Response to referee

1. 1.

   COMMENT: Figure 1 should include $\theta$, $z_{C}$ and $\phi_{C}$. The figures in the manuscript rely heavily on the use of these variables, and the reader needs to better understand what they represent. A clearer definition of $\phi_{C}$ might also be needed in the text, identifying where this angle is being measured from and in what direction the measurement is being made.

   RESPONSE: Figure 1 has been updated and now includes $\theta$, $z_{C}$ and $\phi_{C}$ which can be seen in the highlighted caption on page 3. A much clearer definition of $\phi_{C}$ has also been provided on page 3 in the 3rd block of highlighted text and on page 4 in the 1st block of highlighted text.

2. 2.

   COMMENT: The reader needs a better understanding of how equations 1 to 4 have been derived. Some of this will be made clearer by inclusion of $\theta$, $z_{C}$ and $\phi_{C}$ in Figure 1, but additional discussion and/or intermediary steps (particularly for Equation 2) would be helpful.

   RESPONSE: A much clearer derivation of equation 2 has been provided at page 3 in 3rd highlighted text block. Also, a sentence explaining equation 1 has been added at page 3, highlighted block 2.

3. 3.

   COMMENT: I recommend that the discussion in the last paragraph of Section 2 be expanded, perhaps considerably. Too much information has been compressed into this paragraph, given the non-intuitive nature of the subject matter to a standard audience. The more clarification the authors can provide, the better. What does an equation for the angular velocity look like in terms of the given variables? Can an equation be provided demonstrating how the brightness is calculated?

   RESPONSE: We have expanded the discussion in the last paragraph of Section 2, although not very considerably. We have added extra explanatory words and a new sentence. However, as this paragraph deals primarily with the numerical techniques to calculate angular velocity and brightness, it turns out to be somewhat cumbersome to write out the entire brightness formula, as utilized in Eq. 6, in terms of even more basic variables. However, we do think that the hierarchy of equations is straightforward to code, and the expanded discourse clear.

4. 4.

   COMMENT: In Figure 7, why do the images seen by the central detector and detector 3 go in reverse at angles larger than the RID event (indicated by the solid lines at $\phi_{C}>12.15^{\circ}$ for the central detector and $\phi_{C}>28.31^{\circ}$ for detector 3)? Similarly, detector 3 in Figure 9 exhibits a solid line for $x_{C}>3.64$ (the only reverse motion example I can find for a curve in the classical regime). If these are not plotting errors, then they are non-intuitive and require some additional explanation.

   RESPONSE: Our apologies here. The Python plotting package matplotlib apparently displays figures differently at different magnification levels. Therefore, although Figures 7 and 9 appeared to have the wrong type of plotting lines, this was solely the result of unexpected software display effects. We have now included the same plots, generated with the same software, but at different magnification levels – so that the line styles appear to resolve correctly. This has been done not only for Figures 7 and 9 but also for Figure 8.

5. 5.

   COMMENT: In Figure 7, the non-RID event seen by detector 3 demonstrates why more explanation is needed regarding the dependence of image brightness on relativistic effects and viewing angle: the inverse square effect described in Equation 5 seems to be significantly less than the relativistic contribution, given that the image does not reach maximum brightness when the muon is closest to the detector. But there is no way of determining how these relative contributions are weighted based on the limited information provided in the text.

   RESPONSE: A relevant explanation for this has been given in the highlighted text on page 8.

6. 6.

   COMMENT: I have previously had referees of my own submissions comment on how difficult it can be for colorblind readers to interpret graphs where multiple tracks are delineated only by the colors of lines. Is it possible, in addition to the line colors, to use various arrangements of dots and dashes to delineate event tracks in the figures for colorblind readers?

   RESPONSE: It would be quite confusing to generate so many line styles: at least 7, two for each of the 3 detectors seeing an RID and one for the detector not seeing an RID. Describing all of these line styles would become distracting and almost comical, in our opinion. Rather, to help the colorblind and to increase general plot readability, we have now labelled each line with a detector number. This allows us to keep the same line style meanings – where a solid line means an image moving towards the entry point, and a dashed line means an image moving toward the exit.

In the previous manuscript, the text in Equation 3 did not include arccos. This has been fixed. This was a typo, as the codes and results have always included the arccos function.

Occasional words have been deleted inside the text and therefore not highlighted. None of these deletions have changed any of the science.

The Astrophysics Source Code Library (ASCL) ID for the codes used in the paper has been cited and the corresponding github repository has been included as a footnote on page 13, as suggested by the editor.