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. From the definition of motion it follows that the lengths of the segments are equal 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 an 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 translation. 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. Textbook 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 of pedagogical ideas "Open Lesson" ().

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

Property 1. Let f be the motion of points in the plane, A", B" and C" be the images of points A, B and C during the motion of f. Then the points A", B" and C" lie on one straight line if and only if when points A, B and C are collinear.

Property 4. When moving, it transforms into a segment equal to it. Property 5. When moving, a ray transforms into a ray.

Property 7. Let a circle of radius r centered at a point O be given. Then, when moving, it transforms into a circle of the same radius, centered at a point coinciding with the image of the center O.

By an affine plane frame we mean an ordered triple of noncollinear points. Property 7. When moving, the frame is transformed into a frame, and the orthonormal frame into an orthonormal frame.

Theorem (Basic theorem of motions). Let orthonormal frames u be given on the plane. Then there is a unique move g that takes the frame R to R": .

Consequence. If f is a motion of the plane: translating an orthonormal frame R into an orthonormal frame R", then each point M of the plane with x and y coordinates relative to R corresponds to a point M"= f(M) with the same x and y coordinates relative to R".

"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 ’ figures F ’ , and during the second movement, the point X ’ figures 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 ’ figures 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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Introduction.

Geometric transformations are a rather late branch of mathematics. The first geometric transformations began to be considered in the 17th century, while projective transformations appeared only at the beginning of the 19th century.

In algebra, various functions are considered. The function f assigns to each number x from the domain of the function a certain number f(x) - the value of the function f at the point x. In geometry, functions are considered that have other domains of definition and sets of values. They assign a point to each dot. These functions are called geometric transformations.

Geometric transformations are of great importance in geometry. With the help of geometric transformations, such important geometric concepts as equality and similarity of figures are defined. Thanks to geometric transformations, many disparate facts of geometry fit into a coherent theory.

In the abstract, mainly, we will talk about the transformations of space. All movements, similarities, circular and affine transformations of space, as well as affine and projective transformations of the plane will be considered. For each transformation, its properties and examples of application to the solution of geometric problems will be considered.

First, let's look at some basic concepts that we will need to work with transformations. Let's dwell on two terms: distance and transformation. So what do we mean by these words:

Definition. Distance between two points we will call the length of the segment with ends at these points.

Definition. Transformation set is called a one-to-one mapping of this set onto itself.

Now let's move on to the consideration of certain types of geometric transformations.

Part I. Movements of space.

General properties of movements.

Definition. The space transformation is called movement, if it preserves distances between points.

Movement properties.

1. The transformation inverse to motion is motion.
2. The composition of movements is movement.
3. When moving, a straight line turns into a straight line, a ray into a ray, a segment into a segment, a plane into a plane, a half-plane into a half-plane.
4. The image of a plane angle in motion is a plane angle of the same magnitude.
5. Movement preserves the angle between straight lines, between a straight line and a plane, between planes.
6. Movement preserves the parallelism of straight lines, a straight line and a plane, planes.

Property proofs.

1 and 2. Follow from the definition of motion.

1. Let the points A, X and B lie on the same straight line, and the point X lies between A and B. Then AX + XB = AB. Let the points А´, Х´, В´ be the images of the points А, Х, В during the motion. Then А´Х´+Х´В´=А´В´ (from the definition of motion). And from this it follows that the points A´, X´, B´ lie on one straight line, and X´ lies between A´ and B´.
   From the proven statement it immediately follows that when moving, a straight line turns into a straight line, a ray into a ray, a segment into a segment.

For the plane, the proof can be carried out as follows. Let a, b be two intersecting lines of our plane α, a´, b´ their images. Obviously, a´ and b´ intersect. Let α´ be the plane containing the lines a´, b´. Let us prove that α´ is the image of the plane α. Let М be an arbitrary point of the plane α not lying on the lines a and b. Let us draw a line c through M that intersects lines a and b at different points. The image of this line is the line c´ intersecting the lines a´, b´ at different points. This means that M´, the image of the point M, also lies in the plane α´. Thus, the image of any point of the plane α lies in the plane α´. It is proved similarly that the pre-image of any point of the plane α´ lies in the plane α. Hence α´ is the image of the plane α.

Now it is not difficult to prove the assertion for the half-plane as well. It is only necessary to complete the half-plane to a plane, consider the line a that bounds the half-plane, and its image a´, and then prove by contradiction that the images of any two points of the half-plane lie on the same side of a´.

1. Follows from property 3.
2. It follows from property 4 and the definition of the angle between lines (a line and a plane, two planes) in space.
3. Assume the opposite, i.e. let the images of our parallel lines (a line and a plane, planes) intersect (in the case of parallel lines, it is still necessary to show that their images cannot be skew lines, but this immediately follows from the fact that the plane containing these lines will pass into a plane). Then consider their common point. It will have two prototypes, which is impossible by the definition of transformation.

Definition. Figure F is called equal figure Ф´, if there is a movement that transforms Ф into Ф´.

Types of movements.

3.1. Identity transformation.

Definition. Identity transformation E space is called a transformation in which each point of the space goes into itself.

Obviously, the identical transformation is a movement.

3.2. Parallel transfer.

Definition. Let a vector be given in space. Parallel transfer space onto a vector is called a transformation in which each point M is mapped to a point M´ such that .

Theorem 3.2. Parallel transfer - movement.

Proof. Let А´, В´ be the images of points А, В under parallel transfer to the vector . It suffices to show that AB=A´B´, which follows from the equality:

Transfer property. Parallel translation translates a line (plane) into itself or into a line parallel to it (plane).

Proof. In proving Theorem 3.2, we proved that vectors are preserved under parallel translation. This means that the direction vectors of the lines and the normal vectors of the planes are preserved. This is where our assertion follows.

central symmetry.

Definition. Symmetry with respect to point O ( central symmetry) of space is a space transformation that maps a point O onto itself, and maps any other point M onto a point M´ such that the point O is the midpoint of the segment MM´. Point O is called center of symmetry.

Theorem 3.4. Central symmetry - movement.

Proof.

Let A, B be two arbitrary points, A´, B´ their images, О the center of symmetry. Then .

property of central symmetry. Central symmetry takes a line (plane) into itself or into a line parallel to it (plane).

Proof. When proving Theorem 3.4, we proved that the vectors are reversed under parallel translation. This means that the directing vectors of the lines and the normal vectors of the planes with central symmetry only change directions. This is where our assertion follows.

The theorem on motion assignment.

Theorem 5.1. (theorem about motion specification) Given two tetrahedra ABCD and A´B´C´D´ with respectively equal edges, then there is one and only one motion of space that maps points A, B, C, D respectively to points A´, B´, C´, D ´.

Proof.

I. Existence. If A coincides with A´, B coincides with B´, C coincides with C´, D coincides with D´, then simply the identity transformation is given. If not, then we assume for definiteness that A does not coincide with A´. Consider the plane α of symmetry of the points A and A´. Let the symmetry S α take the tetrahedron ABCD into the tetrahedron A´B 1 C 1 D 1 .

Now, if В 1 coincided with В´, С 1 - with С´, D 1 - with D´, then the proof is complete. If not, then we can assume without loss of generality that the points В´ and В 1 did not coincide. Consider the plane β of symmetry of the points B 1 and B´. Point A´ is equidistant from points B1 and B´, therefore it lies on the plane β. Let the symmetry S β take the tetrahedron A´B 1 C 1 D 1 into the tetrahedron A´B´C 2 D 2 .

Now, if С 2 coincides with С´, and D 2 coincides with D´, then the proof is complete. If not, then we can assume without loss of generality that the points С´ and С 2 did not coincide. Consider the plane γ of symmetry of the points С 2 and С´. Points А´, В´ are equidistant from points С 2 and С´, therefore they lie in the plane γ. Let the symmetry S γ take the tetrahedron A´B´C 2 D 2 into the tetrahedron A´B´C´D 3 .

Now, if D 3 coincides with D´, then the proof is complete. If not, then consider the plane δ of symmetry of the points D 3 and D´. Points А´, В´, С´ are equidistant from points D 3 and D´, therefore they lie in the plane δ. Hence, the symmetry S δ takes the tetrahedron A´B´C´D 3 into the tetrahedron A´B´C´D´.

Thus, the composition of the required number of mirror symmetries reduced transforms the tetrahedron ABCD into the tetrahedron A´B´C´D´. And this transformation is a movement (property 2 of movements).

II. Uniqueness. Let there be 2 movements f and g that take A to A´, B to B´, C to C´, D to D´. Then the motion is an identical transformation, since leaves points A, B, C, D fixed. So f=g.

In the proof of Theorem 5.1 (existence), in fact, the

Theorem 5.2. Any movement of space is a composition of no more than four mirror symmetries.

Homothety of space.

Let us first consider an important particular case of similarity, homothety.

Definition. Homothety with center O and coefficient is a transformation of space, in which the image of each point X is a point X´ such that .

Properties of homothety.

Property proofs.

1 and 2. Follow from the definition of homothety.

3. It is proved similarly to the corresponding theorem on the plane. Indeed, if we consider an arbitrary point X of the space, it will be enough for us to prove our theorem for the plane (AXB).

4. Proved by contradiction.

1. Follows from property 1.

similarity properties.

Theorem 2.1. The similarity of space can be represented by the composition of homothety and movement f:

Proof. Let's make a homothety centered at an arbitrary point. Consider a transformation f such that (the existence of such a transformation follows from the definition of a transformation). The transformation f will be motion by the definition of motion.

Note that by choosing for f the movement , we can get a representation of our similarity in this form as well.

similarity properties.

Property proofs.

1 and 2. Corollaries from Theorem 2.1.

3. Follows from the definition of similarity.

4. For the cube, the theorem is obviously true. For a body consisting of cubes, of course, too.

An arbitrary polyhedron M can be imposed on a cubic lattice. We will grind this lattice. As the side of one cube of our lattice tends to zero, the volumes of two bodies: the body I, consisting of cubes lying completely inside M, and the body S, consisting of cubes that have common points with M, tend to the volume of the polyhedron M (this follows from the fact that that for each face of our polyhedron M, the volume of cubes crossing this face will tend to zero). At the same time, for the image M´ of the polyhedron M with our similarity, the volumes of the bodies I´, S´ (images of the bodies I, S) tend to the volume of the polyhedron M´. For the bodies I and S, our theorem is true, which means that it is also true for the polyhedron M.

The volume of an arbitrary body is determined in terms of the volumes of the corresponding polyhedra, so the theorem is also true for an arbitrary body.

Theorem 2.2. (on setting the similarity of space) If two tetrahedra ABCD and A´B´C´D´ are given such that , then there is exactly one similarity of space for which A→A´, B→B´, С→С´, D→D´.

Proof. That such a similarity exists follows from Theorem 2.1 and the theorem on specifying the motion of space (Part I, Theorem 5.1). Let there be two such transformations: P and Р´. Then the transformation is a motion having fixed points A, B, C, D, i.e. f is the identity transformation. Hence P=P´.

Task 1.

Points M, N, P are located on sides AB, BC, AC of triangle ABC. The points M´, N´, P´ are symmetrical to the points M, N, P with respect to the sides AB, BC, AC. Prove that the areas of triangles MNP and M´N´P´ are equal.

Decision.

For a regular triangle, the assertion is obvious.

In the same way, any trapezoid can be converted into an isosceles one by an affine transformation, i.e. it suffices to prove any affine assertion for an isosceles trapezoid.

Task 2.

In a trapezoid ABCD with bases AD and BC, a line is drawn through point B, parallel to side CD and intersecting diagonal AC at point P, and through point C, a line parallel to side AB and intersecting diagonal BD at point Q. Prove that line PQ is parallel to the bases trapezoid.

Decision.

For an isosceles trapezoid, the assertion is obvious.

Compression to a straight line.

Definition. Compression to a straight lineℓ with coefficient k () is a transformation that takes an arbitrary point M to a point M´ such that and , where .

Theorem 2.1. Contraction to a straight line is an affine transformation.

Proof. By a direct check, we make sure that the straight line goes into a straight line. You can even notice that shrinking to a straight line is a special case of parallel projection (when the projection direction is perpendicular to the line of intersection of the planes).

Theorem 2.2. For any affine transformation, there is a square lattice, which, under this transformation, transforms into a rectangular lattice.

Proof. Let's take an arbitrary square lattice and consider one of its squares OABS. With our transformation, it will turn into a parallelogram О´А´В´С´. If O´A´B´C´ is a rectangle, then our proof is complete. Otherwise, we assume for definiteness that the angle А´О´В´ is acute. We will rotate the square OABS and our entire lattice around the point O. When the square OABS turns on (so point A has moved to point B), point A´ will go to point B´, and B´ to the vertex of the parallelogram adjacent to O´A´ W´S´. Those. angle A´O´B´ becomes obtuse. According to the principle of continuity, at some point he was straight. At this moment, the square OABS turned into a rectangle, and our lattice into a rectangular lattice, etc.

Theorem 2.3. An affine transformation can be represented by a composition of contraction to a straight line and similarity.

Proof. Follows from Theorem 2.2.

Theorem 2.4. An affine transformation that transforms a certain circle into a circle is a similarity.

Proof. We describe a square near our circle and rotate it so that it turns into a rectangle during our transformation (Theorem 2.2.). Our circle will go into a circle inscribed in this rectangle, so this rectangle is a square. Now we can specify the square grid that our transformation will transform into a square grid. Obviously, our transformation is a similarity.

3. Affine transformations of space.

Definition. affine a space transformation is a space transformation that transforms each plane into a plane.

Properties.

1. Under an affine transformation, straight lines become straight lines.
2. An affine transformation of space induces an affine mapping of each plane onto its image.
3. Under an affine transformation, parallel planes (straight lines) pass into parallel planes (straight lines).

Property proofs.

1. It follows from the fact that a straight line is the intersection of two planes, and from the definition of an affine transformation.
2. It follows from the definition of an affine transformation and property 1.
3. For planes it is proved by contradiction, for straight lines - through property 2 and the property of the affine transformation of the plane.

Theorem 3.1. (on specifying an affine space transformation) For any given tetrahedra ABCD and A´B´C´D´ there is a unique affine transformation that takes A to A´, B to B´, C to C´, D to D´.

Proof. The proof is similar to Theorem 1.1. (lattices of parallelepipeds are constructed).

It follows from the proof of Theorem 3.1 that if we have some oblique coordinate system W, and W´ is its image under an affine transformation, then the coordinates of an arbitrary point in space in the W coordinate system are equal to the coordinates of its image in the W´ coordinate system.

From this immediately follows some more properties affine transformation.

1. An affine transformation is affine.
2. Affine transformations preserve the ratios of the lengths of parallel segments.

Now let the coordinate system (O, , , ) be given in space and the affine transformation f takes O to O´ , and the basis vectors to vectors , , respectively. Let us find the coordinates x´, y´, z´ of the image M´(x´,y´,z´) of the point M(x,y,z) under the transformation f.

We will proceed from the fact that the point M in the coordinate system (О, , , ) has the same coordinates as the point М´ in the coordinate system (О´, , , ). From here

Therefore, we have equalities (*):

It is also worth noting that , because the vectors , , are linearly independent.

This determinant is called affine transformation determinant.

Theorem 3.2. The transformation given by equalities (*) at is affine.

Proof. It suffices to check that the transformation inverse to the transformation(*) is affine (property 4). Take an arbitrary plane Аx´+Вy´+Сz´+D=0, where А, В, С are not equal to zero at the same time. Performing substitutions (*), we obtain the equation of its preimage:

It only remains to check that the coefficients at x, y, z in the resulting equation are not simultaneously equal to zero. This is true, because otherwise the system

with a non-zero determinant would have only a zero solution: A=B=C=0, which is not true.

Theorem 3.3. For the volumes V and V´ of the bodies corresponding to the affine transformation, there is a dependence .

Proof. Let non-coplanar vectors , , form a vector basis of the space, and let the vectors , and . Computing the mixed product of these vectors, we get:

.

Let's take advantage of the fact that the volume of an oriented parallelepiped built on vectors as on edges is equal to the mixed product of these vectors:

,

where V 0 is the volume of the parallelepiped built on basis vectors.

An affine transformation does not change the coordinates of the corresponding vectors in the corresponding bases. Therefore, for the volume V´ of the image of the parallelepiped of volume V, we have:

,

where is the volume of a parallelepiped built on vectors as on edges.

From here we get: . Further , so for unoriented volumes we have . This equality can be extended to all bodies in a similar way to the proof of property 4 of similarities (Part II, §2).

Task.

The vertex of the parallelepiped is connected to the centers of three faces that do not contain it. Find the ratio of the volume of the resulting tetrahedron to the volume of the given parallelepiped.

Decision.

Let's calculate this ratio for a cube and, having converted the cube into a parallelepiped by an affine transformation, we will use the fact that the affine transformation preserves the ratio of volumes. For a cube, the ratio is easy to calculate. It is equal to 1:12.

Answer: 1:12.

The relationship of space.

Definition. An affine transformation of space having a plane of fixed points is called related transformation ρ (kinship), and the plane of its fixed points is called kinship plane. Elements that are related are called related.

Definition. The direction of lines connecting related points is called direction of kinship.

kinship properties.

1. Related lines (planes) intersect on the plane of kinship or are parallel to it.
2. (Correctness of determining the direction of kinship) Lines, each of which connects two related points, are parallel.
3. If the direction of relationship is not parallel to the plane of this relationship, then each segment connecting two related points is divided by the plane of relationship in the same ratio.
4. Any plane parallel to the direction of kinship is motionless in this kinship. In it, the relationship of the plane is induced (an affine transformation that has a line of fixed points, called the axis of relationship), the axis of which is the line of its intersection with the plane of the given space relationship.

Property proofs.

1. The proof is similar to the proof of the mirror symmetry property (Part I, §3.5).

2. Let A, B be two distinct points; A´, B´ are their images in relation, α is the plane of relationship. Let be . Then (a property of an affine transformation), i.e. AA´||BB´, etc.

3 and 4. Follow from the proof of property 2.

Definition. The surface represented by the equation , is called ellipsoid. A special case of an ellipsoid is a sphere.

The following fact takes place, which we will not prove, however, in the proof of the following theorems, we will need it:

Theorem 4.1. An affine transformation transforms an ellipsoid into an ellipsoid.

Theorem 4.2. An arbitrary affine transformation of space can be represented by a composition of similarity and relationship.

Proof. Let an affine transformation f map the sphere σ onto the ellipsoid σ´. It follows from Theorem 3.1 that f can be given by these figures. Consider a plane α´ containing the center of the ellipsoid and intersecting it along some circle ω´ (the existence of such a plane can be easily proved from continuity considerations). Let α be the pre-image of α´, be the pre-image of ω´, and β be the sphere having the circle ω´ as its diametral circle. There is a relationship ρ mapping β to σ´ and there is a similarity P mapping σ to β. Then is the required representation.

Theorem 4.3 immediately follows from the proof of the previous theorem:

Theorem 4.3. An affine transformation that preserves the sphere is a similarity.

Part IV. Projective transformations.

1. Projective transformations of the plane.

Definition. Projective plane– an ordinary (Euclidean) plane, completed by points at infinity and a straight line at infinity, also called improper elements. In this case, each straight line is complemented by one improper point, the entire plane - by one improper straight line; parallel lines are complemented by a common improper point, non-parallel - by different ones; improper points complementing all possible lines of the plane belong to the improper line.

Definition. A projective plane transformation that takes any line to a line is called projective.

Consequence. A projective transformation that preserves the line at infinity is affine; any affine transformation is projective, preserving the line at infinity.

Definition. central design the plane α onto the plane β centered at a point O not lying on these planes is called a mapping that associates any point A of the plane α with the point A´ of the intersection of the line OA with the plane β.

Moreover, if the planes α and β are not parallel, then in the plane α there is a line ℓ such that the plane passing through the point O and the line ℓ is parallel to the plane β. We will assume that ℓ during our projection goes to the line at infinity of the plane β (in this case, each point B of the line ℓ goes to that point of the line at infinity, which complements the straight lines parallel to OB). In the plane β there is a line ℓ´ such that the plane passing through the point O and the line ℓ´ is parallel to the plane α. We will consider ℓ´ the image of the straight line α at infinity. The lines ℓ and ℓ´ will be called dedicated.

We can say that a simple transformation of the projective plane is given (if we combine the planes α and β).

It immediately follows from the definition central projection properties:

1. Central design is a projective transformation.
2. The transformation inverse to the central design is the central design with the same center.
3. Lines parallel to the selected ones become parallel.

Definition. Let points A, B, C, D lie on the same line. Double attitude(AB; CD) of these points is called the value. If one of the points is at infinity, then the lengths of the segments, the end of which is this point, can be shortened.

Theorem 1.1. The central projection preserves the dual relationship.

Proof. Let О be the projection center, А, В, С, D – four points lying on one straight line, A´, B´, C´, D´ – their images.

Similarly .

Dividing one equation by the other, we get .

Similarly, instead of point C, considering point D, we get .

From here , i.e. .

To make the proof complete, it remains to note that all segments, areas, and angles can be considered oriented.

Theorem 1.2. Let four points A, B, C, D of the plane π not lie on one line and four points M, N, P, Q of the plane π´ not lie on one line be given. Then there is a composition of central (parallel) projection and similarity that maps A to M, B to N, C to P, D to Q.

Proof.

For convenience, we will say that ABCD and MNPQ are quadrilaterals, although in fact this is not necessary (for example, segments AB and CD can intersect). It will be seen from the proof that we nowhere use that the points A, B, C, D and M, N, P, Q form quadrilaterals in this order.

.

Let us now draw lines AK, BL, CF, DG through points A, B, C, D parallel to X 1 X 2 (K, L lie on DC; G, F lie on AB), and through points N, M - lines NT , MS parallel to Y 1 Y 2 (T, S lie on PQ). Using the central (parallel) projection f, we transform the trapezoid ABLK into the trapezoid A´B´L´K´ of the plane π´, which is similar to the trapezoid MNTS (this is possible according to part I of our proof). Moreover, from the choice of points X 1 , X 2 it follows that the line X 1 X 2 is a distinguished line of the plane π´. Let us mark points С´, D´ on the line L´K´ such that the trapezoid ABCD is similar to the trapezoid A´B´C´D´. Draw lines C´F´, D´G´ parallel to line B´L´ (F´, G´ lie on А´В´) and mark a point Y 1 ´ on line A´B´ such that , . On the line C´D´ mark a point Y 2 ´ such that Y 1 ´Y 2 ´||A´K´ (see the figure). From the choice of points Y 1 ´ and Y 2 ´ it follows that the line Y 1 ´Y 2 ´ is a distinguished line of the plane π´. Under the transformation f, the point E goes to the point E´ of the intersection of the lines A´B´ and L´K´. The point С goes to some point С 0 ´ of the straight line С´D´.

Let us prove that С 0 coincides with С´. From the fact that X 2 under the transformation f goes to the point at infinity of the line C´D´, and Y 2 ´ is the image of the point at infinity of the line CD and the central projection preserves double relations, it follows that , where . Now consider the transformation g, the composition of the central projection and similarity, which takes the trapezoid CDGF to the trapezoid C´D´G´F´. For the transformation g, one can similarly show that . From here it will follow that the points С 0 and С´ coincide. Similarly, one can show that D 0 - the image of the point D under the transformation f - coincides with D´. Thus, the transformation f transforms the quadrilateral ABCD into the quadrilateral A´B´C´D´ similar to the quadrilateral MNPQ, as required.

Theorem 1.3. Let four points be given, no three of which lie on the same straight line: A, B, C, D and A´, B´, C´, D´. Then there is a unique projective transformation taking A to A´, B to B´, C to C´, D to D´.

Existence such a transformation follows from Theorem 1.1.

uniqueness can be proved in the same way as the uniqueness of an affine transformation (Theorem 1.1, Part III): consider a square lattice, build its image, and then refine it. Get around the difficulties that we faced

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 do not 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, according to the property of the internal forces of the system , 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 equal to the geometric sum of 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 of the 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 a 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 influence of internal forces), its fragments 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;

According to the given law of motion of the bodies included in the mechanical system, find the reactions of external constraints;

Based on 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 of a mechanical system are often used.

Corollary 1. If the main vector of external forces applied to a mechanical system is equal to 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 kinetic moment of a mechanical system is the sum of the kinetic moments of the points and bodies included in this system. Consider methods for determining the angular momentum of a material point and a rigid body in various cases of their motion.

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 as 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 - the main moment of 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.