**Updates**:

**2021/04/09**: spherical slicing redone, slightly better than before**2021/04/03**: cylindrical & spherical slices added**2021/03/26**: refocusing to non-planar slicing with planar slicing**2021/03/14**: starting write-up with basic illustrations

## Introduction

After discovering the 4-axis Rotating Tilted Nozzle (RTN) and its prototype of RotBot as developed by ZHAW and their conic slicing method, it became clear to me a **5-axis 3D printer**** with variable tilting nozzle** is the way to go as it is a superset of 4-axis and 3-axis 3D printing.

With that in mind, I realized there was time to explore **non-planar slicing with planar slicers** in more details.

## Slicing Methods

Let’s provide an overview of various slicing methods:

### Horizontal Slices

Vertical slicing creates horizontal slices, the traditional aka **planar slicing** method, so issues and limitations are well known:

- simple to slice
- only challenge is to create support structure for overhangs to ensure all printed layers have layers beneath
- no collision detection needed, as all already printed layers are beneath

### Tilted Slices

Tilted slices are kind of new(er) and became known with belt printers, usually 45° tilted:

Transformation is [ x, y, z – y ]

- simple to slice
- belt-printer: no collision detection is needed
- can print 90° overhangs in one direction only

There are patches for Cura available to slice for belt printer, additional the experimental Slicer4RTN also provides tilted slicing.

### Conical Slices

New slicing method as introduced by ZHAW researchers and announced in 2021/01 utilizing planar slicer:

- requires a center of the conic layers
- can print 90° overhangs, two distinct modes: inside out (outside cone), or outside in (inside cone) depending on direction to a central slicing cone center
- requires rotating and tilted nozzle aka Rotating Tilted Nozzle (RTN)
- angle of conic slicing can be changed from 45° to 20° and models become printable with vertical nozzle with reduced print quality

Transformation is [ x, y, z + sqrt( x^{2} + y^{2}) ]

I implemented a conic slicer named Slicer4RTN in 2021/03. There are more complex conic transformations possible, e.g. map the x/y angle via atan(y/x) but just adding sqrt(x^{2} + y^{2}) to z does achieve a conic slice.

### Cylindrical Slices

Early tests using planar slicers to slice also cylindrical, like this:

Transformation is [ atan(y/x), z, sqrt(x^{2} + y^{2}) ]

It can be printed on a fixed vertical rod, with a rotating and tilting nozzle, or horizontal rotating rod (like a lathe) and vertical nozzle then:

I came up with this way by myself based on the study on conic(al) slicing but I was made aware researchers Coupek, Friedrich, Battran, Riedel back in 2018 published a paper on this method already.

### (Hemi-)Spherical Slices

Early tests using planar slicers to slice also spherical, like this:

Transformation is [ sqrt(x^{2} + y^{2} + z^{2}), atan(y/x), atan(z/sqrt(x^{2} + y^{2})) ]

It can be printed with a 5-axis like PAX printhead, it’s main advantage is to getting close to print continuous overhangs of any angle.

I suspect at least one more suitable and simpler sphere transformation, as soon I came up with such I add it on this blog-post.

## Conclusion

It is possible to slice non-planar with planar slicers by **mapping to and from the space** of the slicing you like to have; yet in the slicing procedure some margins are introduced which need to be compensated – the planar slicer needs to work reliable, **Slic3r** **1.2.9** and **CuraEngine 4.4.1 / cura-slicer**** perform reliable**, whereas PrusaSlicer 2.1.1 makes assumptions of the printability and exits when no printable G-code can be produced, not recommended for this case therefore.

The simpler the transformation forward and backward, the more precise G-code can be obtained, e.g. tilted and conic slices provide precise G-code, whereas cylindrical and spherical slices are harder to tune with the planar slicer.

## References

- Conic Slicing for Rotating Tilted Nozzle (RTN)
- Multi-Axis Printing & Overhangs
**Papers**- Cylinder Slicing: [Coupek, Friedrich, Battran, Riedel] Reduction of support structures and building time by optimized path planning algorithms in multi-axis additive manufacturing (2018) – free PDF
- Conic Slicing: [Wuethrich, Elspass, Bos, Holdener] Novel 4-axis 3D printing process to print overhangs without support material (2020) – non-free PDF, does not reveal actual transformation for conic slicing; future paper in 2021 will explain conic slicing though