Lobachevsky plane

Geometry of Lobachevsky (hyperbolic geometry listen)) is one of the non-Euclidean geometries, a geometric theory based on the same basic premises as ordinary Euclidean geometry, with the exception of the parallel axiom, which is replaced by Lobachevsky's parallel axiom.

The Euclidean axiom of parallels says:

through a point not lying on a given line, there is only one line that lies with the given line in the same plane and does not intersect it.

In Lobachevsky geometry, the following axiom is accepted instead:

through a point not lying on a given line there pass at least two lines that lie with the given line in the same plane and do not intersect it.

Lobachevsky's geometry has extensive applications in both mathematics and physics. Its historical significance lies in the fact that by its construction Lobachevsky showed the possibility of a geometry different from Euclidean, which marked a new era in the development of geometry and mathematics in general.

Story

Attempts to prove the fifth postulate

The starting point of Lobachevsky's geometry was Euclid's fifth postulate, an axiom equivalent to the parallel axiom. It was included in the list of postulates in Euclid's Elements). The relative complexity and non-intuitiveness of its formulation evoked a feeling of its secondary nature and gave rise to attempts to derive it from the rest of Euclid's postulates.

Among those trying to prove were the following scientists:

• ancient Greek mathematicians Ptolemy (II century), Proclus (V century) (based on the assumption that the distance between two parallel ones is finite),
• Ibn al-Haytham from Iraq (late - early centuries) (based on the assumption that the end of a moving perpendicular to a straight line describes a straight line),
• Iranian mathematicians Omar Khayyam (2nd half - beginning of the 12th century) and Nasir ad-Din at-Tusi (XIII century) (based on the assumption that two converging lines cannot continue to diverge without crossing),
• German mathematician Clavius ​​(),
• Italian mathematicians
  • Cataldi (for the first time in 1603 he published a work entirely devoted to the question of parallels),
• English mathematician Wallis ( , published in ) (based on the assumption that for every figure there is a figure similar to it, but not equal to it),
• French mathematician Legendre () (based on the assumption that through each point inside an acute angle you can draw a line that intersects both sides of the angle; he also had other attempts at proof).

In these attempts to prove the fifth postulate, mathematicians introduced some new assertion, which seemed to them more obvious.

Attempts have been made to use proof by contradiction:

• the Italian mathematician Saccheri () (having formulated a statement contradicting the postulate, he deduced a number of consequences and, erroneously recognizing some of them as contradictory, he considered the postulate proven),
• German mathematician Lambert (about, published in) (after conducting research, he admitted that he could not find contradictions in the system he built).

Finally, an understanding began to arise that it is possible to construct a theory based on the opposite postulate:

• German mathematicians F. Schweikart () and Taurinus () (however, they did not realize that such a theory would be just as logically coherent).

Creation of non-Euclidean geometry

Lobachevsky in his work "On the Principles of Geometry" (), his first printed work on non-Euclidean geometry, clearly stated that the V postulate cannot be proved on the basis of other premises of Euclidean geometry, and that the assumption of a postulate opposite to Euclid's postulate allows one to construct a geometry just as meaningful, like Euclidean, and free from contradictions.

Simultaneously and independently, Janos Bolyai came to similar conclusions, and Carl Friedrich Gauss came to such conclusions even earlier. However, Bolyai's writings did not attract attention, and he soon abandoned the subject, while Gauss refrained from publishing at all, and his views can only be judged from a few letters and diary entries. For example, in an 1846 letter to the astronomer G. H. Schumacher, Gauss speaks of Lobachevsky's work as follows:

This work contains the foundations of the geometry that would have to take place and, moreover, would constitute a strictly consistent whole, if Euclidean geometry were not true ... Lobachevsky calls it "imaginary geometry"; You know that for 54 years (since 1792) I have shared the same views with some development of them, which I do not want to mention here; thus, I did not find anything actually new for myself in Lobachevsky's work. But in the development of the subject, the author did not follow the path that I myself followed; it is masterfully done by Lobachevsky in a truly geometric spirit. I consider myself obligated to draw your attention to this work, which will surely give you quite exceptional pleasure.

As a result, Lobachevsky acted as the first brightest and most consistent propagandist of this theory.

Although Lobachevsky's geometry developed as a speculative theory and Lobachevsky himself called it "imaginary geometry", nevertheless, it was Lobachevsky who considered it not as a game of the mind, but as a possible theory of spatial relations. However, the proof of its consistency was given later, when its interpretations were indicated, and thus the question of its real meaning, logical consistency, was completely resolved.

Statement of Lobachevsky's geometry

corner is even more difficult.

Poincaré model

The content of Lobachevsky's geometry

Pencil of parallel lines in Lobachevsky's geometry

Lobachevsky built his geometry, starting from the basic geometric concepts and his axiom, and proved theorems by a geometric method, similar to how it is done in Euclid's geometry. The theory of parallel lines served as the basis, since it is here that the difference between Lobachevsky's geometry and Euclid's geometry begins. All theorems that do not depend on the axiom of parallels are common to both geometries and form the so-called absolute geometry, which includes, for example, theorems on the equality of triangles. Following the theory of parallels, other sections were built, including trigonometry and the principles of analytic and differential geometry.

Let us present (in modern notation) several facts of Lobachevsky's geometry that distinguish it from Euclid's geometry and were established by Lobachevsky himself.

Through the dot P not lying on the given line. R(see figure), there are infinitely many straight lines that do not intersect R and located with it in the same plane; among them there are two extreme x, y, which are called parallel lines R in the sense of Lobachevsky. In Klein's (Poincare's) models they are represented by chords (arcs of circles) having with a chord (arc) R a common end (which, by definition of the model, is excluded, so that these lines have no common points).

Angle between perpendicular PB from P on the R and each of the parallel ones (called angle of parallelism) as the point is removed P decreases from the straight line from 90° to 0° (in the Poincaré model, the angles in the usual sense coincide with the angles in the sense of Lobachevsky, and therefore this fact can be seen directly on it). Parallel x on the one hand (and y opposite) asymptotically approaches a, and on the other hand, it infinitely moves away from it (in models, distances are difficult to determine, and therefore this fact is not directly visible).

For a point located from a given straight line at a distance PB = a(see figure), Lobachevsky gave a formula for the angle of parallelism P(a) :

Here q is some constant related to the curvature of the Lobachevsky space. It can serve as an absolute unit of length in the same way as in spherical geometry the radius of the sphere occupies a special position.

If the lines have a common perpendicular, then they diverge infinitely on both sides of it. To any of them it is possible to restore perpendiculars that do not reach the other line.

In Lobachevsky's geometry there are no similar but unequal triangles; triangles are congruent if their angles are equal.

The sum of the angles of any triangle is less than π and can be arbitrarily close to zero. This is directly visible in the Poincaré model. The difference δ \u003d π - (α + β + γ) , where α , β , γ are the angles of the triangle, is proportional to its area:

It can be seen from the formula that there is a maximum area of ​​a triangle, and this is a finite number: π q 2 .

A line of equal distances from a straight line is not a straight line, but a special curve called an equidistant, or hypercycle.

The limit of circles of infinitely increasing radius is not a straight line, but a special curve called limit circle, or a horocycle.

The limit of spheres of infinitely increasing radius is not a plane, but a special surface - the limit sphere, or horosphere; it is remarkable that Euclidean geometry holds on it. This served Lobachevsky as the basis for the derivation of trigonometry formulas.

The circumference is not proportional to the radius, but grows faster. In particular, in the geometry of Lobachevsky, the number π cannot be defined as the ratio of the circumference of a circle to its diameter.

The smaller the region in space or on the Lobachevsky plane, the less the geometric relations in this region differ from the relations of Euclidean geometry. We can say that in an infinitesimal region, the Euclidean geometry takes place. For example, the smaller the triangle, the less the sum of its angles differs from π; the smaller the circle, the less the ratio of its length to radius differs from 2π, etc. Reducing the area is formally equivalent to increasing the unit length, therefore, with an infinite increase in the unit length, the Lobachevsky geometry formulas turn into formulas of Euclidean geometry. Euclidean geometry is in this sense the "limiting" case of Lobachevsky's geometry.

Applications

• Lobachevsky himself applied his geometry to the calculation of definite integrals.
• In the theory of functions of a complex variable, Lobachevsky's geometry helped build the theory of automorphic functions. The connection with Lobachevsky's geometry was here the starting point of Poincaré's research, who wrote that "non-Euclidean geometry is the key to solving the whole problem."
• Lobachevsky's geometry also finds application in number theory, in its geometric methods, united under the name "geometry of numbers".
• A close connection was established between Lobachevsky's geometry and the kinematics of the special (private) theory of relativity. This connection is based on the fact that the equality expressing the law of propagation of light

when dividing by t 2 , i.e. for the speed of light, gives - the equation of the sphere in space with coordinates v x , v y , v z- velocity components along the axes X, at, z(in "velocity space"). The Lorentz transformations preserve this sphere and, since they are linear, transform the direct velocity spaces into straight lines. Therefore, according to the Klein model, in the space of velocities inside a sphere of radius with, that is, for speeds less than the speed of light, the Lobachevsky geometry takes place.

• Lobachevsky's geometry found a remarkable application in the general theory of relativity. If we consider the distribution of masses of matter in the Universe to be uniform (this approximation is acceptable on a cosmic scale), then it turns out that under certain conditions space has the Lobachevsky geometry. Thus, Lobachevsky's assumption of his geometry as a possible theory of real space was justified.
• Using the Klein model, a very simple and short proof is given

We are used to thinking that the geometry of the observed world is Euclidean, i.e. it fulfills the laws of the geometry that is studied at school. Actually this is not true. In this article, we will consider the manifestations in reality of Lobachevsky's geometry, which, at first glance, is purely abstract.

Lobachevsky's geometry differs from the usual Euclidean one in that in it, through a point not lying on a given line, there pass at least two lines that lie with the given line in the same plane and do not intersect it. It is also called hyperbolic geometry.

1. Euclidean geometry - only one line passes through the white point, which does not intersect the yellow line
2. Riemann geometry - any two lines intersect (there are no parallel lines)
3. Lobachevsky geometry - there are infinitely many straight lines that do not intersect the yellow line and pass through the white point

In order for the reader to visualize this, let us briefly describe the Klein model. In this model, the Lobachevsky plane is realized as the interior of a circle of radius one, where the points of the plane are the points of this circle, and the lines are the chords. A chord is a straight line joining two points on a circle. The distance between two points is difficult to determine, but we do not need it. From the figure above, it becomes clear that through the point P there are infinitely many lines that do not intersect the line a. In standard Euclidean geometry, there is only one line passing through the point P and not intersecting the line a. This line is parallel.

Now let's move on to the main thing - the practical applications of Lobachevsky's geometry.

Satellite navigation systems (GPS and GLONASS) consist of two parts: an orbital constellation of 24-29 satellites evenly spaced around the Earth, and a control segment on Earth, which ensures time synchronization on the satellites and the use of a single coordinate system. The satellites have very accurate atomic clocks, and the receivers (GPS-navigators) have ordinary, quartz clocks. The receivers also have information about the coordinates of all satellites at any given time. Satellites at short intervals transmit a signal containing data on the start time of the transmission. After receiving a signal from at least four satellites, the receiver can adjust its clock and calculate the distances to these satellites using the formula ((time the signal was sent by the satellite) - (the time the signal was received from the satellite)) x (speed of light) = (distance to the satellite). The calculated distances are also corrected according to the formulas built into the receiver. Further, the receiver finds the coordinates of the intersection point of the spheres with centers in the satellites and radii equal to the calculated distances to them. Obviously, these will be the coordinates of the receiver.

The reader is probably aware that due to the effect in Special Relativity, due to the high speed of the satellite, time in orbit is different from time on Earth. But there is still a similar effect in the General Theory of Relativity, connected precisely with the non-Euclidean geometry of space-time. Again, we will not go into mathematical details, since they are rather abstract. But, if we stop taking these effects into account, then within a day of operation, an error of the order of 10 km will accumulate in the readings of the navigation system.

The Lobachevsky geometry formulas are also used in high energy physics, namely, in the calculations of charged particle accelerators. Hyperbolic spaces (that is, spaces in which the laws of hyperbolic geometry operate) are also found in nature itself. Let's give more examples:

The geometry of Lobachevsky can be seen in the structures of corals, in the organization of cellular structures in a plant, in architecture, in some flowers, and so on. By the way, if you remember in the last issue we talked about hexagons in nature, and so, in hyperbolic nature, the alternative is heptagons, which are also widespread.

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”, dedicated to the relationship between Russian and British science, mathematician Valentina Kirichenko tells PostNauka about the revolutionary nature of Lobachevsky’s ideas for the geometry of the 19th century.

Parallel lines do not intersect even in Lobachevsky geometry. Somewhere in the films you can often find the phrase: "But our Lobachevsky's parallel lines intersected." Sounds nice, but it's not true. Nikolai Ivanovich Lobachevsky really came up with an unusual geometry in which parallel lines behave quite differently from what we are used to. However, they do not intersect.

We are accustomed to thinking that two parallel lines do not approach or recede. That is, no matter what point on the first line we take, the distance from it to the second line is the same, it does not depend on the point. But is it really so? And why is it so? And how can this be verified?

If we are talking about physical lines, then only a small section of each line is available to us for observation. And given measurement errors, we can't draw any definitive conclusions about how lines behave very, very far away from us. Similar questions arose already among the ancient Greeks. In the III century BC, the ancient Greek geometer Euclid very accurately stated the main property of parallel lines, which he could neither prove nor disprove. Therefore, he called it a postulate - a statement that should be taken on faith. This is the famous fifth postulate of Euclid: if two straight lines on a plane intersect with a secant, so that the sum of the internal one-sided angles is less than two straight lines, that is, less than 180 degrees, then with sufficient continuation, these two lines will intersect, and it is precisely on the other side of the secant along which the sum is less than two right angles.

The key words in this postulate are "with sufficient continuation". It is because of these words that the postulate cannot be verified empirically. Maybe the lines will intersect in the line of sight. Maybe after 10 kilometers or beyond the orbit of Pluto, or maybe even in another galaxy.

Euclid outlined his postulates and the results that logically follow from them in the famous book "Beginnings". The Russian word "elements" comes from the ancient Greek title of this book, and the word "elements" comes from the Latin title. Euclid's Elements is the most popular textbook of all time. In terms of the number of editions, it is second only to the Bible.

I would especially like to note the wonderful British edition of 1847 with very visual and beautiful infographics. Instead of dull designations on the drawings, color drawings are used there - not like in modern school geometry textbooks.

Until the last century, Euclid's "Beginnings" were mandatory for study in all educational programs that implied intellectual creativity, that is, not just learning a craft, but something more intellectual. The non-obviousness of Euclid's fifth postulate raised a natural question: can it be proved, that is, deduced logically from the rest of Euclid's assumptions? Many mathematicians tried to do this, from the contemporaries of Euclid to the contemporaries of Lobachevsky. As a rule, they reduced the fifth postulate to some more demonstrative statement, which is easier to believe.

For example, in the 17th century, the English mathematician John Wallis reduced the fifth postulate to the following statement: there are two similar but unequal triangles, that is, two triangles whose angles are equal, but the sizes are different. It would seem, what could be easier? Let's just change the scale. But it turns out that the ability to change the scale while maintaining all angles and proportions is an exclusive property of Euclidean geometry, that is, a geometry in which all Euclid's postulates, including the fifth, are fulfilled.

In the 18th century, the Scottish scientist John Playfair reformulated the fifth postulate in the form in which it usually appears in modern school textbooks: two lines intersecting each other cannot be simultaneously parallel to a third line. It is in this form that the fifth postulate appears in modern school textbooks.

By the beginning of the 19th century, many had the impression that proving the fifth postulate was like inventing a perpetual motion machine - a completely useless exercise. But no one had the guts to suggest that Euclid's geometry was not the only possible one: Euclid's authority was too great. In such a situation, Lobachevsky's discoveries were, on the one hand, natural, and on the other, absolutely revolutionary.

Lobachevsky replaced the fifth postulate with a directly opposite statement. Lobachevsky's axiom sounded like this: if from a point that does not lie on a straight line, let out all the rays that intersect this straight line, then on the left and right these rays will be limited by two limiting rays that will no longer cross the straight line, but will become closer and closer to it. Moreover, the angle between these limiting rays will be strictly less than 180 degrees.

It immediately follows from Lobachevsky's axiom that through a point not lying on a given line, one can draw not one line parallel to the given one, as in Euclid's, but as many as you like. But these lines will behave differently from those of Euclid. For example, if we have two parallel lines, then they can first approach and then move away. That is, the distance from a point on the first line to the second line will depend on the point. It will be different for different points.

Lobachevsky's geometry contradicts our intuition partly because at the small distances we usually deal with it differs very little from Euclidean geometry. Similarly, we perceive the curvature of the Earth's surface. When we walk from home to the store, it seems to us that we are walking in a straight line, and the Earth is flat. But if we fly, say, from Moscow to Montreal, then we already notice that the plane flies along an arc of a circle, because this is the shortest path between two points on the surface of the Earth. That is, we notice that the Earth is more like a soccer ball than a pancake.

Lobachevsky's geometry can also be illustrated with a soccer ball, but not an ordinary one, but a hyperbolic one. A hyperbolic soccer ball is glued together just like a normal one. Only in an ordinary ball, white hexagons are glued to black pentagons, and in a hyperbolic ball, instead of pentagons, you need to make heptagons and also glue them with hexagons. In this case, it will turn out, of course, not a ball, but rather a saddle. And on this saddle the geometry of Lobachevsky is realized.

Lobachevsky tried to tell about his discoveries in 1826 at Kazan University. But the text of the report has not survived. In 1829 he published an article on his geometry in a university journal. Lobachevsky's results seemed meaningless to many - not only because they destroyed the usual picture of the world, but because they were not presented in the most understandable way.

However, Lobachevsky also had publications in high-ranking journals, as we call them today. For example, in 1836 he published an article in French entitled "Imaginary Geometry" in the famous journal Krell, in the same issue as the articles of the most famous mathematicians of that time - Dirichlet, Steiner and Jacobi. And in 1840, Lobachevsky published a small and very understandably written book called "Geometric Research on the Theory of Parallel Lines." The book was in German and was published in Germany. There was also a devastating review. The reviewer especially mocked Lobachevsky's phrase: "The further we continue the lines in the direction of their parallelism, the more they approach each other." "This statement alone," the reviewer wrote, "already sufficiently characterizes Mr. Lobachevsky's work and frees the reviewer from the need for further evaluation."

But the book also had one unbiased reader. It was Carl Friedrich Gauss, also known as the King of Mathematicians, one of the greatest mathematicians in history. He highly appreciated Lobachevsky's book in one of his letters. But his review was published only after his death, along with the rest of the correspondence. And that's when the real boom of Lobachevsky's geometry began.

In 1866 his book was translated into French, then into English. Moreover, the English edition was reprinted three more times due to its extraordinary popularity. Unfortunately, Lobachevsky did not live up to this time. He died in 1856. And in 1868, a Russian edition of Lobachevsky's book appeared. It was published not as a book, but as an article in the oldest Russian journal Mathematical Collection. But then this magazine was very young, it was not yet two years old. But the Russian translation of 1945, made by the remarkable Russian and Soviet geometer Veniamin Fedorovich Kagan, is better known.

By the end of the 19th century, mathematicians were divided into two camps. Some immediately accepted Lobachevsky's results and began to develop his ideas further. And others could not give up the belief that Lobachevsky's geometry describes something that does not exist, that is, Euclid's geometry is the only true one and nothing else can be. Unfortunately, the latter included the mathematician, better known as the author of Alice in Wonderland, Lewis Carroll. His real name is Charles Dodgson. In 1890, he published an article entitled "A New Theory of Parallels", where he defended an extremely illustrative version of the fifth postulate. Lewis Carroll's axiom sounds like this: if a regular quadrilateral is inscribed in a circle, then the area of ​​\u200b\u200bthis quadrilateral will be strictly greater than the area of ​​\u200b\u200bany of the segments of the circle that lie outside the quadrilateral. In Lobachevsky geometry this axiom is not true. If we take a sufficiently large circle, then no matter what quadrangle we inscribe in it, no matter how long the sides of this quadrangle may be, the area of ​​the quadrangle will be limited by the universal physical constant. In general, the presence of physical constants and universal measures of length is an advantageous difference between Lobachevsky's geometry and Euclid's geometry.

But Arthur Cayley, another famous English mathematician, in 1859, that is, only three years after the death of Lobachevsky, published an article that later helped to legalize Lobachevsky's postulate. Interestingly, Cayley at that time worked as a lawyer in London and only then received a professorship at Cambridge. In fact, Cayley built the first model of Lobachevsky's geometry, although he solved, at first glance, a completely different problem.

And another remarkable English mathematician, whose name was William Kingdon Clifford, was deeply imbued with the ideas of Lobachevsky. And in particular, he was the first to express the idea, long before the creation of the general theory of relativity, that gravity is caused by the curvature of space. Clifford praised Lobachevsky's contribution to science in one of his lectures on the philosophy of science: "Lobachevsky became for Euclid what Copernicus became for Ptolemy." If before Copernicus mankind believed that we know everything about the Universe, now it is clear to us that we observe only a small part of the Universe. Similarly, before Lobachevsky, mankind believed that there is only one geometry - Euclidean, everything has long been known about it. Now we know that there are many geometries, but we know far from all of them.

geometry theorems of Lobachevsky

1. Basic concepts of Lobachevsky geometry

In Euclidean geometry, according to the fifth postulate, on the plane through a point R, lying outside the line A "A, there is only one straight line B"B, not intersecting A "A. Straight B"B" called parallel to A"A. It suffices to require that there is at most one such line, since the existence of a non-intersecting line can be proved by successively drawing lines PQA"A and PBPQ. In Lobachevsky geometry, the axiom of parallelism requires that through a point R passed more than one straight line that did not intersect A "A.

Non-intersecting lines fill the part of the pencil with a vertex R, lying inside a pair of vertical angles TPU and U"PT", located symmetrically about the perpendicular P.Q. The lines that form the sides of the vertical angles separate the intersecting lines from the non-intersecting ones and are themselves also non-intersecting. These boundary lines are called parallels at point P to a straight line A "A respectively in two directions: T "T parallel A "A in the direction A"A, a UU" parallel A "A in the direction A A". Other non-intersecting lines are called divergent lines with A "A.

Injection , 0< R forms with a perpendicular pQ, QPT=QPU"=, called angle of parallelism segment PQ=a and is denoted by . At a=0 angle =/2; with increasing a the angle decreases so that for each given, 0<a. This dependency is called Lobachevsky function :

P(a)=2arctg (),

where to-- some constant that defines a segment fixed in value. It is called the radius of curvature of the Lobachevsky space. Like spherical geometry, there is an infinite set of Lobachevsky spaces, differing in magnitude to.

Two different straight lines in a plane form a pair of one of three types.

intersecting lines . The distance from the points of one line to another line increases indefinitely as the point moves away from the intersection of the lines. If the lines are not perpendicular, then each is projected orthogonally onto the other into an open segment of finite size.

Parallel lines . In the plane, through a given point, there is a single straight line parallel to the given straight line in the direction given on the latter. Parallel at a point R retains at each of its points the property of being parallel to the same line in the same direction. Parallelism is reciprocal (if a||b in a certain direction, then b||a in the corresponding direction) and transitivity (if a||b and with || b in one direction, then a||s in the corresponding direction). In the direction of parallelism, parallel ones approach indefinitely, in the opposite direction they move away indefinitely (in the sense of the distance from a moving point of one straight line to another straight line). The orthogonal projection of one line onto another is an open half-line.

Divergent lines . They have one common perpendicular, the segment of which gives the minimum distance. On both sides of the perpendicular, the lines diverge indefinitely. Each line is projected onto another into an open segment of finite size.

Three types of lines correspond on the plane to three types of pencils of lines, each of which covers the entire plane: beam of the 1st kind is the set of all lines passing through one point ( Centre beam); beam of the 2nd kind is the set of all lines perpendicular to one line ( base beam); beam of the 3rd kind is the set of all lines parallel to one line in a given direction, including this line.

The orthogonal trajectories of the straight lines of these beams form analogs of the circle of the Euclidean plane: circle in the proper sense; equidistant , or line equal distances (if you do not consider the base), which is concave towards the base; limit line , or horocycle, it can be considered as a circle with an infinitely distant center. Limit lines are congruent. They are not closed and are concave towards parallelism. Two limit lines generated by one bundle are concentric (equal segments are cut out on straight lines of the bundle). The ratio of the lengths of the concentric arcs enclosed between two straight lines of the beam decreases towards parallelism as an exponential function of the distance X between arcs:

s" / s=e.

Each of the analogs of the circle can slide on itself, which gives rise to three types of one-parameter motions of the plane: rotation around its own center; rotation around the ideal center (one trajectory is the base, the rest are equidistant); rotation around an infinitely distant center (all trajectories are limit lines).

Rotation of circle analogues around the straight line of the generating pencil leads to sphere analogues: the sphere proper, the surface of equal distances, and the horosphere, or marginal surfaces .

On the sphere, the geometry of great circles is the usual spherical geometry; on the surface of equal distances - equidistant geometry, which is the Lobachevsky planimetry, but with a larger value to; on the limit surface, the Euclidean geometry of limit lines.

The connection between the lengths of arcs and chords of the limit lines and the Euclidean trigonometric relations on the limit surface make it possible to derive trigonometric relations on the plane, that is, trigonometric formulas for rectilinear triangles.

2. Some theorems of Lobachevsky's geometry

Theorem 1. The sum of the angles of any triangle is less than 2d.

Consider first a right triangle ABC (Fig. 2). His sides a, b, c are depicted respectively as a segment of the Euclidean perpendicular to the line and, arcs of the Euclidean circle with center M and arcs of the Euclidean circle with center N. Injection With--straight. Injection BUT equal to the angle between the tangents to the circles b and with at the point BUT, or, which is the same, the angle between the radii NA and MA these circles. Finally, B = BNM.

Let's build on a segment BN as on the diameter of the Euclidean circle q; she has with circumference with one common point AT, since its diameter is the radius of the circle with. Therefore, the point BUT lies outside the circle bounded by the circle q, hence,

A = MAN< MBN.

Hence, due to the equality MBN+B = d we have:

A + B< d; (1)

so A+B+C< 2d, что и требовалось доказать.

Note that, with the proper hyperbolic motion, any right triangle can be positioned so that one of its legs lies on the Euclidean perpendicular to the line and; therefore, the method we used to derive the inequality (1) applicable to any right triangle.

If an oblique triangle is given, then we divide it by one of the heights into two right-angled triangles. The sum of the acute angles of these right triangles is equal to the sum of the angles of the given oblique triangle. Hence, taking into account the inequality (1) , we conclude that the theorem is valid for any triangle.

Theorem 2 . The sum of the angles of a quadrilateral is less than 4d.

To prove it, it suffices to divide the quadrilateral with a diagonal into two triangles.

Theorem 3 . Two divergent lines have one and only one common perpendicular.

Let one of these diverging straight lines be depicted on the map as a Euclidean perpendicular R to a straight line and at the point M, the other is in the form of a Euclidean semicircle q centered on and, and R and q do not have common points (Fig. 3). Such an arrangement of two divergent hyperbolic lines on a map can always be achieved with proper hyperbolic motion.

Let's spend from M euclidean tangent MN to q and describe from the center M radius MN euclidean semicircle m. It's clear that m--hyperbolic line intersecting and R and q at a right angle. Hence, m depicts on the map the required common perpendicular of the given diverging straight lines.

Two diverging lines cannot have two common perpendiculars, since in this case there would be a quadrilateral with four right angles, which contradicts Theorem 2.

. Theorem 4. The rectangular projection of a side of an acute angle onto its other side is a segment(and not a half-line, as in Euclid's geometry).

The validity of the theorem is obvious from Fig. 4, where the segment AB there is a rectangular projection of the side AB acute angle YOU on his side AS.

In the same figure, the arc DE Euclidean circle with center M is a perpendicular to the hyperbolic line AC. This perpendicular does not intersect with the oblique AB. Therefore, the assumption that the perpendicular and the oblique to the same line always intersect contradicts Lobachevsky's axiom of parallelism; it is equivalent to Euclid's axiom of parallelism.

Theorem 5. If three angles of triangle ABC are equal, respectively, to three angles of triangle A, B, C, then these triangles are congruent.

Assume the opposite and set aside, respectively, on the rays AB and AC segments AB \u003d A "B", AC \u003d A "C". Obviously triangles. ABC and A"B"C" equal in two sides and the angle between them. Dot B does not match with AT, dot C does not match with With, since in any of these cases the equality of these triangles would take place, which contradicts the assumption.

Consider the following possibilities.

a) Point B lies between BUT and AT, dot With-- between BUT and With(Fig. 5); in this and the next figure, hyperbolic lines are conventionally depicted as Euclidean lines). It is easy to verify that the sum of the angles of a quadrilateral SSNE is equal to 4d, which is impossible due to Theorem 2.

6) Point AT lies between BUT and AT, dot With-- between BUT and With(Fig. 6). Denote by D the point of intersection of the segments Sun and BC As C=C" and C" \u003d C, then C= With , which is impossible, since angle C is external to triangle CCD.

Other possible cases are treated similarly.

The theorem is proved because the assumption made has led to a contradiction.

From Theorem 5 it follows that in the geometry of Lobachevsky there is no triangle similar to the given triangle, but not equal to it.