Using the Theory of Functional Connections to Solve Boundary Value Geodesic Problems

Round 1

Reviewer 1 Report

In the reviewed manuscript, the geodesic boundary value problem in curved surfaces is presented and numerically solved using the mathematical framework of the Theory of Functional Connections (TFC), developed by the author Daniele Mortari. The TFC seems to be applied successfully and in the field of geodesic problems, as clearly presented in this work. The manuscript is well-written and the formulae are correct. However, some revisions should be made, in order to improve the context and the readability of the manuscript before publication, as follows:

11)     Lines 69-70: Replace [-π, π] with [-π, π)

22)     Lines 168-169 and 217-218: Replace [0, 2π] with [0, 2π)

33)     In the Abstract and in the start of Section 5 is referred that geodesic equations and velocities are provided for tri-axial ellipsoid and hyperbolic hyperboloid but this holds and for other surfaces.

44)     It is also worth mentioning the singular cases in any surface. For example, in the end of Section 2, in the case of a sphere where u = π/2.

55)     Although a simple algorithm has been developed to mitigate the problem of getting stuck on some local minima, a numerical example for this case is not provided.

66)     In general, I find it excessive to perform numerical validations on so many surfaces. I would prefer some to be omitted and the rest to be enhanced with numerical results e.g. for the tri-axial ellipsoid. Also, the investigation of the convergence behavior in the case of antipodal points of an ellipsoid should be checked by numerical examples.

Author Response

see attache file.

Reviewer 2 Report

Dear author.

Congratulations for this valuable work. Boundary Value Geodesic Problems are important subject. As it is a more general surface, triaxial ellipsoid is more prominent in geodetic applications. Therefore the author can refer to the importance of geodetic calculations on the triaxial ellipsoid. It can also expand with surface fitting issues such as ellipsoid, hyperboloid. Some references should be introduced to expand the above application . Such as Geodetic Computations on Triaxial Ellipsoid, International Journal of Mining Science (IJMS) Volume 1, Issue 1, June 2015, PP 25-34 www.arcjournals.org ©ARC Page | 25 .  Least Squares Fıttıng Of Ellıpsoıd Usıng Orthogonal Dıstances, https://Doi.Org/10.1590/S1982-21702015000200019. Design of Hyperboloid Structures, DOI. Org. 10.26689/jard.v1i2.136

Author Response

see attached file.

Author Response File: Author Response.pdf