How to draw 2 parallel Lobachevsky lines through a point. Practical applications of Lobachevsky geometry

Geometry of Lobachevsky

Introduction

Chapter I. The history of the emergence of non-Euclidean geometry

Chapter II. Geometry of Lobachevsky

2.1 Basic concepts

2.2 Consistency of Lobachevsky geometry

2.3 Models of Lobachevsky geometry

2.4 Triangle and polygon defect

2.5 Absolute unit of length in Lobachevsky geometry

2.6 Definition of a parallel line. Function P(x)

2.7 Poincare model

Practical part

1. The sum of the angles of a triangle

2. The question of the existence of such figures

3. The main property of parallelism

4. Properties of the function P(x)

Conclusion. findings

Applications

List of used literature

Introduction

This work shows the similarities and differences of the two geometries on the example of the proof of one of Euclid's postulates and the continuation of these concepts in Lobachevsky's geometry, taking into account the achievements of science at that time.

Any theory of modern science is considered correct until the next one is created. This is a kind of axiom of the development of science. This fact has been confirmed many times.

Newton's physics grew into relativistic, and that - into quantum. The phlogiston theory became chemistry. Such is the fate of all sciences. This fate did not bypass geometry. The traditional geometry of Euclid has grown into geometry. Lobachevsky. This work is devoted to this branch of science.

The purpose of this work: to consider the difference between Lobachevsky geometry and Euclid's geometry.

The objectives of this work: to compare the theorems of Euclid's geometry with similar theorems of Lobachevsky's geometry;

by solving problems, derive the positions of Lobachevsky's geometry.

Conclusions: 1. Lobachevsky's geometry is built on the rejection of the fifth postulate of Euclid.

2. In Lobachevsky geometry:

there are no similar triangles that are not equal;

two triangles are equal if their angles are equal;

the sum of the angles of a triangle is not equal to 180 0, but less (the sum of the angles of a triangle depends on its size: the larger the area, the more the sum differs from 180 0; and vice versa, the smaller the area, the closer the sum of its angles to 180 0);

through a point outside a line, more than one line parallel to the given line can be drawn.

Chapter 1. The history of the emergence of non-Euclidean geometry

1.1 V postulate of Euclid, attempts to prove it

Euclid is the author of the first rigorous logical construction of geometry that has come down to us. His exposition is so perfect for its time that for two thousand years from the moment of the appearance of his work "Elements" it was the only guide for students of geometry.

"Beginnings" consists of 13 books devoted to geometry and arithmetic in a geometric presentation.

Each book of the Elements begins with a definition of concepts that are encountered for the first time. Following the definitions, Euclid gives postulates and axioms, that is, statements accepted without proof.

Postulate V of Euclid says: and that whenever a line intersects with two other lines, it forms one-sided interior angles with them, the sum of which is less than two lines, these lines intersect on the side on which this sum is less than two lines.

The most important drawback of the system of Euclidean axioms, including its postulates, is its incompleteness, that is, their insufficiency for a strictly logical construction of geometry, in which each sentence, if it does not appear in the list of axioms, must be logically deduced from their last ones. Therefore, Euclid, when proving theorems, was not always based on axioms, but resorted to intuition, visualization and "sensory" perceptions. For example, he attributed a purely visual character to the concept of "between"; he tacitly assumed that a straight line passing through an interior point of a circle must certainly intersect it in two sticks. At the same time, he was based only on visibility, and not on logic; he did not give a proof of this fact anywhere, and could not give it, since he lacked the axioms of continuity. He also lacks some other axioms, without which a strictly logical proof of theorems is not possible.

But no one doubted the truth of the postulates of Euclid, as regards the fifth postulate. Meanwhile, already in antiquity, it was precisely the postulate of parallels that attracted the special attention of a number of geometers, who considered it unnatural to place it among the postulates. This was probably due to the relatively less obviousness and clarity of postulate V: implicitly, it assumes the attainability of any, arbitrarily distant parts of the plane, expressing a property that is found only when straight lines are extended indefinitely.

Euclid himself and many scientists tried to prove the postulate of parallels. Some tried to prove the postulate of parallels, using only other postulates and those theorems that can be deduced from the latter, without using the V postulate itself. All such attempts were unsuccessful. Their common shortcoming is that some assumption, equivalent to the postulate being proved, was implicitly applied in the proof. Others suggested redefining the parallel lines, or replacing the V postulate with something they thought was more obvious.

But centuries-old attempts to prove the fifth postulate of Euclid eventually led to the emergence of a new geometry, which differs in that the fifth postulate is not fulfilled in it. This geometry is now called non-Euclidean, and in Russia it bears the name of Lobachevsky, who first published a work with its presentation.

And one of the prerequisites for the geometric discoveries of N.I. Lobachevsky (1792-1856) was precisely his materialistic approach to the problems of cognition. Lobachevsky, he was firmly convinced of the objective existence of the material world and the possibility of its knowledge, independent of human consciousness. In his speech “On the Most Important Subjects of Education” (Kazan, 1828), Lobachevsky sympathetically quotes the words of F. Bacon: “leave them toiling in vain, trying to extract all wisdom from them alone; ask nature, she keeps all the truths and will answer all your questions without fail and satisfactorily. In his essay “On the Principles of Geometry”, which is the first publication of the geometry he discovered, Lobachevsky wrote: “The first concepts from which any science begins must be clear and reduced to the smallest number. Then only they can serve as a solid and sufficient foundation for the doctrine. Such concepts are acquired by the senses; innate - should not be believed.

Lobachevsky's first attempts to prove the fifth postulate date back to 1823. By 1826, he came to the conclusion that the fifth postulate does not depend on the rest of the axioms of Euclid’s geometry, and on February 11 (23), 1826, at a meeting of the faculty of Kazan University, he made a report “A concise presentation of the principles of geometry with a rigorous proof of the parallel theorem”, in which the beginnings of the “imaginary geometry” discovered by him, as he called the system, which later became known as non-Euclidean geometry, were outlined. The report of 1826 was included in Lobachevsky's first publication on non-Euclidean geometry - the article "On the Principles of Geometry", published in the journal of Kazan University "Kazan Vestnik" in 1829-1830. further development and applications of the geometry discovered by him were devoted to the memoirs "Imaginary Geometry", "The Application of Imaginary Geometry to Some Integrals" and "New Beginnings of Geometry with a Complete Theory of Parallels", published in "Scientific Notes" in 1835, 1836 and 1835-1838, respectively. . A revised text of the "Imaginary Geometry" appeared in a French translation in Berlin, ibid. in 1840. were published as a separate book in German "Geometric studies on the theory of parallel lines" by Lobachevsky. Finally, in 1855 and 1856. he published in Kazan in Russian and French "Pangeometry". He highly appreciated Gauss' "Geometric Studies", who made Lobachevsky (1842) a corresponding member of the Göttingen Scientific Society, which was in essence the Academy of Sciences of the Hanoverian kingdom. However, Gauss did not publish an evaluation of the new geometric system.

1.2 Parallelism postulates of Euclid and Lobachevsky

The main point from which the division of geometry into ordinary Euclidean (common) and non-Euclidean (imaginary geometry or "pangeometry") begins, as you know, is the postulate of parallel lines.

Ordinary geometry is based on the assumption that through a point not lying on a given line, at most one line can be drawn in the plane defined by this point and the line, not intersecting the given line. The fact that through a point not lying on a given line there passes at least one line that does not intersect this line refers to "absolute geometry", i.e. can be proved without the aid of the parallel lines postulate.

The line BB passing through P at right angles to the perpendicular PQ dropped by AA 1 does not intersect the line AA 1 ; this line in Euclidean geometry is called parallel to AA 1 .

In contrast to the postulate of Euclid, Lobachevsky takes the following axiom as the basis for constructing the theory of parallel lines:

Through a point not lying on a given line, in the plane defined by this point and the line, more than one line can be drawn that does not intersect the given line.

This directly implies the existence of an infinite number of lines passing through the same point and not intersecting the given line. Let the line СС 1 does not intersect AA 1; then all the lines passing inside the two vertical angles VRS and B 1 PC 1 also do not intersect with the line AA 1 .

Chapter 2. Geometry of Lobachevsky.

2.1 Basic concepts

In his memoirs On the Principles of Geometry (1829), Lobachevsky first of all reproduced his report of 1826.

On February 7, 1832, Nikolai Lobachevsky presented his first work on non-Euclidean geometry to the judgment of his colleagues. That day was the beginning of a revolution in mathematics, and Lobachevsky's work was the first step towards Einstein's theory of relativity. Today "RG" has collected five of the most common misconceptions about Lobachevsky's theory, which exist among people far from mathematical science

Myth one. Lobachevsky's geometry has nothing in common with Euclidean.

In fact, Lobachevsky's geometry is not too different from the Euclidean geometry we are used to. The fact is that of the five postulates of Euclid, Lobachevsky left the first four without change. That is, he agrees with Euclid that a straight line can be drawn between any two points, that it can always be extended to infinity, that a circle with any radius can be drawn from any center, and that all right angles are equal to each other. Lobachevsky did not agree only with the fifth postulate, the most doubtful from his point of view, of Euclid. His formulation sounds extremely tricky, but if we translate it into a language understandable to a common person, it turns out that, according to Euclid, two non-parallel lines will definitely intersect. Lobachevsky managed to prove the falsity of this message.

Myth two. In Lobachevsky's theory, parallel lines intersect

This is not true. In fact, the fifth postulate of Lobachevsky sounds like this: "On the plane, through a point that does not lie on a given line, there passes more than one line that does not intersect the given one." In other words, for one straight line, it is possible to draw at least two straight lines through one point that will not intersect it. That is, in this postulate of Lobachevsky there is no talk of parallel lines at all! We only talk about the existence of several non-intersecting lines on the same plane. Thus, the assumption about the intersection of parallel lines was born because of the banal ignorance of the essence of the theory of the great Russian mathematician.

Myth three. Lobachevsky geometry is the only non-Euclidean geometry

Non-Euclidean geometries are a whole layer of theories in mathematics, where the basis is the fifth postulate different from Euclidean. Lobachevsky, unlike Euclid, for example, describes a hyperbolic space. There is another theory describing spherical space - this is Riemann's geometry. This is where the parallel lines intersect. A classic example of this from the school curriculum is the meridians on the globe. If you look at the pattern of the globe, it turns out that all the meridians are parallel. Meanwhile, it is worth putting a pattern on the sphere, as we see that all previously parallel meridians converge at two points - at the poles. Together the theories of Euclid, Lobachevsky and Riemann are called "three great geometries".

Myth four. Lobachevsky geometry is not applicable in real life

On the contrary, modern science comes to understand that Euclidean geometry is only a special case of Lobachevsky's geometry, and that the real world is more accurately described by the formulas of the Russian scientist. The strongest impetus for the further development of Lobachevsky's geometry was Albert Einstein's theory of relativity, which showed that the very space of our Universe is not linear, but is a hyperbolic sphere. Meanwhile, Lobachevsky himself, despite the fact that he worked all his life on the development of his theory, called it "imaginary geometry."

Myth five. Lobachevsky was the first to create non-Euclidean geometry

This is not entirely true. In parallel with him and independently of him, the Hungarian mathematician Janos Bolyai and the famous German scientist Carl Friedrich Gauss came to similar conclusions. However, the works of Janos were not noticed by the general public, and Karl Gauss preferred not to be published at all. Therefore, it is our scientist who is considered a pioneer in this theory. However, there is a somewhat paradoxical point of view that Euclid himself was the first to invent non-Euclidean geometry. The fact is that he self-critically considered his fifth postulate not obvious, so he proved most of his theorems without resorting to it.

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||c 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.

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:

Lobachevsky's geometry 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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Lv1. (Axiom of Lobachevsky's parallelism). In any plane there is a line a 0 and a point A 0 that does not belong to this line, such that at least two lines pass through this point that do not intersect a 0 .

The set of points, lines and planes that satisfy the axioms of membership, order, congruence, continuity and the axiom of parallelism of Lobachevsky will be called the three-dimensional space of Lobachevsky and denoted by L 3 . Most of the geometric properties of figures will be considered by us on the plane of the space L 3, i.e. on the Lobachevsky plane. Let us pay attention to the fact that the formal logical negation of the axiom V 1 , the axiom of parallelism in Euclidean geometry, has exactly the same formulation that we have given as the axiom LV 1 . There are at least one point and one line on the plane for which the assertion of the axiom of parallelism of Euclidean geometry does not hold. Let us prove a theorem from which it follows that the assertion of the Lobachevsky parallelism axiom is valid for any point and any straight Lobachevsky plane.

Theorem 13.1.Let a be an arbitrary line, A be a point not lying on this line. Then in the plane defined by the point A and the line a, there are at least two lines passing through A and not intersecting the line a.

Proof. We carry out the proof by the "by contradiction" method, while using Theorem 11.1 (see § 11). Let there be a point A and a line a in Lobachevsky space such that in the plane defined by this point and the line a, the only line that does not intersect a passes through the point A. Let us also drop the points A perpendicular to AB on the line a and at the point A we restore the perpendicular h to the line AB (Fig. 50). As follows from Theorem 4.2 (see § 4), the lines h and a do not intersect. The line h, by virtue of the assumption, is the only line passing through A and not intersecting a. Let us choose an arbitrary point C on the line a. Let us set aside from the ray AC in the half-plane with the boundary AB, which does not contain the point B, the angle CAM equal to ACB. Then, as follows from the same Theorem 4.2, the line AM does not intersect a. It follows from our assumption that it coincides with h. Therefore, the point M belongs to the line h. Triangle ABC is a right triangle. Calculate the sum of the angles of the triangle ABC: . It follows from Theorem 11.1 that the condition of the axiom of parallelism of Euclidean geometry is satisfied. Therefore, in the plane under consideration there cannot exist such points A 0 and a straight line a 0 that at least two straight lines pass through this point that do not intersect a 0 . We have come to a contradiction with the condition of Lobachevsky's parallelism axiom. The theorem has been proven.

It should be noted that in what follows we will use the assertion of precisely Theorem 13.1, essentially replacing with it the assertion of Lobachevsky's parallelism axiom. By the way, in many textbooks, it is this statement that is accepted as an axiom of the parallelism of Lobachevsky's geometry.

It is easy to obtain the following corollary from Theorem 13.1.

Corollary 13.2. In the Lobachevsky plane, through a point not lying on a given line, there are infinitely many lines that do not intersect the given line.

Indeed, let a be a given line, and A a point that does not belong to it, h 1 and h 2 are straight lines passing through A and not intersecting a (Fig. 51). Obviously, all the lines that pass through the point A and lie in one of the angles formed by h 1 and h 2 (see Fig. 51) do not intersect the line a.

In Chapter 2, we proved a number of assertions that are equivalent to the axiom of parallelism in Euclidean geometry. Their logical negations characterize the properties of figures on the Lobachevsky plane.

First, on the Lobachevsky plane, the logical negation of the fifth postulate of Euclid is valid. In Section 9, we formulated the postulate itself and proved a theorem on its equivalence to the axiom of parallelism in Euclidean geometry (see Theorem 9.1). Its logical negation is:

Statement 13.3.There are two non-intersecting lines on the Lobachevsky plane, which, when intersecting with a third line, form one-sided interior angles whose sum is less than two right angles.

In § 12 we formulated the proposal of Posidonius: on the plane there are at least three collinear points located in one half-plane from the given line and equidistant from it. We also proved Theorem 12.6: the proposal of Posidonius is equivalent to the assertion of the axiom of parallelism in Euclidean geometry. Thus, the negation of this assertion acts on the Lobachevsky plane.

Assertion 13.4. The set of points equidistant from the line on the Lobachevsky plane and located in the same half-plane relative to it, in turn, do not lie on the same line.

On the Lobachevsky plane, a set of points equidistant from a straight line and belonging to the same half-plane relative to this straight line form a curved line, the so-called equidistant line. Its properties will be considered by us later.

Consider now Legendre's proposition: Theorem 11.6, which we have proved (see § 11), asserts that It follows that on the Lobachevsky plane the logical negation of this proposition is true.

Assertion 13.5. On the side of any acute angle there is such a point that the perpendicular to it, erected at this point, does not intersect the second side of the angle.

Let us note the properties of triangles and quadrangles in the Lobachevsky plane, which follow directly from the results of Sections 9 and 11. First of all, Theorem 11.1. States that the assumption of the existence of a triangle whose sum of angles coincides with the sum of two right angles is equivalent to the axiom of parallelism of the Euclidean plane. From this and from Legendre's first theorem (see Theorem 10.1, § 10) the following assertion follows

Assertion 13.6. On the Lobachevsky plane, the sum of the angles of any triangle is less than 2d.

From this it follows directly that the sum of the angles of any convex quadrilateral is less than 4d, and the sum of the angles of any convex n-gon is less than 2(n-1)d.

Since on the Euclidean plane the angles adjacent to the upper base of the Saccheri quadrilateral are equal to right angles, which, in accordance with Theorem 12.3 (see § 12), is equivalent to the axiom of parallelism in Euclidean geometry, we can draw the following conclusion.

Statement 13.7. The angles adjacent to the upper base of the Saccheri quadrilateral are acute.

It remains for us to consider two more properties of triangles on the Lobachevsky plane. The first of these is related to Wallis' proposal: there exists at least one pair of triangles in the plane with correspondingly equal angles but not equal sides. In Section 11 we proved that this proposition is equivalent to the axiom of parallelism in Euclidean geometry (see Theorem 11.5). The logical negation of this statement leads us to the following conclusion: there are no triangles on the Lobachevsky plane with equal angles but not equal sides. Thus, the following proposition is true.

Statement 13.8. (the fourth criterion for the equality of triangles on the Lobachevsky plane).Any two triangles on the Lobachevsky plane, having respectively equal angles, are equal to each other.

Consider now the next question. Can a circle be described around any triangle in the Lobachevsky plane? The answer is given by Theorem 9.4 (see § 9). In accordance with this theorem, if a circle can be circumscribed around any triangle on the plane, then the condition of the axiom of parallelism of Euclidean geometry is satisfied on the plane. Therefore, the logical negation of the assertion of this theorem leads us to the following proposition.

Assertion 13.9. There is a triangle on the Lobachevsky plane around which it is impossible to describe a circle.

It is easy to construct an example of such a triangle. We choose some line a and a point A that does not belong to it. Let us drop the perpendicular h from the point A to the line a. By virtue of Lobachevsky's axiom of parallelism, there is a line b passing through A and not perpendicular to h, which does not intersect a (Fig. 52). As you know, if a circle is circumscribed around a triangle, then its center lies at the point of intersection of the perpendicular bisectors of the sides of the triangle. Therefore, it suffices for us to give an example of such a triangle, the perpendicular bisectors of which do not intersect. We choose a point M on the line h, as shown in Figure 52. We display it symmetrically with respect to the lines a and b, we get points N and P. Since the line b is not perpendicular to h, the point P does not belong to h. Therefore, the points M, N and P make up the vertices of the triangle. The lines a and b serve by construction as its perpendicular bisectors. They, as mentioned above, do not intersect. Triangle MNP is the desired one.

It is easy to construct an example of a triangle in the Lobachevsky plane around which a circle can be described. To do this, it is enough to take two intersecting lines, choose a point that does not belong to them, and reflect it relative to these lines. Do the detailed building yourself.

Definition 14.1. Let two directed lines and be given. They are called parallel if the following conditions are met:

1. lines a and b do not intersect;

2. for arbitrary points A and B of the straight lines a and b, any internal ray h of the angle AVB 2 intersects the straight line a (Fig. 52).

We will denote parallel lines in the same way as it is customary in the school geometry course: a || b. Note that parallel lines on the Euclidean plane satisfy this definition.

Theorem 14.3. Let a directed straight line and a point B, which does not belong to it, be given on the Lobachevsky plane. Then a single directed line passes through the given point such that line a is parallel to line b.

Proof. Let us drop the perpendicular BA from point B to the line a and from point B we will restore the perpendicular p to the line BA (Fig. 56 a). The line p, as has been repeatedly noted, does not intersect the given line a. We choose an arbitrary point С on it, divide the points of the segment AC into two classes and . The first class will include such points S of this segment for which the ray BS intersects the ray AA 2 , and the second class includes such points T for which the ray BT does not intersect the ray AA 2 . Let us show that such a division into classes produces a Dedekind section of the segment AC. According to Theorem 4.3 (see § 4) we have to check that:

2. and classes and contain points other than A and C;

3. any point of class other than A lies between point A and any point of class .

The first condition is obvious, all points of the segment belong to one or another class, while the classes themselves, based on their definition, do not have common points.

The second condition is also easy to verify. It is obvious that and . The class contains points other than A, to verify this statement, it is enough to select any point of the ray AA 2 and connect it to point B. This ray will intersect the segment BC at a point of the first class. The class also contains points other than C, otherwise we will come to a contradiction with Lobachevsky's axiom of parallelism.

Let us prove the third condition. Let there be a point S of the first class different from A, and a point T of the second class such that the point T lies between A and S (see Fig. 56 a). Since , then the ray BS intersects the ray AA 2 at some point R. Consider the ray BT. It intersects side AS of triangle ASR at point T. According to Pasha's axiom, this ray must intersect either side AR or side SR of this triangle. Suppose that the ray BT intersects the side SR at some point O. Then two different lines BT and BR pass through the points B and O, which contradicts the axiom of Hilbert's axiomatics. Thus, the ray BT intersects the side AR, which implies that the point T does not belong to the class K 2 . The resulting contradiction leads to the assertion that the point S lies between A and T. The condition of Theorem 4.3 has been verified completely.

In accordance with the conclusion of Theorem 4.3 on the Dedekind section on the segment AC, there is such a point for which any point lying between A and belongs to the class , and any point lying between and C belongs to the class . Let us show that the directed line is parallel to the line . In fact, it remains for us to prove that does not intersect the line a, since, due to the choice of points of class K 1, any internal ray of the angle intersects . Suppose that the line intersects the line a at some point H (Fig. 56 b). We choose an arbitrary point P on the ray HA 2 and consider the ray BP. Then it intersects the segment M 0 C at some point Q (prove this statement yourself). But the interior points of the segment M 0 C belong to the second class, the ray BP cannot have points in common with the line a. Thus, our assumption about the intersection of lines BM 0 and a is incorrect.

It is easy to check that the line is the only directed line passing through point B and parallel to . Indeed, let another directed line pass through point B, which, like and, is parallel to . In this case, we will assume that M 1 is a point of the segment AC. Then, proceeding from the definition of the class K 2 , . Therefore, the ray BM 0 is an internal ray of the angle , therefore, by definition 14.1 it intersects the line . We have arrived at a contradiction with the assertion proved above. Theorem 14.3 is completely proved.

Consider a point B and a directed line that does not contain it. In accordance with the proven theorem 14.3, a directed straight line passes through the point B, parallel to a. Let us drop the perpendicular BH from the point B to the straight line a (Fig. 57). It is easy to see that angle HBB 2 - acute. Indeed, if we assume that this angle is a right angle, then it follows from Definition 14.1 that any line passing through the point B intersects the line a, which contradicts Theorem 13.1, i.e., axiom LV 1 of Lobachevsky's parallelism (see § 13). It is easy to see that the assumption that this angle is obtuse also leads to a contradiction now with Definition 14.1 and Theorem 4.2 (see § 4), since the inner ray of the angle HBB 2 perpendicular to BH does not intersect the ray AA 2 . Thus, the following assertion is true.

Theorem 14.4. Let a directed line be parallel to a directed line. If from the point B of the straight line we drop the perpendicular ВН to the straight line , then the angle HBB 2 is acute.

The following corollary clearly follows from this theorem.

Consequence.If there is a common perpendicular of the directed lines and , then the line is not parallel to the line .

Let us introduce the concept of parallelism for non-directed lines. We will assume that two non-directed lines are parallel if it is possible to choose directions on them so that they satisfy Definition 14.1. As you know, a straight line has two directions. Therefore, from Theorem 14.3 it follows that through the point B, which does not belong to the line a, there pass two undirected lines parallel to the given line. Obviously, they are symmetrical with respect to the perpendicular dropped from the point B to the line a. These two lines are the same boundary lines that separate the pencil of lines passing through the point B and intersecting a from the pencil of lines passing through B and not intersecting the line a (Fig. 57).

Theorem 15.2. (Property of symmetry of parallel lines on the Lobachevsky plane).Let a directed line be parallel to a directed line. Then the directed line is parallel to the line.

The symmetry property of the concept of parallel lines on the Lobachevsky plane allows us not to specify the order of directed parallel lines, i.e. do not specify which line is the first and which is the second. It is obvious that the property of symmetry of the concept of parallel lines also takes place on the Euclidean plane. It follows directly from the definition of parallel lines in Euclidean geometry. In Euclidean geometry, the property of transitivity also holds for parallel lines. If line a is parallel to line b and line b is parallel to line c. then the lines a and c are also parallel to each other. A similar property is also valid for directed lines on the Lobachevsky plane.

Theorem 15.3. (Property of transitivity of parallel lines on the Lobachevsky plane).Let there be given three distinct directed lines , . If a and , then .

Consider a directed line parallel to a directed line. Let's cross them with a straight line. Points A and B, respectively, are the points of intersection of the lines , and , (Fig. 60). The following theorem is true.

Theorem 15.4. The angle is greater than the angle.

Theorem 15.5. An exterior angle of a degenerate triangle is greater than an interior angle not adjacent to it.

The proof follows directly from Theorem 15.4. Pass it on your own.

Consider an arbitrary segment AB. Through point A we draw a line a, perpendicular to AB, and through point B, a line b, parallel to a (Fig. 63). As follows from Theorem 14.4 (see § 14), the line b is not perpendicular to the line AB.

Definition 16.1. The acute angle formed by the lines AB and b is called the angle of parallelism of the segment AB.

It is clear that each segment corresponds to a certain angle of parallelism. The following theorem is true.

Theorem 16.2. Equal segments correspond to equal angles of parallelism.

Proof. Let two equal segments AB and A¢B¢ be given. Let us draw directed lines and through points A and A¢, perpendicular to AB and A¢B¢, respectively, and through points B and B¢, directed straight lines and parallel, respectively, and (Fig. 64). Then and respectively, the parallelism angles of the segments AB and A¢B¢. Let's pretend that

Let us set aside the angle a 2 from the ray BA in the half-plane BAA 2, (see Fig. 64). Due to inequality (1), the ray l is the inner ray of the angle ABB 2 . Since ½1 , then l intersects the ray AA 2 at some point P. Let us plot on the ray A¢A 2 ¢ from the point A¢ the segment A¢P¢ equal to AP. Consider the triangles ABP and A¢B¢P¢. They are rectangular, according to the condition of the theorem they have equal legs AB and A¢B¢, by construction the second pair of legs AR and A¢P¢ are equal. Thus, the right triangle ABP is equal to the triangle A¢B¢P¢. So . On the other hand, the beam B¢P¢ intersects the beam A¢A 2 ¢, and the directed line B 1 ¢B 2 ¢ is parallel to the straight line A 1 ¢A 2 ¢. Therefore, the ray B¢P¢ is the inner ray of the angle A¢B¢B 2 ¢, . The resulting contradiction refutes our assumption, inequality (1) is false. Similarly, it is proved that the angle cannot be less than the angle . The theorem has been proven.

Let us now consider how the angles of parallelism of unequal segments are related to each other.

Theorem 16.3. Let the segment AB be greater than the segment A¢B¢, and the angles and, respectively, their angles of parallelism. Then .

Proof. The proof of this theorem follows directly from Theorem 15.5 (see § 15) on the exterior angle of a degenerate triangle. Consider segment AB. Let us draw through point A directed straight line, perpendicular to AB, and through point B directed straight line, parallel (Fig. 65). Let us plot on the ray AB a segment AP equal to A¢B¢. Since , then P is an interior point of the segment AB. Let's draw a directed straight line C 1 C 2 through R, also parallel. The angle serves as the angle of parallelism of the segment A¢B¢, and the angle serves as the angle of parallelism of the segment AB. On the other hand, from Theorem 15.2 on the symmetry of the concept of parallel lines (see § 15) it follows that the line C 1 C 2 is parallel to the line . Therefore, the triangle RVS 2 A 2 is degenerate, - external, and - its internal angles. Theorem 15.5 implies the truth of the assertion being proved.

It is easy to prove the converse.

Theorem 16.4.Let and be the parallelism angles of the segments AB and A¢B¢. Then, if , then AB > А¢В¢.

Proof. Assume the opposite, . Then it follows from Theorems 16.2 and 16.3 that , which contradicts the condition of the theorem.

And so we proved that each segment has its own angle of parallelism, and a larger segment corresponds to a smaller angle of parallelism. Consider a statement that proves that for any acute angle there is a segment for which this angle is the angle of parallelism. This will establish a one-to-one correspondence between segments and acute angles on the Lobachevsky plane.

Theorem 16.5. For any acute angle, there is a segment for which this angle is the angle of parallelism.

Proof. Let an acute angle ABC be given (Fig. 66). We will assume that all the points considered below on the rays BA and BC lie between the points B and A and B and C. We call a ray admissible if its origin belongs to the side of the angle BA, it is perpendicular to the line BA and is located in the same half-plane with respect to the line BA as the side BC of the given angle. Let's turn to Legendre's suggestion: p A perpendicular drawn to the side of an acute angle at any point on this side intersects the second side of the angle. We have proved Theorem 11.6 (see § 11), which states that Legendre's proposition is equivalent to the parallelism axiom of Euclidean geometry. From this we concluded that on the Lobachevsky plane the logical negation of this statement is true, namely, on the side of any acute angle there is such a point that the perpendicular to it, erected at this point, does not intersect the second side of the angle(see § 13). Thus, there is an admissible ray m with origin at the point M, which does not intersect the side BC of the given angle (see Fig. 66).

Let us divide the points of the BM segment into two classes. class will belong to those points of this segment for which admissible rays with origins at these points intersect the side BC of the given angle, and the class belong those points of the segment BC for which the admissible rays with origins at these points do not intersect the side BC. Let us show that such a partition of the segment VM forms a Dedekind section (see Theorem 4.3, § 4). To do this, you should check that

5. and classes and contain points other than B and M;

6. any point of class , other than B, lies between point B and any point of class .

The first condition is clearly satisfied. Any point of the BM segment belongs either to the class K 1 or to the class K 2 . Moreover, a point, by virtue of the definition of these classes, cannot belong to two classes at the same time. Obviously, we can assume that , the point M belongs to K 2, since the admissible ray with the origin at the point M does not intersect BC. The class K 1 contains at least one point other than B. To construct it, it suffices to choose an arbitrary point P on the side BC and drop the perpendicular PQ from it onto the ray BA. If we assume that the point Q lies between the points M and A, then the points P and Q lie in different half-planes relative to the straight line containing the ray m (see Fig. 66). Therefore, the segment PQ intersects the ray m at some point R. We obtain that two perpendiculars are dropped from the point R to the line BA, which contradicts Theorem 4.2 (see § 4). Thus, the point Q belongs to the segment BM, the class K 1 contains points different from B. It is easy to explain why there is a segment on the ray BA that contains at least one point belonging to the class K 2 and different from its end. Indeed, if the class K 2 of the considered segment BM contains a single point M, then we choose an arbitrary point M¢ between M and A. Consider an admissible ray m¢ with origin at the point M¢. It does not intersect the ray m, otherwise two perpendiculars are dropped from the point to the line AB, so m¢ does not intersect the ray BC. The segment ВМ¢ is the desired one, and all further reasoning should be carried out for the segment ВМ¢.

Let us verify the validity of the third condition of Theorem 4.3. Suppose that there are such points and that the point P lies between the point U and M (Fig. 67). Let us draw admissible rays u and p with origins at points U and P. Since then the ray p intersects the side BC of the given angle at some point Q. The line containing the ray u intersects the side BP of the triangle BPQ, therefore, according to Hilbert's axiom (Pasch's axiom , see § 3) it intersects either the side BQ or the side PQ of this triangle. But, , therefore, the ray u does not intersect the side BQ, therefore, the rays p and u intersect at some point R. We again come to a contradiction, since we have constructed a point from which two perpendiculars are dropped to the line AB. The condition of Theorem 4.3 is completely satisfied.

M. It follows that . We have obtained a contradiction, since we have constructed a point of class K 1 located between the points and M. It remains for us to show that any internal ray of the angle intersects the ray BC. Consider an arbitrary inner ray h of this angle. We choose an arbitrary point K on it, which belongs to the angle , and drop a perpendicular from it to the line BA (Fig. 69). The base S of this perpendicular obviously belongs to the segment VM 0 , i.e. class K 1 (prove this fact yourself). It follows from this that the perpendicular KS intersects the side BC of the given angle at some point T (see Fig. 69). The ray h crossed the ST side of the triangle BST at the point K, according to the axiom (Pasha's axiom), it must intersect either the BS side or the BT side of this triangle. It is clear that h does not intersect the segment BS, otherwise two lines, h and BA, pass through two points and this intersection point. Thus, h intersects the BT side, i.e. beam BA. The theorem is proved completely.

And so, we have established that each segment in Lobachevsky's geometry can be associated with an acute angle - its angle of parallelism. We will assume that we have introduced the measure of angles and segments, we note that the measure of segments will be introduced by us later, in § . We introduce the following definition.

Definition 16.6. If x is the length of the segment, and j is the angle, then the dependence j = P(x), which associates the length of the segment with the value of its angle of parallelism, is called the Lobachevsky function.

It's clear that . Using the properties of the angle of parallelism of a segment proved above (see Theorems 16.3 and 16.4), we can conclude the following: the Lobachevsky function is monotonically decreasing. Nikolai Ivanovich Lobachevsky obtained the following remarkable formula:

,

where k is some positive number. It is of great importance in the geometry of the Lobachevsky space, and is called its radius of curvature. Two Lobachevsky spaces having the same radius of curvature are isometric. From the above formula, as it is easy to see, it also follows that j = P(x) is a monotonically decreasing continuous function whose values ​​belong to the interval .

On the Euclidean plane, we fix a circle w with center at some point O and radius equal to one, which we will call absolute. The set of all points of the circle bounded by the circle w will be denoted by W¢, and the set of all interior points of this circle by W. Thus, . The points of the set W will be called L-points The set W of all L-points is L-plane, on which we will build the Cayley-Klein model of the Lobachevsky plane. We will call L‑straight arbitrary chords of the circle w. We will assume that an L-point X belongs to the L-line x if and only if the point X, as a point of the Euclidean plane, belongs to the chord x of the absolute.

L‑plane, Lobachevsky’s axiom of parallelism holds: through an L-point B that does not lie on the L-line a pass at least two L-lines b and c that do not have common points with the L-line a. Figure 94 illustrates this statement. It is also easy to understand what the parallel directed straight lines of the L-plane are. Consider Figure 95. The L-line b passes through the point of intersection of the L-line a with the absolute. Therefore, the directed L-line A 1 A 2 is parallel to the directed L-line B 1 A 2 . Indeed, these lines do not intersect, and if we choose arbitrary L-points A and B belonging respectively to these lines, then any internal ray h of angle A 2 BA intersects the line a. Thus two L-lines are parallel if they have a common point of intersection with the absolute. It is clear that the property of symmetry and transitivity of the concept of parallelism of L-lines is satisfied. In paragraph 15, we proved the property of symmetry, while the property of transitivity is illustrated in Figure 95. The line A 1 A 2 is parallel to the line B 1 A 2, they intersect the absolute at the point A 2 . The lines B 1 A 2 and C 1 A 2 are also parallel, they also intersect the absolute at the same point A 2 . Therefore, the lines A 1 A 2 and C 1 A 2 are parallel to each other.

Thus, the basic concepts defined above satisfy the requirements of the axioms I 1 -I 3 , II, III, IV of the Hilbert axiomatic groups and the axiom of Lobachevsky's parallelism, therefore they are a model of the Lobachevsky plane. We have proved the meaningful consistency of Lobachevsky's planimetry. We formulate this statement as the following theorem.

Theorem 1. The geometry of Lobachevsky is not contradictory in content.

We have built a model of the Lobachevsky plane, but you can get acquainted with the construction of a spatial model similar to that considered on the plane in the manual.

The most important conclusion follows from Theorem 1. The axiom of parallelism is not a consequence of axioms I–IV of Hilbert's axiomatics. Since the fifth postulate of Euclid is equivalent to the axiom of parallelism of Euclidean geometry, this postulate also does not depend on the rest of Hilbert's axioms.