Volume-Preserving Shear Transformation of an Elliptical Slant Cone to a Right Cone
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsIn this paper, the authors compute the volume of the finite part of a cone after an elliptical section was done on this cone. So in this paper is proved that it is possible to compute that volume just by using the information of the shadow ellipse and the height of cone. The authors remarked that the finite slant cone has the same volume of an elliptic right cone with base the shadow ellipse of the cut portion and with height the distance between the vertex of the cone and the intersection of the height of the original cone with the cutting plane.
The proof of this important result was done by the authors using shear transformations of the elliptical slant cone to a right cone.
The paper is interesting, contains new results and present a way how isometries--- in this case shear transformation could be used for computations of some important results in the theory of quadrics.
In conclusion, the computation of the cone’s volume is thus simplified, since the oblique ellipse, and the vertex’s projection on the cutting plane can be neglected because solely the shadow ellipse must be determined and integrated over, as the authors remarked.
This new method presented by the authors could be also used in future computations for other type of quadrics and represent a new way to compute the volume of the cone using shadow ellipse.
However, some typos are along the paper and I recommend to the authors to check carefully all the paper in this respect. For example, already at the begining of Section 1, is wroted "Thipresents paper" -- I think correct is "The present paper..."
After all the paper is checked carefully in this respect and the English of the paper is improved, I recommend for publication this paper.
Comments on the Quality of English LanguageHowever, some typos are along the paper and I recommend to the authors to check carefully all the paper in this respect. For example, already at the begining of Section 1, is wroted "Thipresents paper" -- I think correct is "The present paper..."
Author Response
We would like to thank the Reviewer for the very useful and constructive comments. We performed a thorough revision of the paper and improved the readability thanks to the Reviewer's comments.
Reviewer 2 Report
Comments and Suggestions for Authors
There are many publications which are devoted to volume –preserving maps. In the present paper, authors consider a nappe of a right circular cone cutting by a transverse plane divides the nappe into a finite volume trunk (an elliptical slant cone) and an infinite frustum. They introduce the shadow ellipse and a new volume-preserving shear transformation of the elliptical slant cone to a right cone. It is shown that the volume of the slanted cone is equal to the volume of the right cone that has basis the shadow ellipse and height the intersection of the height of the original cone with the cutting plane.
As consequence, the volume of the elliptical slant conecan be easily calculated. All figures in the text have an excellent quality.
My opinion: the
Paper is suitable for the journal Axioms
Author Response
We would like to thank the Reviewer for the very useful and constructive comments. We performed a thorough revision of the paper and improved the readability thanks to the Reviewer's comments.
Reviewer 3 Report
Comments and Suggestions for AuthorsThe referee report can be found in the attachment.
Comments for author File: 
Comments.pdf
Moderate editing of the English language is required.
Author Response
We would like to thank the Reviewer for the very useful and constructive comments. We performed a thorough revision of the paper and improved the readability thanks to the Reviewer's comments.
Regarding the suggestion for additional references, we have added a citation to a recent work (Miklavcic, 2020) that provides additional context to our work.
