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┘The Minimal Path to Connect Three Equidistant Points in Hyperbolic Space
By Diana Dimond
RefereeТs report:
I found the paper fairly well-written, well-organized, interesting, and well-suited for Rose-HulmanТs on-line mathematics journal. Nevertheless, I hereby suggest some improvements, which will hopefully make the paper even better.
The author should consult pages 138-145, and in particular, the top of page 140 of MelzakТs book, Companion to Concrete Mathematics, for some historical insights into SteinerТs problem, and in view of them, improve the first part of her abstract.
She should not claim in the second paragraph of her abstract that the Уslicing methodФ was developed by her group. The method seems to date back to at least 1988.
Her paper deals with a restricted Steiner problem (to equidistant points) in the Euclidean and Hyperbolic planes; this should be reflected in the title of the paper.
Many of the diagrams are mere sketches. They should be improved by using the GeometerТs Sketch Pad or some other means. Her Reference 8 provides some excellent examples for professionally drawn figures; she should emulate them.
In particular, there is no reason why the angles marked as right angles are not drawn correctly in Figures 4, 5, 6, 9,10 and 11. Or why the circular arcs are not circular in Figures 7 and 8? On the other hand, since the hyperbolic circles shouldnТt look circular (in the Euclidean sense) and shouldnТt cross the x-axis (which is at infinity in the model), sketching small parts thereof would be more appropriate in Figures 9, 10, and 11.
I would also prefer different notation for the partition lines in Figures 5 and 6 since in Figure 4 the same lines are named differently. ThatТs confusing.
There are several instances of rather sloppy terminology in the paper. For example, in the second paragraph on page 5 it is stated that УOne leg of the triangle is the partition line ЕФ. On page 6 it is claimed that the x-axis is the hypotenuse of a triangle and then it is stated that У one leg is the partition line at C, and the other leg is the line perpendicular to the partition lines.Ф These could be made precise by properly labeling the vertices of the triangles in question.
On page 14, the author may wish to elaborate on what she means by Уn-equidistant pointsФ. I would also prefer a change in her terminology concerning the use of УproveФ one line later, in the last line of the paper, and elsewhere. One proves theorems, claims, conjectures, but not paths and problems.
In the middle of page 9, she needs to elaborate on the claim that У in the upper-half plane model angles are preservedФ. One might ask: By what transformation?
Finally, I think it would be appropriate to explain to the readers that the Уslicing methodФ is basically a method for solving problems by transformation. First, one transforms the problem to a different setting (in this case to the plane with a different metric, to be defined appropriately). Next, one solves the problem in the new space, where it is, ideally, an easier problem. Finally, one transforms the solution back to the original setting, where it should, hopefully, solve the original problem. In the process, one must pay special attention to the invariants (i.e., properties are preserved by the transformation) since they are the keys for the methodТs success.
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