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Our referee has now re-considered your paper and have recommended publication in PeerJ Computer Science. I am pleased to accept your paper in its current form.
Please revise the manuscript based on the suggestions of reviewer 1 and 3. As both of them claimed that, you should give some higher dimensional examples, and it would better to focus on practical usage.
When you revise your manuscript please highlight the changes you make in the manuscript by using the track changes mode or by using bold or coloured text.
In this paper, the authors propose the following three methods to visualize n (>3) dimensional, or n-D, objects as 2-D images:
(1) n-D 'long axis' projection
(2) n-D parallel/perspective projection
(3) (Generalized) spherical projection followed by (generalized) stereographic projection
Among them, the last method (3) is a new one, while (1) and (2) are, in some sense, straightforward generalizations of well-known techniques in 3-D (and 4-D) computer graphics.
I agree with authors on the importance of visualization of 4-dimensional data in the knowledge discovery in general. This paper---with the source codes implementing the proposed methods---would be an important contribution to science and technology.
More detailed description of the prototype program would be desirable. For example: Is it possible to interactively specify the view position and angle in 4-D? Is it possible to interactively rotate the object in 4-D? If the answers are yes, how? Some readers would like to know since those 4-D operations are notoriously difficult with the standard user-interface. And how do the authors apply those operations (rotations, walk-throughs, and others) in higher n-D for n>4?
(I do not insist that the following improvement is necessary. This is just a suggestion for a possible improvement of the manuscript, since the "dimension-independence" is one of the emphasized points in the paper.)
The methods proposed in this paper are formulated in general n-D space for n>=4. On the other hand, the presented examples (Figs. 3, 5, and 9) are all for n=4. It would be nice for readers if authors could include an additional example of higher dimensional visualization, say a simple regular polytope in 5-D that is visualized by recursively applied the proposed method (3) for two times. I know it would lead to a highly complex image, but the complexity itself would convey the challenge of the n-D visualization and the power of the proposed methods to the readers.
--- Much more minor points.
p.3: The matrix T is (n+1)-D, i.e., n-D homogenous coordinates, while the matrix S is written in n-D. This would be confusing for readers who are unfamiliar with computer graphics. A comment would be desirable.
p.12, line 309: I could not understand the sentence "This is necessary because..." and the footnote 8.
The used English is perfect, and clearly readable.
The number of cited references proves that the authors made a comprehensive professional literature research.
The structure of the paper is suitable, figures and tables support the understanding.
In general, the article is well written and clear to understand. The literature, as far as I know, is correct. The article structure is professional, together with figures and tables. The article is self-contained, including relevant results to hypotheses. The formal results include clear definitions of all terms and theorems, and detailed proofs.
This paper presents some original research that fits the Aims and Scope of the journal.
The research question posed in this paper is well defined, relevant & meaningful. Throughout the paper, it is stated how the proposed technique fills the lack of methods to properly visualize nD objects. I think the investigation performed meets all standards about rigorous research to a high technical & ethical standard. The methods are described with sufficient detail and enough information to replicate.
The paper has some important positive aspects, but also some clear dark points. See below.
First of all, let me say that I am not an expert on mathematical visualization, so I cannot assess the importance such approach could have the mathematical community, or to society in general. However, I have a large deal of experience with LoD models, urban and building modeling, GIS, BIM and related technologies, so I will do my review from these points of view.
Now, about the paper itself, I would like to commend the authors for their search for new visualization mechanisms for complex models or situations, like the ones proposed. I think this kind of research should always be welcome.
On the other side, although the paper presents some relevant results, their applicability at its current state is quite limited. First of all, the paper promises a system to visualize data with high dimensionality in 3D (Actually, 2D, as this is displayed on a sheet of paper or on a computer screen), but the higher dimensionality the paper demonstrates is 4D, with examples that are already easy to visualize with common tools. Throughout the paper, there is not a single example with initial dimensionality equal or higher than 5D. For this paper to be truly relevant, more complex dimensions should be explored and demonstrated.
The second serious drawback I can see about this paper at this state, is that all interesting examples are limited to LoD visualization, with the only exception of figure 7, which is a trivial example, and Figure 8, which is taken from somewhere else and that it is not much relevant here. Please, provide a larger variety of examples and applications, besides LoD.
The third drawback, in my point of view, is that LoD, the main application demonstrated in the paper, is mainly used in practice for two purposes: to generate different versions of the model for simulations (like the LoDs in CityGML), or for visualization, where adaptive LoD techniques are often used to select a different model (LoD level) according to some criterion like viewer’s distance. Now, this is basically a matter of selecting the appropriate model, and displaying it, so showing all LoD models can be done, either side by side, or with an animation in the case of a continuous LoD. I really do not see the need of a visualization like that, which can become really confusing for a complex object. Perhaps I might be wrong about this last part, so please, provide an example of LoD for a really complex building, also with several LoD levels. Actually, as the paper should be generic, other LoD applications, like for a tree or a character model, should also be shown and compared (observe here I am referring to mode types of LoD examples, not just simple houses, while in the previous paragraph I referred to non-LoD examples). By the way, how would this technique do to visualize a continuous LoD model, like the ones developed in the past decade (see missing reference below).
* please, add the following seminal reference for LoD:
Level of Detail for 3D Graphics, David Luebke, Martin Reddy, Jonathan D. Cohen, Amitabh Varshney, Benjamin Watson and Robert Huebner, ISBN: 978-1-55860-838-2
* Figures 8 and 11 seem out of place, please, remove.
* Figure 10 is trivial. Can you provide a more complex example?
* The word “cumbersome” in the last sentence before the references looks awkward. Probably, saying “not efficient” would have a better impact on the readers.
* Please, remove the trivial explanation about translations and scaling matrices, these are well known. Also, rotations in 4D are also well known, though homogeneous coordinates, so, in any case, provide an n dimensional formulation. However, I think this part is also trivial and can be safely removed.
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