The topic of this video tutorial will be motion properties, as well as parallel translation. At the beginning of the lesson, we will once again repeat the concept of movement, its main types - axial and central symmetry. After that, we consider all the properties of motion. Let's analyze the concept of "parallel transfer", what it is used for, let's name its properties.

Theme: Movement

Lesson: Movement. Motion Properties

Let's prove the theorem: when moving, the segment passes into the segment.

Let us decipher the formulation of the theorem with the help of Fig. 1. If the ends of a certain segment MN during the movement are displayed at some points M 1 and N 1, respectively, then any point P of the segment MN will necessarily go to some point P 1 of the segment M 1 N 1, and vice versa, to each point Q 1 of the segment M 1 N 1 some point Q of the segment MN will be displayed.

Proof.

As can be seen from the figure, MN = MP + PN.

Let the point P go to some point P 1 "of the plane. The definition of motion implies the equality of the lengths of the segments MN \u003d M 1 N 1, MP \u003d M 1 P 1", PN \u003d P 1 "N 1. From these equalities it follows that M 1 Р 1 ", M 1 Р 1 "+ Р 1 "N 1 = MP + РN = MN = M 1 N 1, that is, the point Р 1 "belongs to the segment M 1 N 1 and coincides with the point P 1, otherwise instead of the above equality, the inequality of the triangle M 1 P 1 "+ P 1" N 1 > M 1 N 1 would be true. That is, we proved that when moving, any point, any point P of the segment MN will necessarily go to some point P 1 of the segment M 1 N 1. The second part of the theorem (concerning the point Q 1) is proved in exactly the same way.

The proved theorem is valid for any motions!

Theorem: when moving, the angle goes into an equal angle.

Let RAOB be given (Fig. 2). And let some movement be given, in which the vertex РО goes to the point О 1 , and the points A and B - respectively to the points А 1 and В 1 .

Consider triangles AOB and A 1 O 1 B 1 . According to the condition of the theorem, points A, O and B move when moving to points A 1, O 1 and B 1, respectively. Therefore, there is an equality of lengths AO \u003d A 1 O 1, OB \u003d O 1 B 1 and AB \u003d A 1 B 1. Thus, AOB \u003d A 1 O 1 B 1 on three sides. From the equality of triangles follows the equality of the corresponding angles O and O 1.

So, any movement preserves angles.

A lot of consequences follow from the basic properties of motion, in particular, that any figure during movement is mapped onto a figure equal to it.

Consider another type of movement - parallel transfer.

Parallel transfer onto some given vector is called such a mapping of the plane onto itself, in which each point M of the plane goes to such a point M 1 of the same plane that (Fig. 3).

Let's prove that parallel translation is a movement.

Proof.

Consider arbitrary segment MN (Fig. 4). Let the point M move to the point M 1 during parallel transfer, and the point N - to the point N 1. In this case, the conditions of parallel transfer are fulfilled: and . Consider a quadrilateral

MM 1 N 1 N. Its two opposite sides (MM 1 and NN 1) are equal and parallel, as dictated by the parallel translation conditions. Therefore, this quadrilateral is a parallelogram according to one of the signs of the latter. This implies that the other two sides (MN and M 1 N 1) of the parallelogram have equal lengths, which was to be proved.

Thus, parallel transfer is indeed a movement.

Let's summarize. We are already familiar with three types of motion: axial symmetry, central symmetry and parallel transfer. We have proved that when moving, a segment passes into a segment, and an angle into an equal angle. In addition, it can be shown that a straight line passes into a straight line when moving, and a circle passes into a circle of the same radius.

1. Atanasyan L. S. and others. Geometry grades 7-9. Tutorial for educational institutions. - M.: Education, 2010.

2. Farkov A. V. Geometry tests: Grade 9. To the textbook of L. S. Atanasyan and others - M .: Exam, 2010.

3. A. V. Pogorelov, Geometry, account. for 7-11 cells. general inst. - M.: Enlightenment, 1995.

1. Russian educational portal ().

2. Festival pedagogical ideas « Public lesson» ().

1. Atanasyan (see references), p. 293, § 1, item 114.

The theorem on the motion of the center of mass.

In some cases, to determine the nature of the motion of a system (especially a rigid body), it is sufficient to know the law of motion of its center of mass. For example, if you throw a stone at a target, you don’t need to know at all how it will tumble during the flight, it is important to establish whether it will hit the target or not. To do this, it is enough to consider the movement of some point of this body.

To find this law, we turn to the equations of motion of the system and add their left and right parts term by term. Then we get:

Let's transform the left side of the equality. From the formula for the radius vector of the center of mass, we have:

Taking from both parts of this equality the second time derivative and noticing that the derivative of the sum is equal to the sum of the derivatives, we find:

where is the acceleration of the center of mass of the system. Since, by the property of internal system forces, then, substituting all the found values, we finally get:

The equation and expresses the theorem on the motion of the center of mass of the system: the product of the mass of the system and the acceleration of its center of mass is geometric sum all external forces acting on the system. Comparing with the equation of motion of a material point, we obtain another expression of the theorem: the center of mass of the system moves as a material point, the mass of which is equal to the mass of the entire system and to which all external forces acting on the system are applied.

Projecting both sides of the equality onto the coordinate axes, we get:

These equations are differential equations of motion of the center of mass in projections on the axes of the Cartesian coordinate system.

The meaning of the proved theorem is as follows.

1) The theorem provides a justification for the methods of point dynamics. It can be seen from the equations that the solutions that we get, considering the given body as a material point, determine the law of motion of the center of mass of this body, those. have a very specific meaning.

In particular, if the body moves forward, then its motion is completely determined by the motion of the center of mass. Thus, a progressively moving body can always be considered as a material point with a mass, equal to the mass body. In other cases, the body can be considered as a material point only when, in practice, to determine the position of the body, it is sufficient to know the position of its center of mass.

2) The theorem allows, when determining the law of motion of the center of mass of any system, to exclude from consideration all previously unknown internal forces. This is its practical value.

So the movement of the car on a horizontal plane can occur only under the action of external forces, friction forces acting on the wheels from the side of the road. And the braking of the car is also possible only by these forces, and not by friction between the brake pads and the brake drum. If the road is smooth, no matter how much the wheels brake, they will slide and will not stop the car.

Or after the explosion of a flying projectile (under the action of internal forces) parts, fragments of it, will scatter so that their center of mass will move along the same trajectory.

The theorem on the motion of the center of mass of a mechanical system should be used to solve problems in mechanics that require:

According to the forces applied to a mechanical system (most often to a solid body), determine the law of motion of the center of mass;

By given law movements of bodies included in the mechanical system, find the reactions of external bonds;

According to the given mutual motion of the bodies included in the mechanical system, determine the law of motion of these bodies relative to some fixed frame of reference.

Using this theorem, one of the equations of motion of a mechanical system with several degrees of freedom can be compiled.

When solving problems, the consequences of the theorem on the motion of the center of mass are often used mechanical system.

Corollary 1. If main vector external forces applied to a mechanical system, zero, then the center of mass of the system is at rest or moves uniformly and rectilinearly. Since the acceleration of the center of mass is zero, .

Corollary 2. If the projection of the main vector of external forces on any axis is equal to zero, then the center of mass of the system either does not change its position relative to this axis, or moves uniformly relative to it.

For example, if two forces begin to act on the body, forming a pair of forces (Fig. 38), then the center of mass With it will move along the same trajectory. And the body itself will rotate around the center of mass. And it doesn't matter where a couple of forces are applied.

By the way, in statics we proved that the effect of a pair on a body does not depend on where it is applied. Here we have shown that the rotation of the body will be around the central axis With.

Fig.38

Theorem on the change of the kinetic moment.

Kinetic moment of a mechanical system relative to a fixed center O is a measure of the motion of the system around this center. When solving problems, it is usually not the vector itself that is used, but its projections on the axes of a fixed coordinate system, which are called kinetic moments about the axis. For example, - the kinetic moment of the system relative to the fixed axis Oz .

The momentum of a mechanical system is the sum of momentum points and bodies included in this system. Consider ways to determine the angular momentum material point and a solid body various occasions their movements.

For a material point with a mass having a velocity, the angular momentum about some axis Oz is defined as the moment of the momentum vector of this point about the selected axis:

The angular momentum of a point is considered positive if, from the side of the positive direction of the axis, the movement of the point occurs counterclockwise.

If a point makes a complex movement, to determine its angular momentum, the momentum vector should be considered as the sum of the quantities of relative and portable movements (Fig. 41)

But , where is the distance from the point to the axis of rotation, and

Rice. 41

The second component of the angular momentum vector can be defined in the same way as the moment of force about the axis. As for the moment of force, the value is zero if the relative velocity vector lies in the same plane as the translational rotation axis.

The kinetic moment of a rigid body relative to a fixed center can be defined as the sum of two components: the first of them characterizes the translational part of the motion of the body together with its center of mass, the second characterizes the movement of the system around the center of mass:

If the body performs translational motion, then the second component is equal to zero

The kinetic moment of a rigid body is most simply calculated when it rotates around a fixed axis

where is the moment of inertia of the body about the axis of rotation.

The theorem on the change in the angular momentum of a mechanical system when it moves around a fixed center is formulated as follows: the total time derivative of the angular momentum vector of a mechanical system with respect to some fixed center O in magnitude and direction is equal to the main moment of external forces applied to the mechanical system, defined relative to the same center

where - main point all external forces about the center O.

When solving problems in which bodies are considered rotating around a fixed axis, they use the theorem on the change in angular momentum relative to a fixed axis

As for the theorem on the motion of the center of mass, the theorem on the change in angular momentum has consequences.

Corollary 1. If the main moment of all external forces relative to some fixed center is equal to zero, then the kinetic moment of the mechanical system relative to this center remains unchanged.

Corollary 2. If the main moment of all external forces about some fixed axis is equal to zero, then the kinetic moment of the mechanical system about this axis remains unchanged.

The momentum change theorem is used to solve problems in which the movement of a mechanical system is considered, consisting of a central body rotating around a fixed axis, and one or more bodies, the movement of which is associated with the central one. Communication can be carried out using threads, bodies can move along the surface of the central body or in its channels due to internal forces. Using this theorem, one can determine the dependence of the law of rotation of the central body on the position or movement of the remaining bodies.

Plane motions and their properties. Movement examples. Classification of movements. Movement group. Applying motion to problem solving

Motion- this is a transformation of figures, in which distances between points are preserved. If two figures are exactly combined with each other by means of movement, then these figures are the same, equal.

Motion is a bijective transformation φ of the plane π, under which for any different points X, Y є π the relation XY  φ(X)φ(Y) is satisfied.

Movement properties:

1.Composition φ ◦ ψ two movements ψ , φ is a movement.

Doc-in: Let the figure F translated by movement ψ into a figure F ', and the figure F ’ is translated by movement φ into a figure F ''. Let the point X figures F goes to the point X ’ shapes F ’ , and during the second movement, the point X ’ shapes F ' goes to point X '' figures F ''. Then the transformation of the figure F into a figure F '', at which an arbitrary point X figures F goes to the point X '' figures F '', preserves the distance between points, and therefore is also a movement.

Song recording always starts from the last movement, because the result of the composition is the final image - it is put in line with the original: X ’’= ψ (X ’) = ψ (φ (X )) = ψ ◦ φ (X )

2. If φ – movement, then transformation φ -1 is also a movement.

Doc-in: Let the shape transformation F into a figure F ' translates various points figures F at various points on the figure F '. Let an arbitrary point X figures F under this transformation goes to a point X ’ shapes F ’.

Shape transformation F ' into a figure F , at which the point X ' goes to point X , is called the inverse transformation of the given . For every move φ it is possible to define the reverse movement, which is denoted φ -1 .

Thus, the transformation reverse motion, is also a movement.

It is obvious that the transformation φ -1 satisfies the equalities: f◦f-1 = f-1◦f = ε , where ε is the identical display.

3. Associativity of compositions: Let φ 1 , φ 2 , φ 3 – voluntary movements. Then φ 1 ◦(φ 2 ◦ φ 3) = (φ 1 ◦φ 2)◦φ 3 .

The fact that the composition of movements has the property of associativity allows us to determine the degree φ with natural indicator n .

Let's put φ 1= φ and φ n +1= φ n◦ φ , if n≥ 1 . Thus the movement φ n obtained by n -multiple consistent application movements φ .

4. Preservation of straightness: Points lying on one straight line, when moving, pass into points lying on one straight line, and the order of their relative position is preserved.

This means that if the points A ,B ,C lying on one straight line (such points are called collinear), go to the points A 1 ,B1 ,C1 , then these points also lie on the line; if point B lies between the points A and C , then the point B1 lies between the points A 1 and C1 .

Doc. Let the point B straight AC lies between the points A and C . Let us prove that the points A 1 ,B1 ,C1 lie on the same line.

If the points A 1 ,B1 ,C1 do not lie on one straight line, then they are the vertices of some triangle A 1 B 1 C 1 . So A 1 C 1 <A 1 B 1 +B 1 C 1 .

By the definition of motion, it follows that AC <AB +BC .

However, by the property of measuring segments AC =AB +BC .

We have come to a contradiction. So the point B1 lies between the points A 1 and C1 .

Let's say the point A 1 lies between the points B1 , and C1 . Then A 1 B 1 +A 1 C 1 =B 1 C 1 , and hence AB +AC =BC . But this is contrary to equality. AB +BC =AC .

Thus, point A 1 does not lie between points B1 , and C1 .

It can be proved similarly that the point C1 cannot lie between points A 1 and B1 . Because from three points A 1 ,B1 ,C1 one lies between two others, then this point can only be B1 . The theorem is proved completely.

Consequence. When moving, a straight line is mapped to a straight line, a ray to a ray, a segment to a segment, and a triangle to an equal triangle.

If we denote by X the set of points of the plane, and by φ(X) the image of the set X under the motion of φ, i.e. the set of all points of the form φ(x), where x є X, then we can give a more correct formulation of this property:

Let φ be a motion, A, B, C three different collinear points.

Then the points φ(A), φ(B), φ(C) are also collinear.

If l is a line, then φ(l) is also a line.

If the set X is a ray (segment, half-plane), then the set φ(X) is also a ray (segment, half-plane).

5. When moving, the angles between the beams are preserved.

Doc. Let be AB and AC - two rays emanating from a point A not lying on the same straight line. When moving, these rays turn into some half-lines (rays) A 1 B 1 and A 1 C 1 . Because motion preserves distances, then triangles ABC and A 1 B 1 C 1 are equal according to the third criterion for the equality of triangles (if the three sides of one triangle are respectively equal to the three sides of another triangle, then these triangles are equal). From the equality of triangles follows the equality of the angles BAC and B 1 A 1 C 1 , which was to be proved.

6. Any movement preserves the co-direction of the rays and the same orientation of the flags.

Rays l A and l B called co-directional(similarly oriented, designation: l A l B ) if one of them is contained in the other, or if they are combined by a parallel transfer. FlagF = (π l , l o) is the union of the half-plane πl and beam lo.

Dot O - the beginning of the flag, beam lo starting at point O - flag pole πl - half-plane with boundary l .

Doc. Let be φ - voluntary movement l A l B -codirectional rays with origins at points BUT and AT respectively. Let's introduce the notation: l A1 = φ (l A ), A 1 = φ (BUT ), l B1= φ (l B ),IN 1 = φ (BUT ).If the rays l A and l B lie on the same straight line, then, by virtue of codirectivity, one of them is contained in the other. Considering that l A  l B , we get φ (l A )  φ (l B ), i.e. l A1 l B1 (the symbol  denotes the inclusion or equality of a subset of elements to a set of elements). If, however, l A, l B lie on different lines, then let n = (AB ). Then there exists such a half-plane n , what l A, l B  n . From here φ (l A ),φ (l B ) φ (n ). Insofar as φ (n ) is a half-plane, and its boundary contains points A 1 and IN 1 , we again get that l A, l B co-directed.

Let's apply the movement φ to identically oriented flags F= (π l ,l A ), G= (πm ,m B ). Consider the case when the points A and B match. If straight l and m are different, then the same orientation of the flags means that either (1) l A πm , m A  π'l , or (2) l A π' m ,m A  πl . Without loss of generality, we can assume that condition (1) is satisfied. Then φ (l A )  φ (πm ), φ (m A )  φ (π'l ). This implies the same orientation of the flags φ (F ) and φ (G ).If the direct l ,m match, then either F=G or F = G'. It follows that the flags φ (F ) and φ (G ) are equally oriented.

Let now dots A and B different. Denote by n straight line ( AB ). It is clear that there are codirectional rays nA and nB and half plane n such that the flag F 1 = (πn, nA ) is co-directed with F , and the flag G 1 = (π n , n B , ) is co-directed with G. Means φ (F ) and φ (G ) are equally oriented. The theorem is proved.

Movement examples:

1) parallel translation - such a transformation of a figure in which all points of the figure move in the same direction by the same distance.

2) symmetry with respect to a straight line (axial or mirror symmetry). transformation σ figures F into a figure F', where each of its points X goes to the point X', which is symmetric with respect to the given line l, is called the symmetry transformation with respect to the line l. At the same time, the figures F and F' called symmetrical with respect to the line l.

3) turn around the point. By turning the plane ρ around this point O is called such a movement in which each ray emanating from this point rotates through the same angle α in the same direction

"Investigation of plane motions and some of their properties". page 21 of 21

Investigation of plane motions

and some of their properties

Content

From the history of the development of the theory of motions.

Definition and properties of motions.

Congruence of figures.

Types of movements.

4.1. Parallel transfer.

4.2. Turn.

4.3. Symmetry about a straight line.

4.4. Sliding symmetry.

5. Study of special properties of axial symmetry.

6. Investigation of the possibility of the existence of other types of movements.

7. Mobility theorem. Two kinds of movements.

8. Classification of movements. Chall's theorem.

Movements as a group of geometric transformations.

Application of movements in problem solving.

Literature.

History of the development of the theory of motions.

The first who began to prove some geometric propositions is considered to be the ancient Greek mathematician Thales of Miletus(625-547 BC). It was thanks to Thales that geometry began to turn from a set of practical rules into a true science. Before Thales, evidence simply did not exist!

How did Thales conduct his proofs? For this purpose, he used movements.

Motion - this is a transformation of figures, in which distances between points are preserved. If two figures are exactly combined with each other by means of movement, then these figures are the same, equal.

It was in this way that Thales proved a number of the first theorems of geometry. If the plane is rotated as a rigid whole around some point O 180 o, beam OA will go to its continuation OA ’ . With such turning (also called central symmetry centered O ) each point BUT moves to a point BUT ’ , what O is the midpoint of the segment AA ’ (Fig. 1).

Fig.1 Fig.2

Let be O - common vertex of vertical corners AOB and BUT ’ OV ’ . But then it is clear that when turning through 180°, the sides of one of the two vertical angles will just pass to the sides of the other, i.e. these two corners are aligned. This means that the vertical angles are equal (Fig. 2).

Proving the equality of angles at the base of an isosceles triangle, Thales used axial symmetry : he combined the two halves of an isosceles triangle by bending the drawing along the bisector of the angle at the apex (Fig. 3). In the same way, Thales proved that the diameter bisects the circle.

Fig.3 Fig.4

Applied Thales and another movement - parallel transfer , at which all points of the figure are displaced in a certain direction by the same distance. With his help, he proved the theorem that now bears his name:

if equal segments are set aside on one side of the angle and parallel lines are drawn through the ends of these segments until they intersect with the second side of the angle, then equal segments will also be obtained on the other side of the angle(Fig. 4).

In ancient times, the idea of ​​movement was also used by the famous Euclid, the author of "Beginnings" - a book that has survived more than two millennia. Euclid was a contemporary of Ptolemy I, who ruled in Egypt, Syria and Macedonia from 305-283 BC.

Movements were implicitly present, for example, in Euclid's reasoning when proving the signs of equality of triangles: "Let's impose one triangle on another in such and such a way." According to Euclid, two figures are called equal if they can be "combined" by all their points, i.e. by moving one figure as a solid whole, one can accurately superimpose it on a second figure. For Euclid, movement was not yet a mathematical concept. The system of axioms first set forth by him in the "Principles" became the basis of a geometric theory called Euclidean geometry.

In modern times, the development of mathematical disciplines continues. Analytical geometry was created in the 11th century. Professor of Mathematics at the University of Bologna Bonaventure Cavalieri(1598-1647) publishes the essay "Geometry, stated in a new way with the help of indivisible continuous." According to Cavalieri, any flat figure can be considered as a set of parallel lines or "traces" that a line leaves when moving parallel to itself. Similarly, an idea is given about bodies: they are formed during the movement of planes.

The further development of the theory of motion is associated with the name of the French mathematician and historian of science Michel Chall(1793-1880). In 1837, he published the work "Historical review of the origin and development of geometric methods." In the process of his own geometric research, Schall proves the most important theorem:

every orientation-preserving motion of a plane is either

parallel translation or rotation,

any orientation-changing motion of a plane is either axial

symmetry or sliding symmetry.

The proof of Chall's theorem is fully carried out in item 8 of this abstract.

An important enrichment that geometry owes to the 19th century is the creation of the theory of geometric transformations, in particular, the mathematical theory of motions (displacements). By this time, there was a need to give a classification of all existing geometric systems. This problem was solved by a German mathematician Christian Felix Klein(1849-1925).

In 1872, assuming the post of professor at the University of Erlangen, Klein gave a lecture on "A Comparative Review of the Newest Geometric Researches". The idea put forward by him of rethinking all geometry on the basis of the theory of motions was called "Erlangen program".

According to Klein, to construct a particular geometry, you need to specify a set of elements and a group of transformations. The task of geometry is to study those relations between elements that remain invariant under all transformations of a given group. For example, Euclid's geometry studies those properties of figures that remain unchanged during movement. In other words, if one figure is obtained from another by movement (such figures are called congruent), then these figures have the same geometric properties.

In this sense, motions form the basis of geometry, and the five axioms of congruence are singled out by an independent group in the system of axioms of modern geometry. This complete and fairly rigorous system of axioms, summing up all previous studies, was proposed by the German mathematician David Gilbert(1862-1943). His system of twenty axioms, divided into five groups, was first published in 1899 in the book "Fundamentals of Geometry".

In 1909 a German mathematician Friedrich Schur(1856-1932), following the ideas of Thales and Klein, developed another system of axioms of geometry - based on the consideration of movements. In his system, in particular, instead of the Hilbert group of axioms of congruence, a group of three axioms of motion.

The types and some important properties of movements are discussed in detail in this essay, but they can be briefly expressed as follows: the motions form a group that defines and determines the Euclidean geometry.

Definition and properties of motions.

By shifting each point of this figure in some way, a new figure is obtained. It is said that this figure is received transformation from this one. The transformation of one figure into another is called a movement if it preserves the distances between points, i.e. translates any two points X and Y one shape per dot X ’ and Y ’ another figure so that XY = X ’Y ’.

Definition. Shape transformation that preserves distance

between points is called the movement of this figure.

! Comment: the concept of movement in geometry is connected with the usual idea of ​​displacement. But if, speaking of displacement, we imagine a continuous process, then in geometry only the initial and final (image) positions of the figure will matter to us. This geometric approach differs from the physical one.

When moving, different points correspond to different images, and each point X one figure is put in correspondence with the only dot X ’ another figure. This type of transformation is called one-to-one or bijective.

With regard to movements, instead of the term "equality" of figures (straight lines, segments, planes, etc.), the term is used "congruence" and the symbol is used  . The symbol є is used to denote belonging. With this in mind, we can give a more correct definition of movement:

Motion is a bijective transformation φ of the plane π, under which for any

various points X, Y є π the relation XY  φ (X ) φ (Y ).

The result of the successive execution of two movements is called composition. If the move is made first φ , followed by movement ψ , then the composition of these motions is denoted by ψ ◦ φ .

The simplest example of movement is the identity display (it is customary to denote - ε ), at which each point X , belonging to the plane, this point itself is compared, i.e. ε (X ) = X .

Let's consider some important properties of motions.

C property 1.

Lemma 2. 1. Compositionφ ◦ ψ two movementsψ , φ is a movement.

Proof.

Let the figure F translated by movement ψ into a figure F ', and the figure F ’ is translated by movement φ into a figure F ''. Let the point X figures F goes to the point X ’ shapes F ’ , and during the second movement, the point X ’ shapes F ' goes to point X '' figures F ''. Then the transformation of the figure F into a figure F '', at which an arbitrary point X figures F goes to the point X '' figures F '', preserves the distance between points, and therefore is also a movement.

Note that the recording of a composition always starts from the last movement, because the result of the composition is the final image - it is put in line with the original:

X ’’= ψ (X ’) = ψ (φ (X )) = ψ ◦ φ (X )

C property 2.

Lemma 2.2 . If aφ – movement, then transformationφ -1 is also a movement.

Proof.

Let the shape transformation F into a figure F ’ translates the various points of the figure F at various points on the figure F '. Let an arbitrary point X figures F under this transformation goes to a point X ’ shapes F ’.

Shape transformation F ' into a figure F , at which the point X ' goes to point X , is called transformation inverse to the given one. For every move φ it is possible to define the reverse movement, which is denoted φ -1 .

Arguing similarly to the proof of property 1, we can verify that a transformation inverse to a motion is also a motion.

It is obvious that the transformation φ -1 satisfies the equalities:

f ◦ f -1 = f -1 ◦ f = ε , where ε is the identical display.

Property 3 (associativity of compositions).

Lemma 2.3. Let φ 1 , φ 2 , φ 3 - voluntary movements. Then φ 1 ◦(φ 2 ◦ φ 3 ) = (φ 1 ◦φ 2 )◦φ 3 .

The fact that the composition of movements has the property of associativity allows us to determine the degree φ with a natural indicator n .

Let's put φ 1 = φ and φ n+1 = φ n ◦ φ , if n ≥ 1 . Thus the movement φ n obtained by n -multiple sequential application of movement φ .

C property 4 (maintaining straightness).

Theorem 2. 1. Points lying on the same straight line, when moving, pass into points,

Motion bodies under the influence of gravity

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Movements preserve distances and therefore preserve all the geometrical properties of figures, since they are determined by distances. At this point, we will get the most general properties movements, citing evidence in cases where it is not obvious.

Property 1. Three points lying on the same straight line, when moving, go into three points lying on the same straight line, and three points not lying on the same straight line, into three points not lying on the same straight line.

Let the motion translate points into points, respectively. Then the equalities hold

If points A, B, C lie on the same straight line, then one of them, for example, point B lies between the other two. In this case and from equalities (1) it follows that . And this equality means that point B lies between points A and C. The first assertion is proved. The second follows from the first and the reversibility of the movement (by contradiction).

Property 2. A segment is transformed into a segment by motion.

Let the ends of the segment AB be associated by the motion f with points A and B. Take any point X of the segment AB. Then, as in the proof of property 1, it can be established that its image - a point lies on the segment AB between points A and B. Further, each point

Y of the segment A B is the image of some point Y of the segment AB. Namely, the point Y, which is removed from the point A at a distance A Y. Therefore, the segment AB is transferred by movement to the segment AB.

Property 3. When moving, a ray becomes a ray, a straight line - into a straight line.

Prove these statements yourself. Property 4. A triangle is translated into a triangle by movement, a half-plane into a half-plane, a plane into a plane, parallel planes- in parallel planes.

Triangle ABC is filled with segments connecting vertex A with points X opposite side BC (Fig. 26.1). The movement will assign to segment BC some segment BC and to point A - point A, not lying on the line BC. To each segment AX, this movement will assign a segment AX, where the point X lies on BC. All these segments AX will fill the triangle ABC.

The triangle goes into it

A half-plane can be represented as a union of infinitely expanding triangles, in which one side lies on the boundary of the half-plane

(Fig. 26.2). Therefore, the half-plane will go over to the half-plane when moving.

Similarly, a plane can be represented as a union of infinitely expanding triangles (Fig. 26.3). Therefore, when moving, a plane is mapped onto a plane.

Since movement preserves distances, the distances between figures do not change when moving. From this it follows, in particular, that during motions parallel planes pass into parallel ones.

Property 5. When moving, the image of a tetrahedron is a tetrahedron, the image of a half-space is a half-space, the image of a space is the whole space.

Tetrahedron ABCD is the union of line segments connecting point D with all possible points X triangle ABC(Fig. 26.4). When moving, the segments are mapped to segments, and therefore the tetrahedron will turn into a tetrahedron.

A half-space can be represented as a union of expanding tetrahedra whose bases lie in the boundary plane of the half-space. Therefore, when moving, the image of a half-space will be a half-space.

Space can be thought of as a union of infinitely expanding tetrahedra. Therefore, when moving, space is mapped onto all space.

Property 6. When moving, the angles are preserved, i.e., every angle is mapped onto an angle of the same type and the same magnitude. The same is true for dihedral angles.

When moving, a half-plane is mapped onto a half-plane. As convex angle is the intersection of two half-planes, and a non-convex angle and a dihedral angle are the union of half-planes, then when moving, the convex angle passes into a convex angle, and the non-convex

angle and dihedral angle, respectively - into a non-convex and dihedral angle.

Let the rays a and b, emanating from the point O, be mapped onto the rays a and b, emanating from the point O. Take the triangle OAB with vertices A on the ray a and B on the ray b (Fig. 26.5). It will appear on equal triangle BAB with vertices A on ray a and B on ray b. Hence, the angles between the rays a, b and a, b are equal. Therefore, when moving, the magnitudes of the angles are preserved.

Consequently, the perpendicularity of the straight lines, and hence the line and the plane, is preserved. Remembering the definitions of the angle between a straight line and a plane and the quantities dihedral angle, we find that the values ​​of these angles are preserved.

Property 7. Movements preserve surface areas and volumes of bodies.

Indeed, since the motion preserves perpendicularity, the motion of the height (triangles, tetrahedra, prisms, etc.) translates into heights (the images of these triangles, tetrahedra, prisms, etc.). In this case, the lengths of these heights will be preserved. Therefore, the areas of triangles and the volumes of tetrahedra are preserved during movements. This means that both the areas of polygons and the volumes of polyhedra will be preserved. The areas of curved surfaces and the volumes of bodies bounded by such surfaces are obtained limit transitions on the areas of polyhedral surfaces and volumes of polyhedral bodies. Therefore, they are preserved during movements.