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Using smartphone photographs of the Moon to acquaint students

with non–Euclidean geometry

Hugo Caerols,1, 2, 3, ∗Rodrigo A. Carrasco,1, 4, †and Felipe A. Asenjo1, ‡

1Facultad de Ingenier´ıa y Ciencias, Universidad Adolfo Ib´a˜nez, Santiago 7941169, Chile.

2Grupo de Observaci´on Astron´omica OAUAI, Universidad Adolfo Ib´a˜nez, Santiago 7941169, Chile.

3IAU National Astronomy Education Coordinator (NAEC) Chile team.

4Data Observatory Foundation, Santiago 7941169, Chile.

(Dated: August 19, 2021)

Non-Euclidean geometry can be taught to students using astronomical images. By using pho-

tographs of the Moon taken with a smartphone through a simple telescope, we were able to in-

troduce these concepts to high-school students and lower-level college students. We teach students

how to calculate lengths of mountain ranges or areas of craters on the Moon’s surface and introduce

ideas of geodesics and spherical triangles. Students can see that accurate measurements cannot be

obtained using ﬂat geometry. Instead, by using three–dimensional curved geometry, estimates of

lengths and areas can be computed with less than 4% error.

I. INTRODUCTION

In classical astronomy and navigation, consideration

of curved spaces and surfaces is essential, and now the

entire Universe is recognized as being non-Euclidean.[15]

However, introducing students to the geometry of curved

surfaces is a challenge.

The use of astronomy-related activities and evidence-

based active engagement have been shown to signiﬁcantly

help students grasp complex concepts, encourage them

to pursue science studies, and reduce the gender gap

in STEM areas.[10–12,14,19] This improvement is fur-

ther enhanced by having students use real data which

they acquire themselves.[1,7] Hence, what better way

to introduce the diﬀerences between Euclidean and non-

Euclidean geometries than with the Moon’s spherical ge-

ometry?

In this work, we demonstrate the use of low-resolution

photographs of the Moon taken with a smartphone and

a simple telescope to acquaint students with concepts of

three–dimensional curved geometry such as distances and

areas, and, at the same time, introduce concepts such

as geodesics. Any student interested in how geological

features on the Moon can be measured can perform their

own calculations with these methods.

We start in Section II by showing that our natural

satellite presents a valuable teaching tool to introduce

curved geometry. In Section III we show that estimates

of distances in craters or valleys on the Moon based on

proportionality arguments are valid only for lengths small

compared to the Moon’s radius. In Section IV we explain

how the methods of non–Euclidean geometry are helpful

for estimating the lengths of mountain chains; we also

show how the results diﬀer when considering a ﬂat geom-

∗Electronic address: hugo.caerols@uai.cl

†Electronic address: rax@uai.cl

‡Electronic address: felip e.asenjo@uai.cl

etry. In the last part of this work we use those methods

to calculate the area of a lunar mare by using a sum of

simple polygons. To perform this task, we introduce in

Section Vconcepts of spherical trigonometry, which al-

low us to obtain any desired area as the sum of the areas

of spherical triangles.

II. PHOTOGRAPHING THE MOON

Earth is unique in the Solar System in having a moon

which is large in comparison to the size of its parent

planet. Furthermore, since we always see the same face

of the Moon due to tidal locking, viewers can see and

photograph the same features in any season.

In principle, any photograph(s) of the Moon can be

used. However, it is important that students participate

in this activity by taking their own photographs. The

Moon passes through a cycle of phases during a month,

which results in varying apparent luminosity. To obtain

the best photographs for these activities, we recommend

against using photos taken when the Moon is full. At full

phase the light reﬂected to Earth is perpendicular to the

Moon’s surface, which tends to ﬂatten the appearance of

features and prevents observation of speciﬁc details. We

suggest planning the observations to take photographs

within a few days of ﬁrst-quarter phase, when the Sun’s

rays highlight the moon’s features. Additionally, the ﬁrst

quarter Moon is always visible in the hours just after

sunset, which is a convenient time.

An example of a good photograph of the Moon is shown

in Fig. 1. The numbered craters in this ﬁgure are used in

the following sections, and they are shown in more detail

in Fig. 2. This was taken using an iPhone 6 attached to

a telescope with a NexYZ 3-Axis Universal Smartphone

adapter. We used a Celestron Nexstar 8SE telescope,

which has a focal length of 2,032 mm and an ocular with

a focal length of 25 mm; this combination gives a magni-

ﬁcation of ∼80. We also used a standard lunar ﬁlter to

cut down the Moon’s glare. A smaller and cheaper tele-

This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS ARTICLE AS DOI: 10.1119/10.0006156

2

scope can certainly be used to obtain the desired images.

After the photograph is taken, a smartphone app such

as Snapseed can be used to enhance the contrast in the

picture.[16]

Although the orientation of the photograph is not es-

sential for the proposed calculations, it should be noted

that telescopes invert images and that they need to be

reversed or rotated to represent what is seen with the

naked eye. Furthermore, images of the Moon taken in the

northern and southern hemispheres are reversed. This

last point is relevant in that the image of the Moon in

Fig. 1was taken from the southern hemisphere.

III. SMALL DISTANCES, FLAT GEOMETRY,

AND PROPORTIONALITY

For distances on the Moon’s surface that are small in

comparison to its radius, Euclidean geometry is suﬃcient

to accurately estimate the distance between two points

that are not near the limb. Near the limb, Euclidean

geometry will fail, as we will see in the case of Plato

FIG. 1: Moon right after ﬁrst quarter, 9 days old and

with a luminosity of 67.4 of that of the full Moon.

Photograph taken with a smartphone and telescope on

April 21, 2021 at 20:27, from Santiago, Chile. The

numbers indicate craters and mare that we analyze in

this study.

Crater dCp cm dCkm dCexact km error %

(1) Plato 0.3212 104.61 100.68 3.91%

(2) Aristillus 0.1653 53.84 54.37 0.98%

(3) Autolycus 0.1197 38.99 38.88 0.27%

(4) Archimedes 0.2476 80.64 81.04 0.49%

(5) Timocharis 0.1032 33.61 34.14 1.55%

(6) Eratosthenes 0.1757 57.22 58.77 2.63%

(7) Copernicus 0.2893 94.22 96.07 1.92%

(8) Eudoxus 0.1445 47.07 67.00 29.7%

TABLE I: Diameters for the seven craters of Fig. 2,

estimated and adopted accurate values.

crater.

Figure 2shows a magniﬁcation of the lower part of

Fig. 1. We can use proportionality to calculate the diam-

eter of these eight craters. The real values of the craters’

diameters dCare small compared to the diameter of the

Moon 2RM, where the Moon’s radius is RM= 1,737.4

km. In the Euclidean approximation, the proportions

between the distances are linear. Thus, a diameter of a

crater can be computed as

dC=RM

RMp

dCp ,(1)

where RMp is the Moon’s radius in the photograph and

dCp is the crater’s diameter in the photograph. To

obtain those values, we use GeoGebra to analyze the

photographs.[8] With this software we can calculate how

large the diameter of the craters and the Moon’s radius

are in the photograph on a centimeter scale. A detailed

procedure to obtain those values can be found in Ref.

[3]. To measure the diameter of a crater in GeoGebra,

place two points that are diametrically opposite at the

crater’s edge. This choice should be made as precisely

as possible, using the zoom-in facility. This pro cedure

helps reduce the errors in the measurements done in the

software.

Applying the above procedure to Figure 2results in

RMp = 5.3345 cm. The same can be performed to ob-

tain the diameters dCp of diﬀerent craters (listed in Table

I). Thus, through this procedure, any crater’s diameter

dCcan be obtained by measuring its equivalent diam-

eter in the photograph and multiplying the result by

RM/RMp = 325.6913 km/cm. In Table Iwe show the

eight diﬀerent values dC p from the photograph (in cm)

and the calculated diameters dC(in km) obtained from

Eq. (1).

We compare the estimated diameters with the exact

values dCexact obtained from the Working Group for

Planetary System Nomenclature (WGPSN) from the In-

ternational Astronomical Union (IAU).[13] The error be-

tween the actual value of the crater’s diameter and the

estimate done through our linear calculation is computed

as 100|dCexact −dC|/dC exact.

Clearly, the best results are obtained for the three

smaller craters at the center of the photograph. Con-

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PLEASE CITE THIS ARTICLE AS DOI: 10.1119/10.0006156

3

FIG. 2: Zoomed in view of Fig. 1. Eight craters with

their respective names: (1) Plato, (2) Aristillus, (3)

Autolycus, (4) Archimedes, (5) Timocharis, (6)

Eratosthenes, (7) Copernicus, (8) Eudoxus. The straight

line between points Aand Brepresents the

approximated Euclidean ﬂat distance of Montes

Apenninus. The straight line between points Cand D

represent the approximated Euclidean ﬂat distance of

Montes Caucasus.

versely, the most signiﬁcant error (∼4%) occurs for the

Plato crater. This error can be partially attributed to

the crater’s size and the lack of good luminosity at its

position close to the edge of the photo. However, this er-

ror is an indicator that the linear Euclidean approxima-

tion for distances of this scale is already breaking down.

For such cases we need to consider the curvature of the

Moon’s surface and use non–Euclidean geometry.

The size of the pixels in the photo determines the

uncertainty of the measurements. Here, this measures

10.6/2448 = 0.00433 centimeters (2448 is the width of

the image in pixels). When scaled to the size of the

Moon, this value gives between 1 and 2 km per pixel.

As we will see in the following, this depends on whether

the crater is near the center or closer to the edge of the

photo. The errors obtained in Tables I and II are in this

range.

IV. LARGE DISTANCES, CURVED

GEOMETRY AND GEODESICS

For greater distances on a curved surface, using pro-

portionality is insuﬃcient to provide estimates with rea-

sonable accuracy, as shown in the previous section. To

calculate curved-surface distances it is essential to intro-

duce the concept of geodesic.

A geodesic is the line of minimum length between two

points on a surface. This can involve any geometry, ﬂat

or curved, and, therefore, a geodesic can be straight or

curved. For example, given two points in a plane, the

geodesic is the straight line joining the points. But for

the curved surface of a sphere, the geodesic is an arc of

a great circle which passes through the two points and

whose center lies at the center of the sphere.

For measuring large distances between two points on

the Moon’s surface, it is suﬃcient to approximate the

Moon as a sphere and calculate the geodesics accordingly.

In what follows, we consider the actual values of lunar

features to be those appearing in Google Earth Pro free

software. [9]

As an example, we measure the length of the Montes

Apenninus mountain chain; see Fig. 2. The distance be-

tween points Aand Bon considering the surface to be

Euclidean and using the proportionality approach of the

preceding section is ∼552 km, but the real distance be-

tween these points is 600 km, an error of ∼8 %. On con-

sidering two points even further apart, the error grows

signiﬁcantly. For example, the correct distance between

craters (1) and (7) is 1,300 km, but using proportionality

gives only ∼1,081 km, an error of about 17%.

To consider non-Euclidean geometry, we ﬁrst note that

surface points on any photograph of the Moon project

into the plane of the sky. This plane, which will in-

clude the Moon’s polar axis, will divide the Mo on into

two. This is illustrated in Fig. 3, where the circle repre-

sents the projection of the Moon onto this plane. Con-

sequently, actual geodesic distances between two points

in a photograph must be calculated by “back-projecting”

them onto the spherical surface.

In Fig. 3,Pand Qrepresent two points in the pho-

tograph of the Moon, P′and Q′are the corresponding

points projected onto the Moon’s surface, and CMis the

center of the Moon. The line P′′Q′is a copy of the line

P Q, but shifted toward the lunar surface; that is, to the

outside observer, P′′Q′overlays P Q. The desired dis-

tance is the length of the curved geodesic between points

P′and Q′. This can be calculated using the Pythagorean

theorem and trigonometry as follows:

Geodesic

z}|{

P′Q′=αRM,(2)

where αis the angle formed by the three points P′,CM,

and Q′,

α=

\

P′CMQ′= 2 arcsin d(P′, Q′)

2RM,(3)

This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS ARTICLE AS DOI: 10.1119/10.0006156

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FIG. 3: Scheme where Pand Qrepresent two points in

the Moon photograph. Points P′and Q′correspond to

their projections in the Moon’s spherical surface. Other

points are depicted in order to perform the calculation.

where d(P′, Q′) is the straight-line distance between

points P′and Q′. Hereafter, d(A, B ) is used to denoted

the straight-line distance between any points Aand B.

This distance can be obtained by the Pythagorean theo-

rem as

d(P′, Q′) = pd(P′′, Q′)2+d(P′′ , P ′)2,

=pd(P, Q)2+d(P′′, P ′)2,(4)

where we have used d(P′′, Q′) = d(P, Q). Now,

d(P′′, P ′) = d(P , P ′)−d(Q, Q′),(5)

and

d(P, P ′) = qR2

M−d(CM, P )2,

d(Q, Q′) = qR2

M−d(CM, Q)2.(6)

Using equation (2), we can obtain the real distance

between two points on the Moon’s surface by using the

linear distance between those two points in the ﬂat pho-

tograph and their distances (again in the photograph) to

the center of the Moon by using the following expression:

geodesic

z}|{

P′Q′= 2RMarcsin

1

2RMsd(P, Q)2+qR2

M−d(CM, P )2−qR2

M−d(CM, Q)22

.(7)

This result can be programmed in GeoGebra to eﬃciently compute estimates for several structures on the Moon’s

surface. We have left a sample version in [4], which can be used if the calculations are too advanced for a group of

students. As an application, we use equation (7) to estimate real distances in Fig. 2for several points and compare

them with the values found in Google Earth Pro software [9].

In Table (II) we summarize the sizes (in km) of several structures and percent errors. We show estimates obtained

considering a ﬂat model, a curved model (computed with equation (7)), and the distance obtained in Google Earth

Pro; the errors are calculated using the values obtained from ref [9]. These values are obtained from measurements

of satellite photographs taken by NASA and partner agencies.

The ﬂat approximation produces errors of 10% or larger, whereas those from the the curved approach are signiﬁcantly

smaller. This error is more signiﬁcant if the feature considered is closer to the Moon’s perimeter in the photograph,

as is the case for the crater Eudoxus and Mare Crisium.

As shown in Table (II), the ﬂat estimate is very poor when this occurs. Like before, there are an uncertainty in

these measurements of 1-2 km per pixel due to the pixel size. This uncertainty is, however, much smaller than the

improvement achieved in the estimates when considering a curved surface over a ﬂat one, which is the central point

of this activity.

V. CALCULATING AREAS AND THE

SPHERICAL TRIANGLE

Now that students have been introduced to the con-

cepts of curved surfaces and how to compute distances

over them, we proceed to calculate the area of a curved-

surface feature using a ﬂat Moon photograph. We apply

this later to obtain the area of a polygonal region of the

Moon, Mare Crisium. But ﬁrst we need to describe some

results from spherical trigonometry.

We start with the basic spherical triangle in Figure 4.

This ﬁgure shows three points on a sphere of radius r, to-

gether with the arcs of the great circles that pass through

them. When the angles α,β, and γare known, the area

can be calculated straightforwardly from the expression

[2]

A(△ABC) = (α+β+γ−π)r2.(8)

This result is known as Girard’s Theorem; it also indi-

cates that the sum of interior angles for a spherical tri-

angle is always greater than πradians, something that

does not occur in Euclidean geometry.

PLEASE CITE THIS ARTICLE AS DOI: 10.1119/10.0006156

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FIG. 4: Spherical triangle ABC in a spherical surface

and its interior angles [2]

To calculate these angles, we use the cosine rule for the

sides of a spherical triangle, which relates the angles to

the side lengths of the triangle.[2] In Figure 4, let ABC

be the spherical triangle with sides AB,B C, and CA,

which have geodesic lengths c,a, and brespectively. The

angles at A,B, and Care α,β, and γ, respectively. The

cosine law gives

α= arccos cos ˜a−cos ˜

bcos ˜c

sin ˜

bsin ˜c!,

β= arccos cos ˜

b−cos ˜acos ˜c

sin ˜asin ˜c!,

γ= arccos cos ˜c−cos ˜acos ˜

b

sin ˜asin ˜

b!.(9)

where ˜a=a/r,˜

b=b/r, and ˜c=c/r. The lengths a,b,

and ccan be calculated with the procedures developed in

Sec. IV, and then, using Eqs. (9), we obtain the angles.

With this information, we can calculate the required area

using Eq. (8).

VI. ESTIMATING AREAS ON THE SURFACE

OF THE MOON USING SPHERICAL

TRIANGLES AND POLYGONS

We are now in a position to compute an area on

the Moon’s surface using the methods from the previ-

ous sections. The previous formulas can be programmed

into GeoGebra to compute the estimates. We provide

sample images and GeoGebra ﬁles in a public reposi-

tory that can be found at https://github.com/raxlab/

moonGeometry[4]. Using this code, one can give three

points in the photograph, and, by using a spreadsheet,

compute the total curved area of interest. Any area can

FIG. 5: (a) Zoomed in view of photograph of Fig. 2

showing the spherical triangle with vertices in the

center of craters Aristillus,Autolycus, and Archimedes.

Results obtained with the GeoGebra code are shown.

(b) A spherical triangle is formed by the same craters

constructed with Google Earth Pro software. The

results for the curved areas using the software and our

calculations are displayed in both ﬁgures for

comparison.

be approximated as a sum of several spherical-triangle

areas. Therefore, the key is to estimate as accurately as

possible the area of each spherical triangle. An example

of the result is given in Fig. 5.

A screenshot of the GeoGebra code is shown in Fig-

ure 5(a). We calculate the area of the spherical triangle

between the centers of the craters Aristillus, Autolycus,

and Archimedes. These correspond to the points F, G,

and H in the GeoGebra code, respectively, and to points

2, 3, and 4 in Figure 2. Part of the code is depicted on the

right-hand side of Figure 5(a). We show the curved dis-

tance between points F and G, which is equal to 141.72

km, and the angles in each vertex, in radians.

The normalized curved area is also shown, which, when

multiplied by the Moon’s radius (Eq. (8)), results in the

area for the spherical triangle, which emerges as 7,135.35

km2. As a comparison, the estimation using the ﬂat

area calculation gives only 5,845.25 km2. To compare

these estimates, we use the Google Earth Pro software

to calculate the same spherical triangle. This pro cedure

is shown in Figure 5(b), and results in a total area of

7,119.2 km2. The estimate using curved areas obtained

with our low-resolution photograph diﬀers by only 0.2%

from the result using satellite photographs from Google

Earth Pro.

PLEASE CITE THIS ARTICLE AS DOI: 10.1119/10.0006156

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Finally, the same two ﬁgures show how the curvature

aﬀects the distances. In Figure 5(a), the perimeter of the

triangle calculated by the sum of the three geodesics of

each side (Eq. (7)) gives 429.12 km. The same perime-

ter using Google Earth Pro results in 429.16 km. These

two results are virtually identical (∼0.009% diﬀerence),

which illustrates how good the calculation of curved ge-

ometrical structures can be when using a low-resolution

photograph and care is taken in specifying the points.

We can use the above spherical triangle approach to

obtain areas from the largest observable Moon structures,

the lunar maria. In particular, we analyze Mare Crisium,

which is shown as element 9 in Figure 1. In Figure 6(a)

we present a magniﬁed view.

The procedure is as follows. We can divide an area

into several spherical triangles; the total area will be the

sum of all the individual areas. This is shown in Figure

6(a). This image shows how we calculate the total area

by estimating the mare’s inner boundary with an inte-

rior triangulation, dividing it into ten triangles. Figure

6(b) shows the same procedure for the outer boundary

with an exterior triangulation; we do this to compute

bounds on the estimate of the total area. Deﬁning the

correct boundaries in these types of photographs is very

challenging.

Using this procedure, we obtain a lower bound for the

total area (the inner boundary estimate) of 133,993 km2.

In comparison, the outer-boundary estimate is 205,684

km2. The average of these is 169,839 km2, which diﬀers

by only about 3.5% from the documented area of 176,000

km2[21]. If we calculate these areas using a proportional

approximation of the ﬂat areas, we obtain lower and up-

per bounds of 48,497 and 70,551 km2; both of these are

very diﬀerent than the documented area.

Figure 6(c) shows the same calculation performed with

Google Earth Pro for the inner boundary. This gives a

value of 138,012 km2, very close (∼3% diﬀerence) to

our estimate. Furthermore, notice that in Figure 6(a),

the distance between points Vand Qseems shorter than

distance between Nand T. That the opposite is actually

the case is shown in Figure 6(c). Our calculation using

curved distances automatically deals with this.

In the above analysis for Mare Crisium, the inner and

outer boundaries result in estimates that diﬀer by 71,691

km2(about 34% of the outer-boundary estimate). It is

important to identify the origin of this variation. To see

and comprehend an image, humans need between approx-

imately 103and 104pixels.[5,18] With the same proce-

dure used to measure the areas in Fig. 1, we can measure

the equivalent area that corresponds to one pixel. We

ﬁnd that each pixel corresponds to roughly between 3

and 8 km2. This implies that the minimum error we

can achieve in an area estimate will be of the order of

∼105km2, which is of the same order as those obtained

for Mare Crisium. Although the errors and uncertainties

can be improved by choosing a better point to estimate

the triangulation or by increasing the resolution of the

captured image, there is always error associated with the

FIG. 6: Mare Crisium. (a) Interior triangulation to

calculate its area, by using a ten-sided polygon. (b)

Exterior triangulation with a ten-sided polygon. (c)

Interior triangulation of the area, by using a ten side

polygon, but with Google Earth Pro software.

PLEASE CITE THIS ARTICLE AS DOI: 10.1119/10.0006156

7

pixel sizes of the adopted image.

VII. CONCLUSIONS

This work shows how Moon photography can be used

to teach non-Euclidean geometry. Although this sub-

ject can be complicated, we believe that by using their

own photographs of the Moon, students can better un-

derstand the importance of curved geometry and gain a

deeper appreciation that celestial objects are not ﬂat.

The activity starts with a photograph of the Moon

taken by students themselves with a smartphone. When

distances and areas of some of the structures are calcu-

lated over the picture; for example, for the lengths of

mountain chains or the sizes of craters, students will re-

alize that the results of ﬂat calculations can diﬀer enor-

mously from the documented values. This insight is the

key to introduce curved geometry, which can be devel-

oped using the same photographs and relatively sim-

ple mathematics. In measuring distances, students learn

about geodesics on the surface of a sphere and how to use

them. We further extended the study to calculate non-

ﬂat areas, for which the concepts of spherical trigonom-

etry are essential.

The activity can also include creating a spreadsheet in

GeoGebra. This free software can automate the proce-

dure of using the points needed to construct distances

and triangles. Students will come to realize that their

decisions regarding selected points will impact their ﬁ-

nal estimates lengths and areas. This diﬃculty will put

the process of measuring geological formations in other

moons or planets into context.

As an initial complementary activity, students can per-

form the same analysis for distances and areas for a pho-

tograph of a soccer ball or basketball. This has the

additional pedagogical beneﬁt that they can compare

their calculations with actual lengths measurements, per-

formed by using a string or ﬂexible tape between two

points. This can be considered as an engaging activity.

Again, students will ﬁnd that to get reasonable estimates

they have to consider the curvature of the surface. If they

can program their spreadsheets and secure measurements

for a ball, they should be able to do it for the Moon as

well.

We have used parts of the approach described here

in courses for high school mathematics teachers, who be-

came motivated to adopt these exercises. We believe that

the best way to use this approach is by challenging teams

of learners. Depending on the context, the teacher can se-

lect either an active expository or a dynamic class, where

the students can use the above equations as an input to

perform their calculations and predictions. In the lat-

ter, teams of students are set to work consecutively the

topics we list in this article. According to the method-

ology used to address the problem, each topic can be

covered in one or two 60-minute classes. For college-level

courses, the teacher can do a guided activity as described

above. In contrast, for summer workshops in astronomy,

the teacher can do a research activity where the professor

can exhibit the main questions or challenges of measuring

distances and areas on spherical objects and then chal-

lenge students to do so with a photo of the Moon, giving

them a copy of this article as a reference.

In summary, we believe that the approach to intro-

ducing curved geometry outlined here is challenging and

motivating for participants. Our experience has been pre-

cisely this for both students and teachers.

An extended version of this paper can be found in [4].

Acknowledgments

This research was partially funded by grants

ASTRO020-0018, ASTRO20-0058, and FONDECYT

1180139 from ANID, Chile. The authors would also like

to thank the reviewers and the editors for their comments

and recommendations, which improved our work.

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38(5):55706, Aug 2017.

[2] David A. Brannan. Geometry. Cambridge University

Press, 2nd edition, 2011.

[3] Hugo Caerols and Felipe A. Asenjo. Estimating the

Moon-to-Earth Radius Ratio with a Smartphone, a Tele-

scope, and an Eclipse. The Physics Teacher, 58(7):497–

501, 2020.

[4] Rodrigo A. Carrasco, Hugo Caerols, and Felipe A.

Asenjo. Moon Geometry Repository v.1.3.https://

github.com/raxlab/moonGeometry, 2021.

[5] Yang Cai. How Many Pixels Do We Need to See Things?

In Sloot P.M.A., Abramson D., Bogdanov A.V., Gor-

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9

Elements (A) (B) (C) (D) (E)

straight-line geodesic accepted % errors % errors

Eudoxus, point 8 in Figs. 1and 247 68 67 1.49 30.9

Montes Caucasus 327 438 443 1.13 25.3

Mare Crisium, point 9 in Fig. 1203 546 555 1.62 62.8

Montes Apenninus 552 596 600 0.67 7.4

Distance between points 1 and 7 in Fig. 21,081 1,302 1,300 0.15 17

TABLE II: Distances (in km) on the Moon’s surface. Column (A) lists lineal (straight-line) distances, column (B)

are geodesic distances based on Eq. (7), column (C) are Google Earth Pro “accepted” values, column (D) are the %

errors between our estimates and the accepted values, and column (E) are the % errors between the straight-line

and geodesic distances. Mare Crisium has an elliptical shape. We use the distance between points Vand Qin ﬁgure

6(a) to calculate its radius, as explained later.

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