From spherical normal to spherical transverse mercator using colatitudes

Abstract:

Geographical challenges have long influenced the development of geometry. Ancient Greek mathematicians like Thales and Ptolemy were also geographers, and later, geometers such as Gauss and Laplace made key contributions to mapmaking. Since the Earth is approximately spheric, no map can perfectly preserve all geographic properties, making map projections essential. These involve two steps: scaling the Earth’s shape (sphere or ellipsoid) and transforming it onto a flat surface (plane, cone, or cylinder). However, all map projections introduce distortions and can be classified into various types, including conformal, equal-area, equidistant, and others, depending on the method and orientation used. The Mercator projection, a conformal cylindrical projection, revolutionized navigation by preserving angles and directions. Its variant, the transverse Mercator projection, introduced by Lambert in 1772, that did not use colatitudes, rotates the Mercator projection to align with a central meridian, minimizing distortions in nearby regions. This paper derives the transverse Mercator projection equations on a sphere from the Mercator projection equations using colatitudes, offering a rigorous yet pedagogically valuable formulation. It fills analytical gaps often left in classical literature, validates its results by deriving the classical equations, and presents a practical application using 5,556 geographic points, confirming no significative differences.

Keywords:
Mathematical Cartography; Mapping; Transverse Cylindrical Projection

1. Introduction

The development of projections is essentially intertwined with the history of cartography, as every cartographer, initially through intuition and later through deliberate effort, strives to reduce distortions when designing a map (Novikova, 2025). It means that geographical problems significantly influenced the development of geometry. In ancient Greece, mathematicians like Thales and Ptolemy were also geographers. Later, geometers such as Gauss and Laplace engaged deeply with geographical challenges, especially in mapmaking (Caddeo and Papadopoulos, 2022). One of these geographical issues arises from the fact that, ideally, a map should fulfill several criteria: it should retain the true shapes of countries, ensure correct area proportions, represent distances accurately, align straight lines with great circles, and facilitate the identification of geographic coordinates. However, because the Earth is round, it is not possible to satisfy all of these conditions simultaneously, making it necessary to prioritize certain aspects over others (Lambert, 2022).

On the methodology of map projection, the shape of the Earth is typically modeled as a solid of revolution, either an ellipsoid or a sphere, serving as a reference for all points (Richardus and Adler, 1972). Nevertheless, an ellipsoid or a sphere is not a developable surface like a cone or a cylinder. Consequently, it is impossible to wrap a flat sheet of paper around the Earth in a way that precisely represents the shapes of countries and continents and then seamlessly unfold it into a two-dimensional map (Melluish, 2014).

In this sense, the map projection process involves two steps: first, the model for the Earth (sphere or ellipsoid) is scaled down using a fixed ratio, known as the principal scale. According to Pearson (1990, p. 05), “principal scale refers to the ratio of a true length distance on the map to the equivalent distance on the model for the Earth”. Second, this reduced model is transformed into a flat representation through geometric methods or mathematical equations on a projection surface (plane, cone, or cylinder) (Gaspar, 2005).

Theoretically, map projections are infinite in number and must be classified. Since no projection can be entirely free of distortion, as demonstrated by Leonhard Euler in 1777, they are typically grouped into conformal (preserving angles), equal-area (preserving areas), equidistant (preserving specific distances), and others (Lapaine, 2019). However, in addition to this classification, other schemes can also be employed. According to Richardus and Adler (1972, p. 04) “the projection surface is considered as the extrinsic problem and the process of projection or representation as the intrinsic problem”.

The extrinsic aspect concerns the geometric relationship between the projection surface and the reference surface and is divided into three classes: the nature of the projection surface (plane, cone, or cylinder); the type of contact with the reference surface (tangent, secant, or polysuperficial); and the orientation of the projection surface (normal, transverse, or oblique). The intrinsic aspect focuses on the properties of the cartographic projection and its method of generation. It includes the preservation of specific properties (equidistance, conformality, or equivalency) and the mode of construction, whether geometric, semi-geometric, or conventional. These classifications provide a systematic framework for understanding both the geometric structure and functional purpose of map projections (Richardus and Adler, 1972).

In 1569, the Dutch cosmographer, mathematician, and cartographer Gerard Mercator introduced A New and Enlarged Description of the Earth with Corrections for Use in Navigation, which presented a groundbreaking map projection that transformed maritime navigation by enabling sailors to plot and maintain a steady course using compass bearings over long distances (Abee, 2019). This projection became one of the most important and well-known map projections and came to be called the Mercator projection. It revolutionized maritime navigation, making route planning with a fixed azimuth straightforward. As maritime travel expanded globally, precise maps became essential, and Mercator’s projection emerged as a significant mathematical and cartographic achievement. It continues to be widely used in modern mapmaking (Smetanová et al., 2016).

The Mercator projection preserves angles between intersecting curves, making it the only conformal cylindrical projection. This property is particularly useful for mapping, as it ensures that directions on the map accurately reflect those in the real world. Since meridians and parallels can be determined through celestial observations and directions can be established with a compass, this projection is very effective for navigation (Vermeer and Rasila, 2020).

Derived from Mercator projection, the transverse Mercator projection was introduced by Johann Heinrich Lambert in 1772. This projection rotates the Mercator projection 90° so that its equator aligns with the desired central meridian. This effectively wraps the cylinder around the Earth, ensuring the central meridian remains true to scale. As a result, the projection maintains low distortion in nearby regions while preserving conformality, similar to the Mercator projection (Snyder, 1987).

Nevertheless, many people without deep knowledge of map projections struggle to understand their construction due to complex equations, especially when based on an ellipsoidal Earth. As a result, most users have only a basic understanding of these projections. Simplifying the equations using a spherical model can make the concepts more accessible to non-specialists, even if it does not fully meet the needs of geodesists (Lauf, 1975).

Therefore, this paper aims to develop a mathematical derivation of the functions of the transverse Mercator projection considering a sphere as a reference surface, based on the equations of the Mercator projection also considering a sphere as a reference surface, using colatitudes. Moreover, this formulation offers pedagogical value, as classical literature often presents the final equations without derivation or, when derivations are included, tends to omit intermediate analytical steps, potentially hindering comprehension for readers without advanced mathematical expertise. As an algebraic validation, the classical equations for the transverse Mercator projection will be deduced from the expressions derived in this paper. Finally, using both sets of equations, a practical application will be presented, in which 5,556 points with known latitudes and longitudes are calculated and the differences in the X and Y coordinates are analyzed, confirming no significative discrepancies.

2. Rotation of the Graticule

In map projections, the projected image of a grid of parallels and meridians is called a graticule, and a system of lines parallel to the x and y-axes forms a map projection grid on the projection surface (Meyer, 2018). Specifically, the Mercator projection must satisfy the following conditions: (1) The scale remains accurate along the equator; and (2) The y-coordinate origin is set at the equator (Krakiwsky, 1973).

As derived from the Mercator projection, the transverse Mercator projection can be obtained, as it is a variation of the Mercator projection rotated 90°, aligning it with a central meridian, just as the Mercator projection aligns with the equator. Since the cylinder of the transverse Mercator projection touches the sphere along a meridian, the scale is true along that meridian, known as the central meridian, which serves as the reference for the map’s x-coordinate, while the starting point for the y-coordinate is the equator (Anderson and Mikhail, 1998).

Based on these considerations, the equations of the transverse Mercator projection can be derived from the Mercator projection by considering the sphere as the reference surface and applying a rotation of the graticule and the normal cylinder, with the aid of spherical trigonometry equations.

Graticule rotation in map projections involves selecting an arbitrary axis of revolution instead of the Earth’s actual rotation axis. The designated poles are called metapoles, and their coordinates, known as metacoordinates, differ from geographic coordinates by a prime symbol. The system’s orientation is set by the geographic coordinates of the metapole, with the prime metameridian always passing through the meta north pole (Kerkovits, 2023).

Figure 1 shows a spherical model of the Earth, depicting the original system and the metasystem rotated clockwise through a right angle.

Figure 1:
Coordinates and metacoordinates on the sphere.

In Figure 1 one can observe the following points, angles, and lines:

O is the center of rotation;

N is the North Pole;

S is the South Pole;

N’ is the meta north pole;

S’ is the meta south pole;

NS represents the Greenwich meridian, and the metaequator;

N’S’ represents the equator, and the Greenwich metameridian;

P is any point on the surface of the sphere;

NP is the colatitude ∆ of P;

N’P is the metacolatitude ∆’ of P;

$O\widehat{N}P$ is the longitude λ of P;

$O\widehat{N\text{'}}P$ is the metalongitude λ’ of P.

Thus, the first problem is to find the metalongitude and metacolatitude of a point P from its longitude and colatitude, because, according to Melluish (2014, p. 02), “mathematical calculations can often be simplified by using colatitude instead of latitude”. The colatitude Δ of any point P, also called polar distance, is represented as a function of its latitude φ according to equation (1):

$\Delta =\frac{\pi }{2}-\phi$ (1)

Algebraically, the first problem can be described in two steps. In the first step, we can write equations (2) and (3), where the first represents the metacolatitude of P as a function of the colatitude and longitude of P, and the second represents the metalongitude of P as a function of the colatitude and longitude of P.

${\Delta }^{\text{'}}=f\left(\Delta ,\lambda \right)$ (2)

${\lambda }^{\text{'}}=g\left(\Delta ,\lambda \right)$ (3)

From equations (2) and (3), it is possible to extract from Figure 1 the spherical triangle that, when solved, can lead to the solution of the first problem. To clarify, Figure 2 highlights the spherical triangle NN’P, which will be the basis of the deductions that will follow.

Figure 2:
Spherical triangle NN’P.

Thus, since the three sides of the spherical triangle in Figure 2 are arcs of great circles, using an application of the equations of spherical trigonometry found in Olsen (2024), we have the equations (4) to (12). Thus,

By the cosine formula:

$cos\Delta =cos\frac{\pi }{2}cos{\Delta }^{\text{'}}+sin\frac{\pi }{2}sin{\Delta }^{\text{'}}\cos \left(\frac{\pi }{2}-{\lambda }^{\text{'}}\right)$ (4)

$cos\Delta =sin{\Delta }^{\text{'}}sin{\lambda }^{\text{'}}$ (5)

$sin{\lambda }^{\text{'}}=\frac{cos\Delta }{sin\Delta \text{'}}$ (6)

$cos{\Delta }^{\text{'}}=cos\frac{\pi }{2}cos\Delta +sin\frac{\pi }{2}sin\Delta \cos \left(\frac{\pi }{2}-\lambda \right)$ (7)

$cos{\Delta }^{\text{'}}=sin\Delta sin\lambda$ (8)

${\Delta }^{\text{'}}=co{s}^{-1}\left(sin\Delta sin\lambda \right)$ (9)

By the sine formula:

$\frac{\sin \left(\frac{\pi }{2}-{\lambda }^{\text{'}}\right)}{sin\Delta }=\frac{\sin \left(\frac{\pi }{2}-\lambda \right)}{sin\Delta \text{'}}$ (10)

$\frac{cos\lambda \text{'}}{sin\Delta }=\frac{cos\lambda }{sin\Delta \text{'}}$ (11)

$cos{\lambda }^{\text{'}}=\frac{sin\Delta cos\lambda }{sin\Delta \text{'}}$ (12)

Dividing equation (6) by equation (12) and expanding, we arrive at equations (13) to (16):

$tan{\lambda }^{\text{'}}=\frac{\frac{cos\Delta }{sin\Delta \text{'}}}{\frac{sin\Delta cos\lambda }{sin\Delta \text{'}}}$ (13)

$tan{\lambda }^{\text{'}}=\frac{cos\Delta }{sin\Delta }\frac{1}{cos\lambda }$ (14)

$tan{\lambda }^{\text{'}}=cotg\Delta sec\lambda$ (15)

$\lambda \text{'}=ta{n}^{-1}\left(cotg\Delta sec\lambda \right)$ (16)

3. Rotation of the Normal Cylinder

Now that the issue of the graticule’s rotation has been resolved, we can proceed to the second problem: the rotation of the normal cylinder of the Mercator projection. To address this, we begin with the equations of the Mercator projection in one of its most traditional forms, as shown in equations (17) and (18) (Pearson, 1990):

$X_N=R\mu \left(\lambda -{\lambda }_0\right)$ (17)

$Y_N=R\mu \left\{\ln \left[\tan \left(\frac{\pi }{4}+\frac{\phi }{2}\right)\right]\right\}$ (18)

In equations (17) and (18):

X_N is the abscissa in the Mercator projection;

Y_N is the ordinate in the Mercator projection;

R is the radius of the sphere;

µ is the principal scale of the map;

λ is the longitude of any point P, in radians;

λ₀ is the longitude of the central meridian, in radians;

φ is the latitude of any point P, in radians.

For deduction purposes, considering R=1, µ=1 and with the central meridian at Greenwich, (λ ₀ =0), equations (17) and (18) can be written as equations (19) and (20):

$X_N=\lambda$ (19)

$Y_N=\ln \left[\tan \left(\frac{\pi }{4}+\frac{\phi }{2}\right)\right]$ (20)

To express equations (19) and (20) as functions of colatitude and longitude, equation (19) can be renumbered, and equation (1) can be substituted into (20) and developed, as shown in equations (21) to (25):

$X_N=\lambda$ (21)

$Y_N=\ln \left[\tan \left(\frac{\pi }{4}+\frac{\frac{\pi }{2}-\Delta }{2}\right)\right]$ (22)

$Y_N=\ln \left[\tan \left(\frac{\pi }{4}+\frac{\pi -2\Delta }{4}\right)\right]$ (23)

$Y_N=\ln \left[\tan \left(\frac{2\pi -2\Delta }{4}\right)\right]$ (24)

$Y_N=\ln \left[\tan \left(\frac{\pi -\Delta }{2}\right)\right]$ (25)

Equations (21) and (25) represent the Cartesian coordinates of any point in the Mercator projection as a function of its colatitude and longitude, for a sphere with a radius equal to one unit, with the central meridian in Greenwich, which starts the x-axis count, and whose y-axis count begins at the equator. Figure 3 illustrates this system.

Figure 3:
Normal cylinder of the Mercator projection.

Applying a rotation to the Cartesian system (X _N , Y _N ) in Figure 3 equal to a right angle, in a clockwise direction, centered on point O, we arrive at a new Cartesian system (x’, y’), which can be considered as a Cartesian metasystem, as shown in Figure 4.

Figure 4:
Normal cylinder rotated.

Considering that in the Cartesian metasystem of Figure 4, the coordinates of x’ and y’ are a function of the metacolatitude and metalongitude, in a way analogous to equations (21) and (25) we can write equations (26) and (27):

$x^{\text{'}}={\lambda }^{\text{'}}$ (26)

$y^{\text{'}}=\ln \left[\tan \left(\frac{\pi -{\Delta }^{\text{'}}}{2}\right)\right]$ (27)

However, equations (26) and (27) are still not sufficient to correctly represent what is desired. This is because, as observed in Figure 1, the longitude of P has a positive value, while the metalongitude of P has a negative value. This means that equation (3) has the opposite direction, taking point P, which is in the east in the original system, to the west in the metasystem. Therefore, to find the correct direction, equation (26) will have the opposite direction, and equation (27) will be repeated with new numeration, as in equations (28) and (29):

$x^{\text{'}}={\lambda }^{\text{'}}$ (28)

$y^{\text{'}}=\ln \left[\tan \left(\frac{\pi -{\Delta }^{\text{'}}}{2}\right)\right]$ (29)

Even with this correction, as can be seen in Figure 4, the Cartesian metasystem presented therein still does not comply with the conditions of the transverse Mercator projection, since according to Krakiwsky (1973, p. 51) “the requirements for the transverse Mercator projection are: (l) The scale is true along the central meridian; (2) The origin of the ordinate y is at the equator; (3) The origin of the abscissa X is at central meridian”.

Figure 5 shows how the transverse cylinder system of the transverse Mercator projection should be, where X _T represents the abscissa and Y _T the ordinate. Therefore, a difference can be seen in the axes of Figures 4 and 5, which must be corrected.

Figure 5:
Transverse cylinder of the transverse Mercator projection.

Thus, a solution to fulfill the conditions of the transverse Mercator projection from the Cartesian metasystem of Figure 5 can be given by equations (30) and (31):

$X_T=y^{\text{'}}$ (30)

$Y_T=-x^{\text{'}}$ (31)

Substituting equations (28) and (29) into equations (30) and (31) we arrive at equations (32) and (33):

$X_T=\ln \left[\tan \left(\frac{\pi -{\Delta }^{\text{'}}}{2}\right)\right]$ (32)

$Y_T={\lambda }^{\text{'}}$ (33)

Equations (32) and (33) are in function of metacolatitude and metalongitude. To express them in terms of colatitude and longitude, substitutions must be made using equations (9) and (16), resulting in equations (34) and (35):

$X_T=\ln \left[\tan \left(\frac{\pi -co{s}^{-1}\left(sin\Delta sin\lambda \right)}{2}\right)\right]$ (34)

$Y_T=ta{n}^{-1}\left(cotg\Delta sec\lambda \right)$ (35)

Equations (34) and (35) were derived considering R=1, µ=1 and λ ₀ =0. Thus, for a sphere of radius equal to R, a central meridian equal to λ ₀ and a principal scale equal to µ, equations (34) and (35) can be rewritten as in equations (36) and (37):

$X_T=R\mu \ln \left[\tan \left(\frac{\pi -co{s}^{-1}(sin\Delta \mathrm{s}\mathrm{i}\mathrm{n}(\lambda -{\lambda }_0\left)\right)}{2}\right)\right]$ (36)

$Y_T=R\mu ta{n}^{-1}\left(cotg\Delta \sec \left(\lambda -{\lambda }_0\right)\right)$ (37)

4. Algebraic Validation

Equations (36) and (37) represent the equations of the transverse Mercator projection considering a sphere as the reference surface that we aimed to derive. It is now necessary, through algebraic deduction, to show that they lead to one of the classical forms of these equations.

To perform this task, equation (36) will initially be developed, replacing colatitude with latitude, obtaining equations (38) to (45):

$X_T=R\mu \ln \left[\tan \left(\frac{\pi -co{s}^{-1}(sin\Delta \mathrm{s}\mathrm{i}\mathrm{n}(\lambda -{\lambda }_0\left)\right)}{2}\right)\right]$ (38)

$X_T=R\mu \ln \left[\tan \left(\frac{\pi -co{s}^{-1}(sin\Delta \mathrm{s}\mathrm{i}\mathrm{n}(\lambda -{\lambda }_0\left)\right)}{2}\right)\right]$ (39)

$X_T=R\mu \ln \left[\tan \left(\frac{\pi -co{s}^{-1}(cos\phi \mathrm{s}\mathrm{i}\mathrm{n}(\lambda -{\lambda }_0\left)\right)}{2}\right)\right]$ (40)

$X_T=R\mu \ln \left[\tan \left(\frac{\pi }{2}-\frac{co{s}^{-1}(cos\phi \mathrm{s}\mathrm{i}\mathrm{n}(\lambda -{\lambda }_0\left)\right)}{2}\right)\right]$ (41)

$X_T=R\mu \ln \left[\frac{\sin \left(\frac{\pi }{2}-\frac{co{s}^{-1}(cos\phi \mathrm{s}\mathrm{i}\mathrm{n}(\lambda -{\lambda }_0\left)\right)}{2}\right)}{\cos \left(\frac{\pi }{2}-\frac{co{s}^{-1}(cos\phi \mathrm{s}\mathrm{i}\mathrm{n}(\lambda -{\lambda }_0\left)\right)}{2}\right)}\right]$ (42)

$X_T=R\mu \ln \left[\frac{\cos \left(\frac{co{s}^{-1}(cos\phi \mathrm{s}\mathrm{i}\mathrm{n}(\lambda -{\lambda }_0\left)\right)}{2}\right)}{\sin \left(\frac{co{s}^{-1}(cos\phi \mathrm{s}\mathrm{i}\mathrm{n}(\lambda -{\lambda }_0\left)\right)}{2}\right)}\right]$ (43)

$X_T=R\mu \ln \left[\frac{\sqrt{\frac{1+\mathrm{c}\mathrm{o}\mathrm{s}\left(co{s}^{-1}\right(cos\phi \mathrm{s}\mathrm{i}\mathrm{n}(\lambda -{\lambda }_0)\left)\right)}{2}}}{\sqrt{\frac{1-\mathrm{c}\mathrm{o}\mathrm{s}\left(co{s}^{-1}\right(cos\phi \mathrm{s}\mathrm{i}\mathrm{n}(\lambda -{\lambda }_0)\left)\right)}{2}}}\right]$ (44)

$X_T=R\mu \ln \left[\frac{\sqrt{\frac{1+cos\phi \mathrm{s}\mathrm{i}\mathrm{n}(\lambda -{\lambda }_0)}{2}}}{\sqrt{\frac{1-cos\phi \mathrm{s}\mathrm{i}\mathrm{n}(\lambda -{\lambda }_0)}{2}}}\right]$ (45)

Now, equation (37) will be developed, replacing colatitude with latitude, obtaining equations (46) to (48):

$X_T=\frac{R\mu }{2}\ln \left(\frac{1+cos\phi \mathrm{s}\mathrm{i}\mathrm{n}(\lambda -{\lambda }_0)}{1-cos\phi \mathrm{s}\mathrm{i}\mathrm{n}(\lambda -{\lambda }_0)}\right)$ (46)

$Y_T=R\mu ta{n}^{-1}\left(cotg\Delta \sec \left(\lambda -{\lambda }_0\right)\right)$ (47)

$Y_T=R\mu ta{n}^{-1}\left(cotg\left(\frac{\pi }{2}-\phi \right)\sec \left(\lambda -{\lambda }_0\right)\right)$ (48)

Thus, from equations (45) and (48), we obtain the direct equations for the transverse Mercator projection considering the sphere as reference surface, in one of its classic forms, according to the equations (49) and (50) (Pearson, 1990):

$X_T=\frac{R\mu }{2}\ln \left(\frac{1+cos\phi \mathrm{s}\mathrm{i}\mathrm{n}(\lambda -{\lambda }_0)}{1-cos\phi \mathrm{s}\mathrm{i}\mathrm{n}(\lambda -{\lambda }_0)}\right)$ (49)

$Y_T=R\mu ta{n}^{-1}\left(\frac{tan\phi }{cos(\lambda -{\lambda }_0)}\right)$ (50)

In equations (49) and (50), the latitude φ and the longitudes λ and λ ₀ are expressed in radians.

5. Example of a practical application

As an example of the application of equations (36) and (37), a comparison can be made between the coordinates of points calculated from them, with the coordinates calculated using the classical equations (49) and (50). To perform this task, the official latitudes and longitudes of 5,556 points were selected, all located in Brazil, from the Geodetic Database of the Brazilian Institute of Geography and Statistics (IBGE, 2025).

From the latitudes and longitudes, the X and Y coordinates of the transverse Mercator projection were calculated for all points using equations (49) and (50) and equations (36) and (37), with a Python program specially developed to perform calculations with different significant digits after the decimal point. Thus, calculations were made for 50, 75, and 100 significant digits, and the largest (max) and smallest (min) values for each one were selected for the differences between X and Y, as can be seen in Table 1.

Table 1:
Differences between equations.

Type of difference	∆X (m)	∆Y (m)
∆ max50	4.484e-44	4.484e-44
∆ min50	-8.968e-44	-3.363e-44
∆ max75	1.390e-68	3.477e-69
∆ min75	-6.954e-69	-3.477e-69
∆ max100	5.990e-94	1.797e-94
∆ min100	-2.397e-94	-1.797e-94

The results presented in Table 1 demonstrate that the discrepancies between the equations are influenced by the level of precision applied in the computations. However, these differences are not significant in all cases. In this way, both latitude-based and colatitude-based equations can be reliably employed in various applications, such as precision agriculture, environmental monitoring, civil defense, disaster management, smart cities, logistics, marketing and market analysis, augmented reality, virtual reality, public health and epidemiology, and so on.

6. Final Considerations

The development of map projections was motivated by the fundamental difficulty of representing the curved surface of the Earth on a flat surface, namely, a map. Throughout history, mathematicians and geographers have refined projection methods to meet various mapping requirements, resulting in the development of crucial tools such as the Mercator and the transverse Mercator projections. While these projections provide considerable benefits, especially for navigation, their mathematical underpinnings can be intricate and difficult for non-specialists to understand.

The algebraic deduction carried out demonstrated that, from a mathematical point of view, the equations presented here using colatitudes, namely equations (36) and (37), produce equivalent results from an application standpoint to the classical equations, that use latitudes, namely equations (49) and (50). The comparisons made about the X, Y coordinates for the 5,556 points in Brazil, with known latitudes and longitudes, also showed no significant differences between coordinates for all points, allowing all applications in the most diverse areas in mapping and engineering.

Thus, the use of colatitude allows the Mercator projection formulation to be used to develop the Transverse Mercator Projection with a sphere as a reference surface.

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• DATA AVAILABILITY
  This study did not generate new datasets. All data analyzed were obtained from publicly available sources, which are cited in the References section.

Data availability

This study did not generate new datasets. All data analyzed were obtained from publicly available sources, which are cited in the References section.

Publication Dates

• Publication in this collection
  03 Nov 2025
• Date of issue
  2025

History

• Received
  13 Apr 2025
• Accepted
  04 Sept 2025