schwarz P
- Schwarz D – Neovius
- Schwarz P – Neovius
- thickness: IWP Skeletal – Schwarz P
- thickness: IWP Skeletal – Schwarz D

Discontinueous Transitions:

- IWP Skeletal – P Skeletal
- IWP Skeletal – Neovius
- solid: IWP Skeletal – Schwarz P
- solid: IWP Skeletal – Schwarz D

## Combining Implicit Surfaces

### Additions

Algebraic addition has the effect of **apply one geometry within another**, alike recursion:

### Multiplications

Algebraic multiplication has the **effect of clipping, or geometrical intersection**:

## Mapping Implicit Surfaces

One can map the coordinates, and create a cylindrical gyroid, where former X & Y become distance and rotation angle, and Z remains as is, and so spherical projection is possible as well, or even feed coordinates through implicit formula itself:

Next blog-post(s) I will go into further details utilizing TPMS in Additive Manufacturing (AM) like FDM/FFF, SLA, MSLA, SLS, MJF or SLM – each one of them have unique features and limitation for using those Parametric Generative Infill Geometries.

## Appendix: Visualization

In case you wondered of the different styled visualization through this blog-post, let me show you the different approaches to discretize implicit defined surfaces.

### Voxels

The code is rather simple with OpenSCAD yet rather slow: either `skin`

is true or false, and `delta`

determines the thickness of the skin if enable:

```
t = 1;
r = 20*t;
st = 1/2;
delta = 0.2;
function schwarz_p(x,y,z,s=1) = cos(x*s) + cos(y*s) + cos(z*s);
skin = true;
for(x=[-r:st:r])
for(y=[-r:st:r])
for(z=[-r:st:r]) {
f = schwarz_p(x,y,z,360/20/2);
if(skin && abs(f)<delta) // -- skin only
translate([x,y,z]) cube(st);
else if(!skin && f<delta) // -- inside/outside
translate([x,y,z]) cube(st);
}
```

Rendered via voxelation:

### Surface

Rendered in OpenSCAD via marching cube algorithm with Level Surfaces:

### Volumes & Surfaces in OpenGL GLSL

Following experiments were done with **Spirula/Implicit3** within the browser, the implicit formulas are rendered in realtime at 100-500 fps using OpenGL’s GLSL (GL Shader Language):

One has to clip the formulas with a cube in order to have a limited set, otherwise you get a full screen looking at infinite X, Y & Z, here Schwarz P:

### Meshs with Marching Cube

In order to create a mesh, I developed *PyImplicit* which utilizes *Numpy* library to calculate the implicit formula fast, and then run a Marching Cube algorithm over the result in order to get a discrete mesh like STL, OBJ, or 3MF to process further for 3D printing.

^{*},

surface gradient between Schwarz D to Schwarz P (top) at certain thickness or solid,

2nd row: 2x IWP skeletal 90mm cubes at different frequency;

Background: various 30/40mm cube clipped Triply Periodic Minimal Surfaces

*) some of my larger prints I attach RFID tags, e.g. as on top of the variable frequency Schwarz P print, which I store the print UID from my Prynt3r job which logs all my prints with all settings and webcam snapshots. In future blog-post I will illustrate my NFC/RFID setup.

And *Polyviuw* is a small mesh viewer using *Polyscope Python* as backend to display it as mesh:

It is easy to create huge files when exporting an implicit generative infill geometry and one ends up with a 700MB binary STL file, which becomes hard to view at least on my system. To handle complex outer forms, with complex inner geometries I estimate reaching multiple gigabytes large files – let’s see.

## References

**3D-Meier.de**: Juergen Meier’s excellent write-up on*Triply Periodic Minimal Surfaces*(german only) with formulas & references**Commercial Software**:**nTopology.com**: outer and inner geometry incl. simulation**GeoDict.com**: digital material development, deep dive into designing interior geometries**Spherene.ch**: Adaptive Density Minimal Surfaces (ADMS) software framework, startup**Metafold3D.com**: volumetric design, combining interior (infill) and exterior structures (mesh boundaries); cloud API, startup

**Papers**:- [Panesar, Abdi, Hickmann, Ashcroft]
**Strategies for functionally graded lattice structures derived using topology optimisation for Additive Manufacturing**(PDF) - [Singh, Narayanan]
**Real-Time Ray-Tracing of Implicit Surfaces on the**(PDF)

GPU - [Feng, Fu, Yao, He]
**Triply periodic minimal surface (TPMS) porous structures**(PDF) - [Ntintakis, Stavroulakis]
**Infill Microstructures for Additive Manufacturing**(PDF) - and many more

- [Panesar, Abdi, Hickmann, Ashcroft]
**Github**:- Level Surfaces: OpenSCAD based 3D surface grapher (OpenSCAD)
- Spirula (WebGL), instant preview of implicit surfaces in browser using GPU
- PyMCubes, marching cubes
- Scaffolder, Python / Blender plugin mixing meshs with TPMS

**Miscellaneous**:- From Triply Periodic Minimal Surface to Architecture by Catherine Zhang
- Inigo Quilez: Distance Functions, the jewels of SDFs / implicit formula