The Algebra of Projective Spheres on Plane, Sphere and Hemisphere

Numerous authors studied polarities in incidence structures or algebrization of projective geometry [1] [2]. The purpose of the present work is to establish an algebraic system based on elementary concepts of spherical geometry, extended to hyperbolic and plane geometry. The guiding principle is: “The point and the straight line are one and the same”. Points and straight lines are not treated as dual elements in two separate sets, but identical elements within a single set endowed with a binary operation and appropriate axioms. It consists of three sections. In Section 1 I build an algebraic system based on spherical constructions with two axioms: ab ba = and ( )( ) ab ac a = , pro-viding finite and infinite models and proving classical theorems that are adapted to the new system. In Section Two I arrange hyperbolic points and straight lines into a model of a projective sphere, show the connection between the spherical Napier pentagram and the hyperbolic Napier pentagon, and describe new synthetic and trigonometric findings between spherical and hyperbolic geometry. In Section Three I create another model of a projective sphere in the Cartesian coordinate system of the plane, and give methods and techniques for using the model in the theory of functions.


Introduction
Since the beginning of my research in 1969 I have always tried to combine abstract theory with direct experimentation on real spheres in order to construct geometric figures and configurations. When studying the theoretical background, I was surprised at the stark contrast between the simple and elegant handling of points and the equatorial line on the sphere.
What is the advantage of the basic element so chosen? Let ( ) , , X X x ′ x de-note a basic element with opposite points X, X', and equator x. Two basic ele- uniquely determine a third basic element In this way, an algebraic structure with a binary operation is established in the set of spherical basic elements. The resulting element can geometrically be constructed from the factor elements in several ways. It is up to us to choose whatever representation is most convenient for us.

Basic Algebraic Properties of the Operation
Notation: Given elements x and y, denote the operation by writing the two factors without any operation sign between them: = xy z .
At this stage the algebraic properties are proved by referring to the geometry of the sphere. Later on an axiomatic system is created to build the proofs on the axioms whenever possible.                    Call a set that satisfies these axioms a projective sphere. Note that the product xx is not disposed of in the axioms. The axioms only claim that ab ba

Choose Axioms from the above Properties
The xx case will be examined below.

Binary Operation vs. Binary Relation
The above definition of projective spheres is built on a binary operation. Another option is to start with a binary relation based on the notion of incidence between points and lines [5] [6] [7].
Consider a set of elements with a binary relation R defined between any two elements of the set, including the x R x case, for which the following two axioms hold: 1) = x R y y R x (symmetric relation); (any two different elements ≠ x y determine one element z for which z R x and z R y) The operation can be deduced from the relation or conversely. The present paper focuses on the operation. That is why the main axiom ( )( ) = ab ac a plays a key role, similar to the associative axiom ( ) ( ) = a bc ab c in the theory of groups and fields. Nevertheless, the relation is also frequently used when it is more viable, mainly in Section Two and Section Three.

Theorems Deduced from the Axioms
Note that the theorems and proofs are based only on the algebra of projective spheres, without reference to geometric properties.

I. Lénárt Journal of Applied Mathematics and Physics
Theorem 1 ( Figure 8): aa cannot be defined for all elements of the set with at least two different elements. At first sight, this property seems a trifle, no more than a remark. Actually, it is of fundamental importance for the whole system. In his theory that relates to the present one in several ways, Devidé ranked this property among the axioms [5]. I deduce it as a theorem from the two axioms.
Proof: I prove the following: If there is at least two different elements a b ≠ in the set, the expression (ab)(ab) cannot be well defined for all elements of the set. The proof is short: Consequence: This is a partial binary operation with aa undefined (Cf. division by zero among numbers). It cannot be extended into a non-partial operation that defines the product of any two elements, including identical elements.
Note that the proof does not preclude the option that aa is interpreted for a subset of the set.
Theorem 2 ( Figure 10): From ab ac = it follows that ab ac bc = = . Proof: The assumption needs to be refined. If b c = , then ab ac = , but bc cannot be performed because of Theorem 1 that was already proved.
The only escape from the contradiction is to assume that ab cd bd = = , for in this case Theorem 1 forbids to execute operations (ab)(bd) and (cd)(bd).
Same for any other equality in ab cd ad bc ac bd = = = = = . Q.E.D. Theorem 4 ( Figure 13): From ab c = and bc a = follows ca b = .
Proof (almost trivial): Theorem 5 ( Figure 14): a, ab, (ab)a make an octant, a triangle with three right angles. This means that the product of any two elements among a, ab, (ab)a gives the third element.
Proof: Apply Theorem 2: ( ) The theorem remains valid for c a The expression (ab)a or (ab)b cannot be simplified in the general case.

Other Properties in Subsection 2
This property cannot be derived from the axioms.

Models of Projective Spheres
Model 1 consists of three elements of symmetric roles. In the operation (Cayley) table the main diagonal is empty due the "aa undefined" property. The product of two different elements is the third element. No solution for the equation ux x = . I call this set a Klein projective sphere or K-sphere because it is similar to the Klein group deprived of the unit element.
Another form of displaying the set is the incidence table. I call an element in the last row the title element of the column above it which contains all elements incident to the title element below the column. If element a is in the column of element b, then b is in the column of a.The product of any two elements above a title element is the title element. However, the roles of the three elements are not symmetric, and ux = x does have solutions for x in this set: ab b = ; ac c = .
An element x is reflexive if there is at least one solution of equation ux x = (The names "self-conjugated" or "self-incident" are also used).
Elements ab b = and ac c = are reflexive in this model, while a is non-reflexive. Table 1 and Table 2 show a Klein sphere displayed in an operation table and an incidence table. Table 3 and Table 4 show a Galois sphere of three elements with two reflexive (self-incident) elements also in an operation table and an incidence table.
Model 3: A set of seven elements: Check the two axioms: In the operation table (Table 5)  In the incidence table (Table 6) the last row contains the title elements in the reverse order. The product of two elements can be determined in two ways: Find the title element of the only column that contains the two elements together; or, Find the common element of the two columns whose title elements are the two factors.
Model 4: A set of 13 elements: The number of reflexive elements in a Galois sphere is the same as the number of elements incident to one and the same element, that is, the number of elements in the column above the title element. Table 7 shows a Galois sphere, reflexive elements shown in italics, as with Table 4 and Table 6.      Table 5. Fano sphere in an operation table.  Table 7. Galois sphere with four reflexives in an incidence table. , , , , ; 0 a a a la la la l = ≠ . The product ( ) 1 2 3 , , x x x of elements ( ) , , a a a and ( ) 1 2 3 , , b b b is given by an indeterminate system of two linear equations: Model 7: Pencil of Euclidean straight lines through a point in 3D space [6].
The product of two lines is their common perpendicular in the pencil (Cf. vectorial product).
Model 8: Real and ideal points and real straight lines on the hyperbolic surface (Section Two).
Model 9: The Cartesian coordinate system of the plane with twofold polarity (Section Three).

Theorems about Reflexive Elements
(x is reflexive if ux x = has at least one solution.) Theorem 9: Two reflexive elements cannot be incident to each other (In the column of a reflective element there is no other reflective element).
This is a fundamental property with important consequences for the structure of a finite or infinite projective sphere that contains reflexive elements. ( ) c bc b ab a = = = which is a contradiction. Q.E.D.

Connection of Projective Spheres with Projective Planes
Many theorems and proofs in the theory of projective planes can be formulated and proved for projective spheres, such as: Consequences of the existence of a triangle or a quadrilateral in the set; Theorems on perspectivity and projectivity; Number of elements in finite non-trivial sets [9]; Theorem of Gleason on Fano planes [10]; Theorem of Bose on ovals; Theorem of Baer on polarities, n-chains, n-cycles, and number of reflexive elements; Theorem of Bruck and Ryser on subsets; etc.

Change of Notation
One of the greatest obstacles to the acceptance of a new theory is a new type of notation. Grassmann used a new symbol in his epoch-marking work that made his thoughts unaccessible for most of his contemporaries. Still, I take the risk, because I agree with Struik's remark [11] on the development of mathematics: "The improvement in technique was a result of the improvement in notation".
The binary operation in projective spheres is not associative: ( ) ( ) ab c a bc ≠ in the general case. The term abc is therefore meaningless in this form. A complex expression involves a forest of parentheses and brackets which make the notation awkward and confusing. It took me decades to realize that change was necessary, and another couple of years to get used to my own reforms. However, once getting used to the new notation, I found that the benefits outweighed the inconveniences.
In what follows, the operations are performed strictly from the left to the right. Any variance from this order is indicated by vertical bars. There is no bar between the factors for the strongest priority; one bar for the less strong priority; two bars for the next, even less strong priority, and so on. It is often useful to rearrange the order of the elements to reach minimal number of bars. The expressions in the present paper contain at most three bars between two factors. Examples: The meaning of expression 1234 with traditional brackets is ( )

Some Noteworthy Configurations
Suppose that the expressions are defined. For example, 123|231 implies that 1, 2, 3 are all different, 12 3 ≠ , 23 1 ≠ , 123 231 ≠ . Figure 22 shows the three concurrent altitudes and the orthocentre of the  Figure 23 shows the Pappos configuration in a premissa-conclusion form: If a a a a a a = = ; 1 2 b a b a b a b a b a b a b a b a b a b a b Figure 24 shows that this configuration can be built on five independent elements: 13 Figure 25 denotes the vertices of two perspective triangles of the Desargues configuration: If 1 1 2 2 a a b b a a b b a a b b a a b b a a b b a a Figure 26 displays one triangle denoted by vertices, the other by sides: Figure 27 demonstrates that the Desargues configuration needs six elements. Five elements are independent, the sixth element is only for generating element 16 (The structure remains the same if element 6 is moving anywhere along the product 16). With this notation, we have: The Little-Desargues configuration ( Figure 28) is derived from the third definition of the Desargues configuration ( Figure 27) replacing "16" or "61" by 1, and "1" by 341. For example: 61 | 23 | 4 | 5 1 | 2 | 34 1| 23 | 4 || 5 | 341| 2 | 34 2314 | 3415 | 2 | 34 = = " " " " . Element 1 is shown as a line, elements 2, 3, 4, 5 are points in Figure 28.

Proving Theorems
Following are some examples of applying the algebra of projective spheres on the equivalents of classical theorems in projective planes. The proof shows the three steps that many other proofs follow in projective spheres and projective planes (cf. Hilbert's proofs about Desarguesian planes in [9]). First step: Lemma.
Second step: Dual of the lemma. Third step: Final proof, based on the two dual lemmas.
The following proofs make extended use of Theorem 2 using the short form

Theorem of Hessenberg
Theorem 11: A Pappos sphere is a Desargues sphere.
Assumption: Given a Pappos sphere (in which the Pappos configuration is added as the third axiom to the two axioms in subsection 3), prove that it is also a Desargues sphere.
Following is Hessenberg's proof [12] translated to the algebra of the present system.
Apply the first form of Pappos configuration ( Figure 23) according to the following pattern: a a a a a a = = , 1 2 Conclusion: b a b a b a b a b a b a b a b a b a b a b a Apply Definition 3 of the the Desargues configuration ( Figure 27
A system which is built on two dual sets of "points" and "lines" does not deal with the product of a "point" and a "line", two elements taken from two different sets. The projective spheres are built on one set of elements. Therefore the product of a "point" and a "line" is also defined, for example, as a perpendicular dropped from the point to the line.
The Hesse-Chasles or Hesse configuration only requires three elements, in contrast with the four independent elements of the Fano configuration, or five elements of Pappos, or six elements of the Desargues configuration.
Many classical algebraic systems use axioms involving three elements, such as semigroups, Clifford algebras, Lie-Killing or Jordan algebras, Gatial algebra, etc. [13]. In like manner, we can study projective spheres with the Hesse configuration as the third axiom to the two basic axioms. An algebraic system with only three elements used in the axioms may prove to be easier to handle than another system of axioms with five or six defining elements.
Following are two simple theorems about well-known geometric facts in a Hesse sphere, that is, in a projective sphere endowed with the Hesse configuration as the third axiom in the set.

Theorem about Quadrilaterals in a Hesse Sphere
Q.E.D.

Two Non-Isomorphic Projective Spheres with 21 Elements
The theory of finite projective planes admits only one projective plane with 21 points and 21 lines. In contrast, there are two non-isomorphic projective spheres with 21 elements shown in Table 9 and Table 10. Their incidence tables are isomorphic, but the title elements in the last row are arranged differently [6] [7].
The Fano property applies both sets, but the Hesse property only in the first.

Projective Spheres with 91 Elements
Devidé [14] proved formulas for the possible number of reflexive elements.

What Type of Line in Planar, Spherical, or Hyperbolic Geometry Corresponds to the Transversal of Two Skew Straight Lines in Euclidean 3D Space?
Gábor Gévay and Lajos Szilassi [17]

Hyperbolic Rectangular Hexagon on 2D Surface as the Equivalent of Skew Euclidean Rectangular Hexagon in 3D Space
Three Euclidean skew lines with three common perpendiculars form a skew I. Lénárt Journal of Applied Mathematics and Physics hexagon in the Euclidean 3D space. Any two adjacent sides are perpendicular to each other. Three hyperbolic skew lines with three common perpendiculars form a hyperbolic convex hexagon on the hyperbolic 2D surface with six interior angles 90˚ each (Figure 30).

Hyperbolic Rectangular Hexagon vs. a Pair of Spherical Polar Triangles
The hyperbolic rectangular hexagon can be interpreted as the hyperbolic counterpart of a pair of spherical polar triangles which determine a spherical rectangular hexagram with intersecting sides.         As seen in Figure 33 and Figure 34, the vertices of the spherical hexagram do not coincide with the vertices of the spherical polar triangles. The sides of the rectangular spherical hexagon correspond to the equatorial arcs of the angles of the polar triangles. For example, the angle at vertex P 6 P 1 |P 2 P 3 corresponds to the equatorial arc P 1 P 2 . Consequently, the sides of the hyperbolic rectangular hexagon correspond to the angles of spherical polar triangles.

Hyperbolic Napier Pentagon with Five Right Angles
The hyperbolic rectangular hexagon has six right angles. Are there hyperbolic rectangular polygons with fewer than six right angles?
A biangle with two right angles or a triangle with three right angles or a quadrilateral with four right angles cannot be constructed on the hyperbolic surface, but a hyperbolic convex pentagon with five right angles does exist, and corresponds to the spherical Napier pentagram. Figure 35 shows a spherical rectangular Napier pentagram with right angles at each vertex 1 2 3 4 5 , P , P , P P , P . The points of intersection | | | P P P P , P P P P , P P P P , P P P P , P | P P P | are the vertices of a spherical convex pentagon.

Cycles of Incidence on the Sphere
The incidence relation was discussed in Section One ( Figure 16).    the limiting case will be P itself. This is not a proof, just a way to visualize the process. Still, the only reasonable choice for the result of line l and one of its ideal points P is P itself. The ideal point P is a reflexive element that can be viewed as either a point at infinity or a line degenerated into a point.

Cycles of Incidence on the Hyperbolic Surface
The arrangement of the hyperbolic surface into a projective sphere allows to apply cycles of incidence in the same manner as on the sphere. The incidence con-

I. Lénárt
dition for a hyperbolic point and a straight line is that the point is on the line; for two straight lines is that they are perpendiculars; the only difference is that I was unable to extend the incidence relation to the case of two hyperbolic points.
A hyperbolic Napier pentagon in Figure 41

Hyperbolic 5-Cycles Are the Simplexes of Hyperbolic Geometry
The 5-hyperbolic cycles, including the Napier pentagon and the Lambert quadrilateral, play a much more important role than just transferring certain polygons from the sphere to the hyperbolic surface. It is not the general triangle (which proves to be a 6-cycle), but the 5-cycle that can be viewed as the simplex of 2D hyperbolic geometry.
Hyperbolic geometry reveals the true meaning and significance of the Napier construction that was originally discovered 400 years ago in spherical geometry.

The Fundamental Formula for the Hyperbolic Napier Pentagon
Theorem 14: In a hyperbolic Napier pentagon any two adjacent sides k, l determine the opposite side n of the pentagon by the fundamental formula: cosh sinh sinh = n k l (1) Figure 44 shows a right-angled triangle in the Napier pentagon with two adjacent sides denoted by leg1 and leg2 and their hypotenuse. In contrast, the Napier hypotenuse is the side of the pentagon opposite the hypotenuse defined by leg1 and leg2. Proof: Several authors, including Gauss [19], studied the trigonometry of the spherical pentagon ABCDE (Figure 46). However, the present proof relates to the spherical Napier pentagram 1 2 3 4 5 P P P P P , because the spherical pentagram is the counterpart of the hyperbolic Napier pentagon.    Now consider the right-angled spherical triangle P 5 P 1 P 2 with legs 5 1 P P 90 = + k , 2 1 P P 90 = + l , and hypotenuse 2 5 P P = n . Here

Napier Pentagon Decomposed into Two Lambert Quadrilaterals
Drop a perpendicular from a vertex of the Napier pentagon to the opposite side that divides the pentagon into two Lambert quadrilaterals. Conversely, any Lambert quadrilateral with angle α < 90˚ can be supplemented by another Lambert quadrilateral with angle (90˚ -α) to form a Napier pentagon ( Figure 47).
This construction reminds of the decomposition of a spherical biangle into two supplementary triangles.
By notation of Figure 47 we have:

Hyperbolic Rectangular Hexagon Decomposed into Two Napier Pentagons
The properties of the rectangular hexagon can be deduced from formula (1) of the Napier pentagon. Figure 48 shows a hyperbolic rectangular hexagon with sides 1 2 6 61 61 1 6 1 P F F P P P a a a = + = = + ,

Two Triads of Non-Adjacent Sides of the Hyperbolic Rectangular Hexagon as the Counterpart of a Pair of Spherical Polar Triangles
The sides of the hyperbolic hexagon can be divided into two triads of non-adjacent elements 6 1 2 3 4 5 P P , P P , P P a c e = = = and 1 2 3 4 5 6 P P , P P , Figure 48). The two triads ("odd sides" and "even sides") can be viewed the counterpart of a pair of spherical polar triangles (Figure 31), or the rectangular spherical hexagram (Figure 33). From another perspective, they also correspond to the angles and sides of a spherical triangle. The role of spherical angles can be attributed to either of the hyperbolic triads, while the other hyperbolic triad corresponds to the sides of the spherical triangle. Alternatively, the two triads can be taken for the sides of a pair of polar triangles, or for the angles of the same pair of triangles. A pair of polar triangles ( Figure 31 and Figure 33) can also be viewed as a rectangular spherical hexagram. It follows that the three common altitudes of the polar triangles are transformed into the common perpendiculars of the opposite sides of the spherical hexagram.

Noteworthy Lines in the Hyperbolic Rectangular Hexagon
This interpretation can be extended to the hyperbolic rectangular hexagon in Figure 50. On the sphere and on the hyperbolic hemisphere, the three altitudes or common perpendiculars meet in the orthocentre denoted by H in Figure 49 and Figure 50. Figure 51 shows the inscribed circle in one triad of sides of the rectangular hyperbolic hexagon. It is tangential to the "odd sides", but not to the "even sides". Figure 52 shows the inscribed circle in the other triad of sides, tangential to the "even sides", but not to the "odd sides". This property corresponds to a similar property of spherical circles inscribed in a pair of polar triangles. Figure 53 and Figure 54 show the Theorem of Fagnano applied to the hyperbolic rectangular hexagon: The feet of altitudes in a triad of non-adjacent sides are the vertices of an inscribed triangle in which the original altitudes are angle bisectors. The two figures display the inscribed triangles in the "odd triad" and the "even triad, respectively. Again, this property corresponds to the similar statement in a pair of polar triangles on the sphere.

The Desargues Configuration on the Hyperbolic Surface
The Desargues configuration can be displayed in novel forms by the present algebraic approach. Figure 55 displays the two 6-cycles ("triangles") of the Desargues configuration in the form where all the elements of the 6-cycles are lines, while the "centre" S and "axis" T appear as points. Figure 56 shows two perspective rectangular hyperbolic hexagons in the form of regular triangles. "Perspective" means in this case that the three products of the corresponding three pairs of sides are concurrent. The "centre" S of perspectivity coincides with the "axis" T of perspectivity, because a a b b a b a a b b a b a a b b b An interesting application of the form of the Desargues theorem in Figure 55 is the Veblen-Young dual theorem as cited by Gévay [21]: If three triangles are perspective from the same point, the three axes of perspectivity of the three pairs of triangles are concurrent. Conversely, if three triangles are perspective from the same line, the three centres of perspectivity of the three pairs of triangles are collinear.
These two theorems can be unified in one theorem in the following way: If the common perpendiculars of the odd sides of three rectangular hexagons are concurrent, the common perpendiculars of the even sides are also concurrent.
This form is more suitable for generalization in higher dimensions than the point-line version.

Pairs of Numbers-Points and Straight Lines
As shown in Section One and Section Two, the theory of projective spheres treats points and straight lines as identical elements within a single set, not as dual elements in two separate sets.
In If no distinction is made between points and straight lines, it seems reasonable to say that the ordered pair of real numbers (a, b) represents both the point (a, b) and the straight line ax b y + =.
The problem lies in defining the relation of incidence between a point and a There are two ways to escape this trap.
We can accept that the incidence relation is antisymmetric in the above sense:

Classification of Poles and Polars of Elements in a Cartesian Projective Sphere
In order to create a projective sphere, we have to add ideal elements (ideal points and the ideal line) to the real points and lines of the coordinate system. Follow-

I. Lénárt Journal of Applied Mathematics and Physics
ing is a list of possible cases with real and ideal elements.
1) Two real points and two real straight lines that are mirror images to the y-axis (Figure 57). The point of intersection of the two lines is the midpoint on the y-axis between the two points. This is not a reflexive element, because the pole points are not on (not incident to) the polar lines.

Algebraic Condition of the Relation of Incidence
Two elements Similar proofs for other combinations of (±a), (±c), (±e). Q.E.D.

Klein Spheres (K-Spheres)
As shown in Section One, the Klein sphere or K-sphere is a closed set in the theory of projective spheres (subsection 7, Model 1). It consists of three different elements, each of which is incident to the other two. It follows from the definition that reflexive elements cannot occur in a K-sphere.
The condition of incidence in the K-sphere in the Cartesian system:  The expression xx cannot be uniquely determined for all elements in the set with at least two different elements.

Differentiable Functions
When a point in the plane approaches another point, the two points define a unique straight line through them. If the two points coincide, an infinite number of straight lines can be drawn through the point. So there is no unique result of the xx operation.
However, a differentiable function has a precisely defined tangent line in each of its points. The tangent is the limit of an infinite series of secants each of which passes through the given point and another point of the function. By the operation as defined in the projective sphere, this means that if x is an element of the differentiable function, operation xx is uniquely defined by the tangent at the given point of the function. The classical calculus determines the slope of the tangent as the limit of division with an arbitrarily small number Δx. The algebra of projective spheres defines the tangent as the product of an element of a differentiable function multiplied by itself.
How can we determine the coordinates of the tangent? By definition, the tangent is incident to the point. There is no difference between points and lines in a projective sphere, so the condition of incidence can be applied to determine the coordinates of the tangent.
Before giving the general formula, we begin with concrete examples: 1) Given a point of the function y = x 2 : first coordinate x, second coordinate x 2 .
We want to determine the second coordinate of the tangent in this point. Figure 62 shows the polar lines of the tangent elements. Figure 63 shows the pole points of the same tangent elements.  The tangent is incident to the point, so we can apply the condition of incidence: The product of the first coordinates is equal to the difference of the second coordinates. The product of the first coordinates in this case is 2x 2 . The second coordinate of the tangent is either −x 2 or 3x 2 , but the geometric image of the function shows that the only possible solution is −x 2 .
Furthermore, we can construct the normal to the parabola which is the product of the point and the tangent in the projective sphere. The three elements form a K-sphere at almost every point of the differentiable function. Table 12 shows the coordinates of these elements displayed as points.
Exceptions are the points in which the tangents are parallel with the x-axis.
hese points are reflective elements of the perspective sphere representing local extrema (maximum or minimum points), or certain inflection points of the function (Figure 64).  Table 13 shows the respective coordinates.
1) Another example is the unit circle in Figure 66. The red curves show the pole points of the tangent function.
2) The hyperbola in Figure 67 is another example. The third row of Table 14 shows the coordinates of the normal, that is, the product of the point and the tangent, thus establishing K-spheres to every point. Figure 67 only displays the hyperbola and the tangent function.      Table 13. Coordinates of the unit circle and its tangents.

Closing Remarks
I am far from assigning any kind of exclusivity or superiority to the present algebraic-axiomatic system compared to other systems. I am just saying that this is a meaningful construction among many other possible options. It is up to the user to decide which system is more suitable for solving a mathematical or educational problem.
Certainly, this work poses many unresolved issues and incomplete analogies.
After five decades of research, however, it is high time that I summarized the results so far in the hope that future researchers will be able to use and improve what I have done.
I have been immersed in the theory of projective spheres for many years because they have given me joy and satisfaction. I ask future teachers of this topic to share this joy, satisfaction and self-confidence with their students.