Updates:
- 2024/01/22: published without much reflection & conclusion as research is ongoing
- 2023/12/02: adding more examples and refining details
- 2023/10/22: start writeup
Introduction
While studying continuous fiber 3D printing and its main nature is to find ways to lay fiber without interruption. In order to refresh my memory I revisited the Lissajous forms, which until recently only knew in their 2D form, the swirling strings or lines – and now extending it into 3D as well.
The main idea is to realize how a line, string or fiber can be used to fill non-planar and circumvent a 3D structure and how angular shifting in Lissajous context affects such form.
3D Lissajous
- angle: 0 .. 2pi or 0 .. 360°
- p, n, m: 0 .. 1000, the amount of loops
- phi0, phi1, phi2: the angular offsets 0 … 2pi or 0 .. 360°
- X = sin(angle*p+phi0)*r
- Y = sin(angle*n+phi1)*r
- Z = sin(angle*m+phi2)*r
I did a lot of experimenting – I could post hundreds of forms – but let me focus on one a bit closer, which got my attention:
![](https://xyzdims.com/wp-content/uploads/2023/10/Screenshot-from-2023-10-11-14-11-48-1024x835.png)
![](https://xyzdims.com/wp-content/uploads/2023/10/Screenshot-from-2023-10-11-14-11-54-1024x835.png)
![](https://xyzdims.com/wp-content/uploads/2023/10/Screenshot-from-2023-10-12-05-48-28-1024x835.png)
![](https://xyzdims.com/wp-content/uploads/2023/10/Screenshot-from-2023-10-12-05-48-38-1024x835.png)
It is a very interesting transition, 8/13/21 with phi0=0° gives almost a cube-like structure, and shifting the X loop to 90° we get a tetrahedron:
![](https://xyzdims.com/wp-content/uploads/2023/10/20231014_094133-1024x576.jpg)
80mm side length
![](https://xyzdims.com/wp-content/uploads/2023/10/20231014_094105-1024x576.jpg)
80mm side length
![](https://xyzdims.com/wp-content/uploads/2023/10/20231014_094403-1024x576.jpg)
p=8, n=13, m=21, phi0=0° & 90°
![](https://xyzdims.com/wp-content/uploads/2023/10/20231023_054817-1024x576.jpg)
p=8, n=13, m=21, phi0=0° & 90°
Spherical Lissajous
While playing with 3D Lissajous, I thought to adapt the cyclic nature, but apply it to a circle laying in the XY plane and then rotate in X axis, and Y axis as well, and optionally cyclic translation as well:
- d: diameter
- angle: 0 .. 2pi or 0 .. 360°
- p: amount of loops as in X=sin(angle*p)*d/2, Y=cos(angle*p)*d/2
- q: amount of X rotations: rotateX(angle*q)
- r: amount of Y rotations: rotateY(angle*r)
The model was printed with MSLA white resin at XYZ 50um resolution with 120mm diameter, with a few support structures near the bottom:
![](https://xyzdims.com/wp-content/uploads/2023/10/20231020_133002-1024x576.jpg)
MSLA printed at 120mm diameter
![](https://xyzdims.com/wp-content/uploads/2023/10/20231020_133956-1024x576.jpg)
![](https://xyzdims.com/wp-content/uploads/2023/10/20231020_133043-1024x576.jpg)
closeup 1
![](https://xyzdims.com/wp-content/uploads/2023/10/20231020_133122-1024x576.jpg)
closeup 2
![](https://xyzdims.com/wp-content/uploads/2023/10/20231020_133205-1024x576.jpg)
closeup 3
![](https://xyzdims.com/wp-content/uploads/2023/10/20231020_134057-1024x576.jpg)
Spherical Lissajous with Translations
Using the Spherical Lissajous and extend it slightly:
- [A,B,C]loop/offset/radius: translate([ sin(angle*AL+AO)*AR], sin(angle*BL+BO)*BR], sin(angle*CL+CO)*CR ])
which spreads the ribbons away from the spherical surface origins.
Spherical Lissajous 5.11 AL=3, AR=5
It’s symmetric X- and Z-wise, in Y-axis it isn’t.
The model was printed with MSLA white resin at XY 35um / Z 50um, at 60mm in Z height, ~94mm width; with a some support structures:
![](https://xyzdims.com/wp-content/uploads/2023/10/20231026_183134-1024x576.jpg)
with spreading struts
![](https://xyzdims.com/wp-content/uploads/2023/10/20231026_175502-1024x576.jpg)
![](https://xyzdims.com/wp-content/uploads/2023/10/20231026_175611-1024x576.jpg)
![](https://xyzdims.com/wp-content/uploads/2023/10/20231026_175641-1024x576.jpg)
![](https://xyzdims.com/wp-content/uploads/2023/10/20231026_183315-1024x576.jpg)
with spreading struts
Spherical Lissajous 11.15 AL=2, AR=5
A more elaborate form is 11.15 AL=2, AR=5:
So, there is no X-, Y- or Z-wise symmetry.
The model was printed with MSLA white resin at XY 35um / Z 50um, at 60mm in Z height, ~94mm width; with a some support structures:
![](https://xyzdims.com/wp-content/uploads/2023/10/20231023_092158-1-1024x576.jpg)
with spreading struts
![](https://xyzdims.com/wp-content/uploads/2023/10/20231023_092301-1024x576.jpg)
closeup 1
![](https://xyzdims.com/wp-content/uploads/2023/10/20231023_092305-1024x576.jpg)
closeup 2
![](https://xyzdims.com/wp-content/uploads/2023/10/20231023_092309-1024x576.jpg)
closeup 3
![](https://xyzdims.com/wp-content/uploads/2023/10/20231023_092350-1024x576.jpg)
with spreading struts
and printing it larger with ~200mm width with manually positioned support:
![](https://xyzdims.com/wp-content/uploads/2024/01/Screenshot-from-2024-01-22-08-22-46-1024x810.png)
![](https://xyzdims.com/wp-content/uploads/2024/01/20240122_080803-1-576x1024.jpg)
![](https://xyzdims.com/wp-content/uploads/2024/01/20240122_111359-1-1024x576.jpg)
That’s it (for now).