DOI: https://doi.org/10.62204/2336-498X-2023-4-10

SCIENTIFIC PRINCIPLES OF LEARNING SPHERICAL

GEOMETRY IN INSTITUTIONS OF HIGHER EDUCATION

Natalia Shapovalova,
Ph.D. in Physico-mathematical Sciences, Associate Professor,
Mykhailo Dragomanov State University of Ukraine,
n.v.shapovalova@udu.edu.ua; ORCID: 0009-0000-7084-1460
Larisa Panchenko,
Ph.D. in Pedagogy, Associate Professor,
Mykhailo Dragomanov State University of Ukraine,
larpan97@gmail.com; ORCID: 0009-0001-8156-286X
Olga Mihulova,
 Master of Mathematics,
Mykhailo Dragomanov State University of Ukraine,
rukovoditel10b@gmail.com; ORCID: 0009-0008-9916-0560

Annotation. The article provides historical information about the emergence and development of spherical geometry, considers the basic concepts of spherical geometry, reveals and improves the scientific principles of teaching spherical geometry in institutions of higher education. The interdisciplinary connections of spherical geometry with geography, astronomy, navigation, cosmology, and others are described. The determined potential of spherical geometry in the field of astronomy, where determining the coordinates of objects on the celestial sphere is important for understanding cosmic phenomena. Since spherical geometry is an important tool for the analysis and modeling of real phenomena in three-dimensional space, its application in modern GPS technologies and geospatial information systems emphasizes the relevance of this area of research.

The peculiarities of teaching spherical geometry in institutions of higher education are analyzed and the main methodical aspects of this process are revealed. The purpose, content, basic provisions of spherical geometry are considered, and modern approaches and methods of its teaching are proposed. The proposed use in the educational process of practical and applied applications of facts of spherical geometry, tools of dynamic geometry, interdisciplinary connections of spherical geometry with physics, biology, astronomy, cosmology.

Keywords: spherical geometry, basics of geometry, sphere, competence, interdisciplinary connections, educational process, learning, scientific approach, physics, astronomy.

Problem statement. In the educational process, both in secondary education institutions and in institutions of higher education, mathematics acts as one of the key subjects that contributes to the mastery of other disciplines and finds wide practical application. Particular importance is attached to the study of geometry, which plays a specific role in learning and practical activities.

Knowledge of geometry is of great importance, because in modern society, the development of spatial concepts and logical thinking become decisive for the professional competence of specialists in many fields. The geometry course determines the level of development of these skills, which is a key criterion for professional success.

The study of the properties of geometric bodies and the study of their connections not only form the scientific worldview of students and pupils, but also contribute to the development of logic, systematicity and the ability to express one’s thoughts in a reasoned way. Knowledge of geometry opens up wide opportunities for the development of the culture of thinking, spatial ideas, creativity and independence in learning.

The development of spatial thinking, in particular in the study of spherical geometry, is an important component of education. Logical thinking acquired through geometric studies is necessary not only for mathematicians, but also for representatives of other sciences, contributing to the correct analysis, formulation of hypotheses and solving problems in various fields of knowledge.

Like the geometry of Euclid, spherical geometry is used for practical purposes, in ancient times it was required by the science of astronomy. This knowledge was important, for example, to travelers and sailors who charted their routes by the stars. Since during astronomical observations, it was believed that the Sun, Moon and stars move on an imaginary “heavenly sphere”, therefore, knowledge of the geometry of the sphere was necessary to study their movement.

Currently, spherical geometry is widely used in the following areas:

  1. Geodesy and navigation. Spherical geometry is the foundation for solving problems of location, geodesy and navigation.
  2. To calculate the path, distances and angles in navigation and aviation;
  3. To describe the method of finding the shortest distance between two points along

the Earth’s surface, if their geographic coordinates are known;

  1. To determine the initial course of the ship when moving from one point to another,

if the geographical coordinates of these points are known.

  1. Spherical geometry is used in cartography: in particular, in the construction of map projections.
  2. Geoinformation systems (GIS). Nowadays, GIS have become an integral part of many industries, including geology, urban planning, transport logistics, and others. The use of spherical geometry in GIS helps to correctly display and analyze information for the further correct implementation of the tasks.
  3. In space research. Calculation of distances, angles and coordinates of objects in outer space is important.
  4. In the modern world, mobility and globalization are increasing, information about the modern world and its processes are becoming increasingly important. Understanding spherical geometry helps to effectively interact with global aspects of life, including international travel, communication, trade.

Therefore, research and application of the facts of spherical geometry remain relevant due to their applied importance in various fields of science, technology and everyday life. Therefore, the development of the scientific foundations of teaching spherical geometry in institutions of higher education is an urgent need.

Analysis of recent research and publications. Hipparchus of Nice (180-185 BC)

and Leonard Euler (1707-1783) are considered to be the founders of spherical geometry. Spherical geometry arose in the 1st-2nd centuries AD, when, after the Roman conquests, close contact was established between Greek and Alexandrian geometers and Babylonian astronomers. In the 1st century the “spherical” of Menelaus appeared, which was used for astronomy by the famous Claudius Ptolemy. Later, with the development of navigation and geography, spherical geometry began to be used for the surface of the globe.

Questions related to the study of spherical geometry are very closely intertwined with the features of psychology and the theory of cognition in general, with questions about how spatial imagination and intuition arise. Famous scientists Leonard Euler, Georg Friedrich Bernhard Riemann, Felix Klein, Henri Poincaré, A.D. Aleksandrov, I.P. Yegorov, O.S. Smogorzhevskyi, N.V. Yefimov, L.S. Atanasyan, O.V. Manturov, V.P. Yakovets and others.

The purpose of the article. The purpose of the article is to use available sources, scientific literature and own experience to reveal and describe the results of the study of the specifics and features of teaching spherical geometry to students of higher education institutions. To reveal the main methodological aspects of this process and to improve the scientific principles of teaching spherical geometry in institutions of higher education. To analyze the practical application of the facts of spherical geometry for solving real tasks and problems in various fields of science and practice.

Research methods in spherical geometry. Studies of spherical geometry include the use of various methods that allow studying the properties and relationships between objects on the sphere.

Here are some basic methods used in the study of spherical geometry.

  1. The analytical method includes the use of analytical coordinates on the sphere to express geometric objects and their properties. Solving systems of equations that correspond to the equations of spherical geometry.
  2. The synthetic method consists in constructing geometric figures and proving statements with the help of special spherical constructions and mutual positions of objects and using spherical lune or spherical digons or figure having two angles, spherical triangles and their properties to study geometric problems.
  3. The methods of differential geometry include the use of concepts and statements, theorems and criteria of differential geometry to study the properties of curves and surfaces on a sphere, the determination of vectors and curves for objects of spherical geometry.
  4. Methods of projective geometry – the use of projective transformations to study the properties of spherical shapes when transforming spherical objects in order to study their characteristics.
  5. Geodetic methods consist in the use of geodetic measurements and triangulation to determine distances and angles between points on the sphere, solving navigational and cartographic tasks using spherical calculations.

These methods are often combined for a more effective and comprehensive study of the facts of spherical geometry and taking into account their specificity.

Presenting main material. There is much in common between the geometry of the Euclidean plane and spherical geometries; this is explained by the fact that the sphere has the same “mobility” as the plane: an arbitrary point of the plane and the directions emanating from it can be combined by the movement of the plane with any other point of the plane and the direction emanating from it, and also an arbitrary point of the sphere and the direction emanating from it can be combined by the rotation of this sphere with any other point of the sphere and the direction emanating from it.

If the main concepts of geometry on the Euclidean plane are a point, a straight line and the movement of a plane, then in spherical geometry the point of a sphere, a great circle and the movement of a sphere play the same role [8, p.49].

The cross-section of a sphere by an arbitrary plane is a circle, because if you drop a perpendicular from the center of the sphere onto this plane and rotate the space around this perpendicular by an arbitrary angle, then when you rotate, both the sphere and the plane and the line of their intersection will pass into itself; therefore, an arbitrary point of this line of intersection is at the same distance from the point of intersection of the plane with the perpendicular, and this line of intersection is a circle.

When the plane passes through the center of the sphere, i.e. represents a diametrical plane, the circle on the sphere is called a great circle; all other circles on the sphere are called small circles.

Since a single plane passes through arbitrary three points of space that do not lie on the same straight line, then a single diametrical plane passes through two arbitrary points of the sphere that are not diametrically opposite.

Therefore, a single great circle passes through two points of the sphere that are not diametrically opposite. This fact is analogous to the fact that a single straight line passes through two points on the Euclidean plane. An infinite number of large circles can be drawn through two diametrically opposite points. Since any two diametrical planes of the sphere intersect along its diameter, then any two great circles intersect at two diametrically opposite points of the sphere.

Since a plane divides space into two regions, a great circle divides a sphere into two regions, these regions are called hemispheres. Because two intersecting planes divide space into four regions, two great circles divide the sphere into four regions. As three planes intersecting at one point divide space into eight regions, so three great circles not intersecting at one point divide the sphere into eight regions.

If the first two of these properties are analogous to the properties of straight lines on the Euclidean plane, which is divided into two regions by a straight line and into four regions by two intersecting straight lines, then the third property is not analogous to the corresponding property of straight lines on the Euclidean plane, since three pairwise intersecting straight lines do not pass through one point, divide the plane into seven parts.

The great circle corresponds to two diametrically opposite points of the sphere cut out of it with a diameter that is perpendicular to the plane of the great circle. These two points are called the poles of the great circle.

Two diametrically opposite points A and B on the sphere correspond to a single great circle, for which points A and B are poles; this great circle is called the pole of a pair of diametrically opposite points A and B. Each point of the pole is called polar conjugate to each of its poles, in other words, the points P and Q of the sphere are polar conjugate if the radii OP and OQ are perpendicular (O is the center of the sphere).

The concept of motion on the sphere can be introduced analogously to the corresponding concept on the Euclidean plane. The movement of the sphere is such a transformation of the sphere in which the distance between the points is preserved. In other words, the transformation φ of the sphere is a movement if, for arbitrary points A, B of the sphere, the distance between the points A’=φ(A) and B’=φ(B) is equal to the distance between the points A and B: AB = A’B’.

The basic properties of motions in the Euclidean plane carry over accordingly to motions on a sphere, but motions on a sphere have some distinctive properties that motions on the Euclidean plane do not. In particular, since two points A and B are diametrically opposite if and only if the distance between them is the largest possible value equal to 2r (where r is the radius of the sphere), then it follows from the definition of motion that in arbitrary motion the spheres are diametrically opposite points of the sphere pass into diametrically opposite points. This property has no analogue in the Euclidean geometry of the plane, since there are no such pairs of points on the Euclidean plane that the movement of one of these points determines the movement of the other. Therefore, if the movement of a plane is a transformation of a set of points of this plane, then the movement of a sphere is essentially a transformation of a set of pairs of diametrically opposite points of the sphere.

The simplest motions of the sphere are the rotation of the sphere about any axis passing through the center of the sphere, the symmetry of the sphere with respect to any plane passing through the center of the sphere, the symmetry of the sphere with respect to its center.

As in planimetry, the composition of any two motions of a sphere is also a motion of a sphere. Spherical geometry studies those properties of figures on the sphere that are preserved during arbitrary movements of the sphere.

Figures on the sphere that can be translated into each other by some movement of the sphere are called equal figures, the geometric properties of equal figures are the same.

Since the arbitrary motion of the sphere translates a pair of diametrically opposite points into a pair of diametrically opposite points, the pair of diametrically opposite points in spherical geometry is an independent geometric object.

It is appropriate to draw students’ attention to one remarkable property of these pairs of points: each theorem of spherical geometry corresponds to another theorem of this geometry, which is obtained with the first substitution of the words: “pair of diametrically opposite points” and “great circle”, “lies on” and “passes through” , “connect” and “cross”. Example:

Two pairs of diametrically opposite points of the sphere are connected by one big circle. Two great circles on the sphere intersect at one pair of diametrically opposite points.

This property of the theorems of spherical geometry is a consequence of the fact that a pair of its poles mutually uniquely corresponds to each great circle on the sphere, and to any pair of diametrically opposite points of the sphere, their polar corresponds mutually uniquely, and if a great circle passes through a pair of diametrically opposite points, then the poles of this circle lie on the polar of this pair of points. And this property is called the principle of duality, and the theorems obtained from each other by the specified substitution are called dual theorems. If one of the dual theorems is proved, then the proof of the second theorem can be obtained from the proof of the first theorem by going from each great circle to its poles, and from each pair of diametrically opposite points to its poles.

The angle between two intersecting lines in space is called the angle between the tangents to these lines at the point of their intersection. A partial case of the general concept of an angle between two lines is the angle between two great circles on a sphere. The angle on the sphere is equal to the length of the arc of the great circle between the points of the sides of the angle, polar conjugate to the apex of the angle, divided by the radius of the sphere.

Lines, angles, triangles, curves and other figures on the sphere have specific properties.

Two great circles define four angles between two semicircles, pairwise equal to each other. Those of these angles, both sides of which are continuations of the sides of the second angle, are equal and are called vertical angles, those of these angles that have one common side, and in the sum make up the expanded angle π are called adjacent angles. The angle between the two great circles is equal to the length of the arc connecting the poles divided by the radius of the sphere. Great circles, one of which passes through the pole of the other, intersect at right angles. Such large circles are called perpendicular. Each of the two perpendicular great circles passes through the pole of the second great circle. From this it follows that a great circle is the pole of the point of intersection of two great circles, perpendicular to the two great circles, that is, two great circles always have a single great circle, which is perpendicular to both of them.

For comparison, note that on the Euclidean plane, a common perpendicular can be drawn only to parallel straight lines, and not one, but many common perpendiculars can be drawn to two parallel straight lines.

Three great circles on the sphere, not intersecting at one point, divide the sphere into eight regions. Each of these areas, bounded by arcs of three great circles, is called a spherical triangle. That is, a spherical triangle is a figure formed by three arcs of large circles that intersect at three points, for example, triangle ABC. Arcs of great circles bounding a spherical triangle are called its sides, the ends of these arcs are called its vertices, and the angles formed by the sides of a spherical triangle at its vertices are called angles of a spherical triangle [1, p.111-112].

It is clear that each side of a spherical triangle is less than half of a great circle.

Considering the properties of triangles on a sphere, it should be noted that while the sides of a triangle on the Euclidean plane are straight line segments and are measured in linear units, the sides of a spherical triangle are arcs of great circles and are measured in arc units – degrees or radians.

Each side of a spherical triangle is less than the sum of the other two and greater than their difference. The semi-perimeter of a spherical triangle is always larger than each of its sides. The sum of the sides (perimeter) of a spherical triangle is always less than 360o and greater than zero.

In spherical geometry, there is such a figure as a spherical lune or spherical digons or  figure having two anglest, which does not exist on the Euclidean plane. A dihedral is a part of a sphere bounded by two halves of large circles with common ends; these common ends, called vertices of the dihedral, are diametrically opposite points of the sphere.

Two spherical triangles are called equal if they can be aligned with each other by the movement of the sphere. It is obvious that a correspondence can be established between the vertices of two equal spherical triangles, in which both the corresponding sides and the corresponding angles of these spherical triangles are equal: for this, each vertex of the first spherical triangle must be matched with the vertex of the second spherical triangle into which it passes at combinations of these spherical triangles.

We have six signs of the equality of spherical triangles: two spherical triangles are congruent if:

  • two sides of one spherical triangle are equal to two sides of another spherical triangle and the angles between these sides are equal;
  • two angles of one spherical triangle are equal to two corresponding angles of another spherical triangle and equal sides between these angles;
  • all sides of one spherical triangle are equal to the corresponding sides of another spherical triangle;
  • two sides of one spherical triangle are equal to two corresponding sides of another spherical triangle, the angles opposite the other two sides are simultaneously acute or obtuse;
  • two angles of one spherical triangle are equal to two corresponding angles of another spherical triangle, the sides lying opposite two equal angles are equal, and the sides lying opposite two other equal angles are simultaneously smaller or larger than πr;
  • all three angles of one spherical triangle are equal to the corresponding angles of another spherical triangle.

The first four of these signs of equality are analogous to the signs of equality of triangles on the Euclidean plane. The fifth sign of the equality of spherical triangles also has an analogue in geometry on the Euclidean plane, but with the difference that the fifth criterion for the equality of plane triangles, that is, triangles on the Euclidean plane, does not have a condition similar to the condition formulated at the end of sign of equality of spherical triangles. The sixth sign of the equality of spherical triangles has no analogue at all in the geometry of the Euclidean plane, where the equality of the corresponding angles of two triangles is a sign not of equality, but of the similarity of triangles.

The concept of equality of figures on the sphere can be introduced in the same way as it is done for figures on the Euclidean plane. First, two spherical figures are said to be equal if they have equal corresponding elements. Secondly, two spherical figures are called equal if, by some movement on the sphere, one of them is reflected on the other, while the vertices of one figure become the vertices of the other so that the order of the vertices is preserved. In the school course of geometry, it is proved that for a plane such two definitions of the equality of figures are equivalent.

Note that the equality of figures is also defined as follows: two spherical figures are called equal if one of them can be combined with the other by overlapping. It is clear that all the corresponding elements are equal in the shapes that overlap when superimposed.

But the reverse statement for spherical figures is not always correct. For example, let us have two triangles ABC and A’B’C’ such that all the elements of the triangle ABC are equal to the corresponding elements of the triangle A’B’C’, but they cannot be combined by overlapping, because they have the opposite orientation. For such triangles, instead of the term “equal”, the term “symmetric” is used.

In an arbitrary spherical triangle, each side is less than the sum of the other two sides, and greater than their difference.

From this, as in the geometry of the Euclidean plane, the consequence follows that in an arbitrary spherical triangle the larger side lies opposite the larger angle, and the larger side lies opposite the larger side.

Due to the fact that the length of any continuous line on the sphere can be replaced with a very small error by the length of the line consisting of arcs of great circles connecting the points of the given lines, then the arc of a great circle, a smaller semicircle, is shorter than any continuous line on of the sphere connecting the same points of the sphere, that is, this arc of the great circle is the shortest line on the sphere. In this respect, a great circle is analogous to a straight line on a plane. We can see from here. That this line on the earth’s (spherical) surface, which is obtained on it by hanging and which in small areas is taken for a straight line, when sufficiently extended is an arc of a great circle. Since these lines are drawn by surveyors, great circles are also called geodetic lines on the sphere.

In any spherical triangle, the difference of the sum of any two angles and the third is always less than two right angles. In any spherical triangle, the sum of the angles and the third is always less than 540 о and greater than 180 о, i.e.

180 о < ∟А+∟В+∟С < 540 о.

The sum of the angles in a spherical triangle is a variable value and is always greater than 180о, i.e. ∟А+∟В+∟С= 180 о+ ε [1, p.114-115].

It should be noted that in spherical geometry there is even a geodesic triangle with three right angles.

Note that in a spherical triangle, the concepts of bisectors, medians and heights, as well as the ratio between sides and angles have the same meaning as in a triangle on a plane.

In particular, te following statements apply:

  1. Opposite equal sides of a spherical triangle lie equal angles and vice versa.
  2. In an arbitrary spherical triangle, the larger side lies against the larger angle and vice versa.
  3. In an isosceles spherical triangle, the angles lying opposite equal sides are equal.

Therefore, the formulas of spherical geometry for figures with small linear dimensions compared to the radius of the sphere coincide with the corresponding formulas of Euclidean geometry.

The sum of the angles of a spherical triangle is greater than the extended angle. This is a significant difference between geometry on a sphere and geometry on the Euclid plane and geometry on the Lobachevsky plane.

In spherical geometry, there are no similar triangles because the angles of a triangle uniquely determine its sides.

Parallel “straight lines” cannot be drawn in spherical geometry, while there are parallel straight lines on the Euclid plane and the Lobachevsky plane.

Conclusions. In the process of teaching spherical geometry, which is one of the nonEuclidean geometries, it is advisable to use comparative analysis, namely to compare the statements of parabolic geometry or Euclid’s geometry, hyperbolic geometry or Lobachevsky’s geometry, projective geometry, spherical geometry, elliptic geometry or Riemann’s geometry, activating facts known to students , and identify their common or distinctive features. The most effective methods of teaching non-Euclidean geometries are the explanatory and illustrative method and the heuristic conversation. It is during the heuristic conversation that students compare statements of non-Euclidean geometries with their counterparts from Euclidean geometry.

In order to increase the level of educational activity, it is necessary to continue to form general mental actions and methods of mental activity in students, to strengthen the motivation of learning and to use traditional and new technologies, modern information technologies that activate educational and cognitive activity.

In the process of learning spherical geometry, it is advisable to create and use reference notes as one of the teaching aids.

Reference abstract (RA) is a visual structural and logical diagram, with the help of which educational material is presented in a condensed form, taking into account essential connections and relationships.

One of the main advantages of using spherical geometry is that it enables the researcher to model and analyze phenomena in three-dimensional space, making it an important tool in fields related to geography, astronomy, navigation, cosmology and other fields.

Having analyzed a variety of applied problems, we determined that spherical geometry is successfully used in solving problems related to determining distances, directions, angles and other parameters on a sphere. Another important aspect is the possibility of taking into account the curvature of the Earth when modeling geographical phenomena. For example, in navigation systems, where determining the exact location and determining the optimal routes become critical tasks.

The research shows that spherical geometry allows to effectively solve problems related to global positioning and navigation on Earth, as evidenced by its use in modern GPS technologies and geospatial information systems.

Also useful for the educational process will be the use of tools of dynamic geometry, interdisciplinary connections of spherical geometry with physics, geography, biology, astronomy, seafaring, cosmology.

In further research, it is possible to expand the scientific principles of learning spherical geometry in institutions of higher education and the application of facts of spherical geometry in other fields of science and technology, developing new methods and approaches for solving real problems. Using interdisciplinary connections, it is important to consider the prospects of using spherical geometry in innovative technologies and new scientific research.

Thus, the study of the properties of geometric figures in non-Euclidean geometries expands students’ understanding of the modern picture of the universe, increases the competence of future specialists and stimulates their own search for new mathematical, geometric and physical ideas and theories [8, p.52].

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