Notes on 3D CAD

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# Good morning!  Hummm... all three suggested options are important.  What 'parametric nomenclature' will suffice for description for slicing into print layers?

For describing and slicing the torus into printable “bagel-slice” layers, a compact **parametric vocabulary** around the standard torus / torus-knot equations is enough.

## Core Parametric Nomenclature

Use the standard torus + torus‑knot parameters:

- **Major radius**: $R$
Distance from torus center to centerline of tube.[^1]
- **Minor radius**: $r$
Radius of the tube cross‑section.[^1]
- **Torus angle** (around the big circle): $\theta \in [0, 2\pi)$
Parameter for position along the torus axis.[^1]
- **Tube angle** (around the small circle): $\phi \in [0, 2\pi)$
Parameter for position around the tube cross‑section.[^1]
- **Torus knot integers**: $(p, q)$
For you, $(p, q) = (13, 8)$.[^2][^3]
- **Global rotation angle for each printed third**: $\alpha_k$
With $k \in \{0,1,2\}$ and $\alpha_k = k\cdot 2\pi/3$ for the three 120° “bagel slices”.

These define both the shape and how you slice it.

## Torus and Torus Knot Parametric Equations

**Torus (no knot)** in 3D:[^1]

$$
\begin{aligned}
x(\theta,\phi) &= (R + r\cos\phi)\cos\theta \\
y(\theta,\phi) &= (R + r\cos\phi)\sin\theta \\
z(\theta,\phi) &= r\sin\phi
\end{aligned}
$$

**(p, q) torus knot** (centerline of a winding groove):[^3][^2]

$$
\begin{aligned}
x(t) &= (R + r\cos(qt))\cos(pt) \\
y(t) &= (R + r\cos(qt))\sin(pt) \\
z(t) &= r\sin(qt), \quad t\in[0, 2\pi)
\end{aligned}
$$

For your 13:8: set $p = 13$, $q = 8$.

## Slicing Nomenclature for the 3 Printed Thirds

To describe the printable thirds cleanly, define:

- **Torus‑angle interval per third**:
    - Third 1: $\theta \in [0, 2\pi/3]$
    - Third 2: $\theta \in [2\pi/3, 4\pi/3]$
    - Third 3: $\theta \in [4\pi/3, 2\pi]$

or equivalently, use a **segment index** $k\in\{0,1,2\}$ and define each third as:

$$
\theta \in \left[\frac{2\pi k}{3}, \frac{2\pi (k+1)}{3}\right].
$$

You can then say in your documentation:

> “Segment $k$ is the torus patch defined by $\theta \in [2\pi k/3, 2\pi (k+1)/3]$, $\phi \in [0, 2\pi)$.”

This is precise enough for CAD scripting and for human readers.

## Nomenclature for Parallel Knot Grooves

For **parallel 13:8 windings**:

- **Base knot**: $C_0(t) = (x(t), y(t), z(t))$ as above.
- **Parallel copy index**: $n \in \{0,1,\dots,N-1\}$.
- **Phase‑offset parameter**: $\delta_n$ (start position shift along the knot).
- **Groove radius offset** (in normal direction): $\Delta r$ or $\Delta n$.

You can define each groove centerline as:

$$
C_n(t) = C_0(t + \delta_n).
$$

For equal spacing around the torus:

- $\delta_n = \frac{2\pi n}{N}$ (N parallel grooves, evenly phased).[^4][^5]

Then you specify groove **cross‑section**:

- Groove width: $w_g$
- Groove depth: $d_g$

In CAD terms:

> “Each winding groove is a sweep of a circular profile of radius $w_g/2$ along $C_n(t)$ with normal offset giving depth $d_g$.”

## Nomenclature for Print Segments and Orientation

For your print/assembly description, minimum useful symbols:

- **Segment index**: $k$ (third index 0,1,2).
- **Segment torus interval**: $\Theta_k = [2\pi k/3, 2\pi (k+1)/3]$.
- **Print orientation rotation**: $\mathcal{R}_k$
The rigid transform that puts one cut face of segment $k$ onto the build plane.

You can write:

> “Printable segment $S_k$ is the restriction of the torus to $\theta \in \Theta_k$; a rigid transform $\mathcal{R}_k$ is applied so that one radial cut face lies in the build plane (Z = 0).”

For joints:

- **Joint planes**: $\Pi_k$ at $\theta = 2\pi k/3$.
- **Dovetail features on $\Pi_k$**: parameterized by arc‑length $s$ along $\Pi_k$.


## Layer / Slicing Description

At the slicer / G‑code level you rarely need more than:

- Part’s **global orientation** (defined by $\mathcal{R}_k$).
- **Layer height** $h$.
- **Z‑extent** of the rotated segment: $Z \in [0, H_k]$.

If you want to be explicit in a spec:

> “Each third‑segment $S_k$ is printed with build direction $\hat{z}$ orthogonal to the torus plane; slicing is at constant $z = nh$, $n \in \mathbb{Z}$, over the transformed geometry $\mathcal{R}_k(S_k)$.”

That’s all the nomenclature you really need:

- $R, r, \theta, \phi, (p,q), t$
- Segment index $k$ with $\theta$-intervals
- Groove index $n$ with phase offsets $\delta_n$
- Rotation $\mathcal{R}_k$ to define print orientation
- Joint planes $\Pi_k$

If you’d like, next step can be: map those symbols directly into a Fusion 360 or OpenSCAD script skeleton so this nomenclature becomes live geometry instead of just documentation.[^6][^7][^3]

<div align="center">⁂</div>

[^1]: https://en.wikipedia.org/wiki/Torus

[^2]: https://en.wikipedia.org/wiki/Torus_knot

[^3]: https://mathworld.wolfram.com/TorusKnot.html

[^4]: https://www.mi.sanu.ac.rs/vismath/taylorapril2011/Taylor.pdf

[^5]: https://www.nature.com/articles/s41598-019-53548-w

[^6]: https://community.xometry.com/kb/articles/755-whats-the-maximum-size-of-a-part-you-can-print

[^7]: https://sinterit.com/3d-printing-guide/design-for-3d-printing/how-big-can-3d-printers-print/