Family of Enneper Minimal Surfaces

Round  1

Reviewer 1 Report

In this paper, the author introduced the idea of family of Enneper's algebraic surfaces in three-dimensional Euclidean space. The author has defined the family of Enneper's minimal surfaces Em and obtained irreducible implicit equations and degrees of the surfaces E1, E2 and E3 by using polynomial eliminate method on Maple algorithm. Also, by using the integral free form of Weierstrass, the author obtained some algebraic functions for the family of Enneper's surfaces of integer order. Minimal surfaces have applications in the areas of architecture, molecular engineering and materials science. Therefore, any new result regarding the minimal surfaces it is welcome.

Author Response

English language and style were checked.

Reviewer 2 Report

I think that the manuscript is interesting and worthy to appear in mathematics. 

Author Response

 English language and style were checked.                                   

Reviewer 3 Report

The paper considers the family of the higher order Enneper minimal surface in the 3D Euclidean space.

The paper seem technically sound and is written in a professional style.

The reviewer recommends:

- A more detailed and extended abstract. The 2-line text is almost a copy of the title and is clearly not sufficiently descriptive.

- A smoother introduction to the topic. In the present form the so-called introduction is just the usual section 2. The reviewer recommends having an introduction discussing the motivation of the study, including a review of the literature and the progress in the topic, the relevance of the topic, and the organisation of the paper.

- A more complete review of the literature. In fact, only 8 references seems very limited. Furthermore, there is no recent references, since they are only from the years 1866-1998 !! 

- Adding a small section at the end with conclusions, since the manuscript ends abruptly in the present form. It would help the paper readability having some text outlining the main conclusions. This problem is even more relevant, since the abstract and introduction are far from being sufficiently descriptive

- Some reworking of the writing style. In fact, the grammar seems correct but the style is very "telegraphic". As mentioned before, the absence of abstract, introduction and conclusions does not helps either. Furthermore, Section titles such as 3. Degree of E1, 4. Degree of E2, 5. Degree of E3 are very short. Consider having some more informative text.

Author Response

- A more detailed and extended abstract. The 2-line text is almost a copy of the title and is clearly not sufficiently descriptive.

- A smoother introduction to the topic. In the present form the so-called introduction is just the usual section 2. The reviewer recommends having an introduction discussing the motivation of the study, including a review of the literature and the progress in the topic, the relevance of the topic, and the organisation of the paper.

- A more complete review of the literature. In fact, only 8 references seems very limited. Furthermore, there is no recent references, since they are only from the years 1866-1998 !! 

- Adding a small section at the end with conclusions, since the manuscript ends abruptly in the present form. It would help the paper readability having some text outlining the main conclusions. This problem is even more relevant, since the abstract and introduction are far from being sufficiently descriptive

- Some reworking of the writing style. In fact, the grammar seems correct but the style is very "telegraphic". As mentioned before, the absence of abstract, introduction and conclusions does not helps either. Furthermore, Section titles such as 3. Degree of E1, 4. Degree of E2, 5. Degree of E3 are very short. Consider having some more informative text.

Regarding all above comments and suggestions of the Reviewer, paper was written again, carefully. Also, English language and style were checked, and some figures were added.

Author Response File: Author Response.pdf

Round  2

Reviewer 3 Report

The revised version can be accepted.

The reviewer has some doubts about the graphical quality of the figures. The authors should check that detail.