The previous posts on screw theory and (dual) quaternions are not consistent with each other and need to be made so before they are combined. This post will start on establishing a notation that will be consistent across them all.
First, let's start with the posts on quaternions, Plucker coordinates and dual quaternions. The proposed notation will be as follows.
Regular quaterions will use bold italic lower case lettering. They are composed of a scalar part in lower case italic with a $w$ subscript and a vector part in lower case italic with an arrow above. The vector part can be broken down into it's $i, j, k$ vector components. Subscripts $x, y, z$ on the three scalars will associated them with their $\vec{i},\vec{j},\vec{k}$ vector components as illustrated in the following example:
$$\begin{align} \pmb{p} &= p_w + \vec{p} \\ &= p_w + p_x\vec{i} + p_y\vec{j} + p_z\vec{k} \end{align}$$The conjugate for the regular quaternion will use a $*$ superscript and the vector components simply use a sign change.
$$\begin{align} \pmb{p^{*}} &= p_w - \vec{p} \\ &= p_w - p_x\vec{i} - p_y\vec{j} - p_z\vec{k} \end{align}$$Dual quaternions are in bold italic lower case letters with a "hat" symbol above. They are made up of two regular quaternions but one is multiplied by the dual number $\epsilon$. Also, the dual part of the dual quaternion will used an $\epsilon$ subscript on it's regular quaternion scalars as shown below:
$$\begin{align} \pmb{\hat{\sigma}} &= \pmb{\sigma} + \epsilon\pmb{\sigma_{\epsilon}} \\ &= \sigma_w + \sigma_x\vec{i} + \sigma_y\vec{j} + \sigma_z\vec{k} + \epsilon\big(\sigma_{{\epsilon}w} + \sigma_{{\epsilon}x}\vec{i} + \sigma_{{\epsilon}y}\vec{j} + \sigma_{{\epsilon}z}\vec{k}\big) \end{align}$$There are three types of conjugates associate with dual quaternions and we will use superscripts as shown below to distinguish each type:
The dual number is conjugated
$$\pmb{\bar{\hat{\sigma}}} = \pmb{\sigma} - \epsilon\pmb{\sigma_{\epsilon}}$$-
The quaternion components are conjugated
$$\pmb{\hat{\sigma}^{*}} = \pmb{\sigma^{*}} + \epsilon\pmb{\sigma_{\epsilon}^{*}}$$ -
The dual number and quaternion components are both conjugated.
$$\pmb{\bar{\hat{\sigma}}^{*}} = \pmb{\sigma^{*}} - \epsilon\pmb{\sigma_{\epsilon}^{*}}$$
For the post on screw theory, several images along with the text need new labels. The translation vector $\vec{t}$ needs to be introduced and added to figures. One that requires this addition also uses the capital $\mathsf{T}$ on the screw axis line which could be changed to $\mathsf{S^{'}}$. The scalar $t$ appears later in the article when looking at half-turns and the screw triangle. This can be changed to the letter $h$ instead. Also, $\vec{n}$ is used as the unit direction vector for a screw axis. This will be changed to $\vec{l}$. The letter $l$ appears in the discussion of the half-turn axes and can remain the same since these are scalars in the $\vec{l}$ direction. Figures 15 and 19, change scalars $t$'s to $h$'s. Figures 20, 21, 22 and 23, change $\vec{n}$'s to $\vec{l}$'s. There are also inconsistencies in the use of $\theta$ and $\phi$ between the Mozzi-Chasles paper and the rest that lead to a dual quaternion formulation. It seems easier to use the Mozzi-Chasles convention and change the quaternion and dual quaternion articles to use $\phi$ instead of $\theta$ since much fewer images need changing. Nine images versus two or three images. Text changes are easy since one can just use the replace function on a text editor. The Plucker article can be left with $\theta$'s because these are unrelated or they could be changes to something else like $\alpha$'s.
