Chapter 1. Supplement. Transformation of Cartesian rectangular coordinates in the plane and in space. Special coordinate systems on the plane and in space.
The rules for constructing coordinate systems on a plane and in space are discussed in the main part of Chapter 1. The convenience of using rectangular coordinate systems was noted. In the practical use of analytical geometry tools, it is often necessary to transform the adopted coordinate system. This is usually dictated by considerations of convenience: geometric images are simplified, analytical models and algebraic expressions used in calculations become clearer.
The construction and use of special coordinate systems: polar, cylindrical and spherical is dictated by the geometric meaning of the problem being solved. Modeling with special coordinate systems often facilitates the development and use of analytical models for solving practical problems.
The results obtained in the Appendix of Chapter 1 will be used in linear algebra, most of them in calculus and physics.
Transformation of Cartesian rectangular coordinates in the plane and in space.
When considering the problem of constructing a coordinate system on a plane and in space, it was noted that the coordinate system is formed by numerical axes intersecting at one point: two axes are required on the plane, and three in space. In connection with the construction of analytical models of vectors, the introduction of the operation of the scalar product of vectors and the solution of problems of geometric content, it was shown that the use of rectangular coordinate systems is most preferable.
If we consider the problem of transforming a specific coordinate system in the abstract, then in the general case it would be possible to admit arbitrary movement in a given space of the coordinate axes with the right to arbitrarily rename the axes.
We will proceed from the primary concept frame of reference accepted in physics. Observing the movement of bodies, it was found that the movement of an isolated body cannot be determined by itself. It is necessary to have at least one more body, relative to which the movement is observed, that is, its change relative provisions. To obtain analytical models, laws, motion with this second body, as with a frame of reference, a coordinate system was connected, and so that the coordinate system was solid !
Since the arbitrary movement of a rigid body from one point in space to another can be represented by two independent movements: translational and rotational, the options for transforming the coordinate system were limited to two movements:
1). Parallel transfer: we follow only one point - point.
2). Rotation of the axes of the coordinate system relative to a point: as a rigid body.
Convert cartesian rectangular coordinates on a plane.
Let we have on the plane of the coordinate system:, and. The coordinate system is obtained by translating the system in parallel. The coordinate system is obtained by rotating the system by an angle, and counterclockwise rotation of the axis is taken as the positive direction of rotation.
Let us define the basis vectors for the adopted coordinate systems. Since the system is obtained by parallel translation of the system, then for both of these systems we will take the basis vectors:, and unit and coinciding in direction with the coordinate axes, respectively. For the system, as the basis vectors, we will take unit vectors that coincide in direction with the axes,.
Let a coordinate system be given and point \u003d is defined in it. We will assume that before the transformation we have coinciding coordinate systems and. Apply a parallel translation defined by a vector to the coordinate system. It is required to define the transformation of the coordinates of the point. Let's use the vector equality: \u003d +, or:
Let us illustrate the parallel translation transformation by an example known in elementary algebra.
Example D–1 : The equation of the parabola is given: \u003d \u003d. Reduce the equation of this parabola to its simplest form.
Decision:
1). Let's use the trick selection of a full square : \u003d, which can be easily represented as: –3 \u003d.
2). Let's apply a coordinate transformation - parallel transfer : \u003d. After that, the parabola equation takes the form:. This transformation in algebra is defined as follows: parabola \u003d is obtained by shifting the simplest parabola to the right by 2, and up by 3 units.
Answer: the simplest form of a parabola:.
Let a coordinate system be given and point \u003d is defined in it. We will assume that before the transformation we have coinciding coordinate systems and. We apply a rotation transformation to the coordinate system so that relative to its original position, that is, relative to the system, it turns out to be rotated by an angle. It is required to define the transformation of the coordinates of the point \u003d. Let's write the vector in coordinate systems and: \u003d. (2) \u003d 1. From the theory of second-order lines it follows that the simplest (canonical!) Equation of an ellipse has been obtained.
Answer: the simplest form of a given line: \u003d 1 - the canonical equation of an ellipse.
Let two arbitrary Cartesian rectangular coordinate systems be given on the plane. The first is determined by the beginning O and the basis vectors i j , the second - by the center ABOUT' and basis vectors i ’ j ’ .
Let us set the goal to express the x y coordinates of some point M relative to the first coordinate system in terms of x’ and y’ - coordinates of the same point relative to the second system.
notice, that
Let's denote the coordinates of the point O 'relative to the first system through a and b:
Let us expand vectors i ’ and j ’ on the basis i j :
(*)
In addition, we have: 
... We introduce here the expansions of vectors in the basis i
’
j
’
:
from here 
We can conclude that no matter what two arbitrary Cartesian systems on the plane are, the coordinates of any point in the plane relative to the first system are linear functions of the coordinates of the same point relative to the second system.
We scalarly multiply the equations (*) by i then on j :
Let denote the angle between the vectors i and i ’ ... Coordinate system i j can be combined with the system i ’ j ’ by parallel transfer and subsequent rotation through an angle . But here an arc option is also possible: the angle between the basis vectors i i ’ also , and the angle between the basis vectors j ’ j ’ is equal to - . These systems cannot be combined with parallel translation and rotation. It is also necessary to change the direction of the axis. atto the opposite.
From the formula (**) we obtain in the first case:
In the second case
Conversion formulas are:
We will not consider the second case. Let us agree to consider both systems to be right.
Those. conclusion: whatever the two right-handed coordinate systems, the first of them can be aligned with the second by parallel translation and subsequent rotation around the origin by some angle .
Parallel transfer formulas: 
Axis rotation formulas: 
Inverse transformations:
Transforming Cartesian rectangular coordinates in space.
In space, arguing in a similar way, you can write:
(***)
And for coordinates, get:
(****)
So, whatever two arbitrary coordinate systems in space, the x y z coordinates of some point relative to the first system are linear functions of the coordinates x’ y’ z’ the same point relative to the second coordinate system.
Multiplying each of the equalities (***) scalarly by i ’ j ’ k ’ we get:
AT 
let us clarify the geometric meaning of the transformation formulas (****). To do this, assume that both systems have a common origin: a
=
b
=
c
= 0
.
Let us introduce into consideration three angles that fully characterize the position of the axes of the second system relative to the first.
The first angle is formed by the x-axis and the u-axis, which is the intersection of the xOy and x'Oy 'planes. Angle direction is the shortest rotation from the x-axis to the y-axis. Let's denote the angle by . The second angle is not exceeding the angle between the axes Oz and Oz ’. Finally, the third angle is the angle between the u-axis and Ox ’, measured from the u-axis in the direction of the shortest rotation from Ox’ to Oy ’. These angles are called Euler angles.
The transformation of the first system into the second can be represented in the form of sequential three rotations: by an angle relative to the Oz axis; at an angle relative to the Ox ’axis; and at an angle relative to the Oz 'axis.
The numbers ij can be expressed in terms of the Euler angles. We will not write these formulas because of their cumbersomeness.
The transformation itself is a superposition of a parallel translation and three consecutive rotations through Euler angles.
All these considerations can be carried out for the case when both systems are left-handed, or with different orientations.
If we have two arbitrary systems, then, generally speaking, they can be combined by means of parallel transfer and one rotation in space around a certain axis. We will not search for her.
1) Transition from one Cartesian rectangular coordinate system on the plane to another Cartesian rectangular coordinate system with the same orientation and with the same origin.
Suppose that two Cartesian rectangular coordinate systems are introduced on the plane hoy and with a common origin ABOUThaving the same orientation (fig. 145). We denote the unit vectors of the axes Oh and OU respectively through and, and the unit vectors of the axes and through and. Finally, let is the angle from the axis Oh to the axis. Let be x and at - coordinates of an arbitrary point M in system hoy, and and are the coordinates of the same point M in system .
Since the angle from the axis Oh before the vector is, then the coordinates of the vector
Angle from axis Oh before the vector is; so the coordinates of the vector are equal.
Formulas (3) § 97 take the form
Transition matrix from one Cartesian hoy rectangular coordinate system to another rectangular coordinate system with the same orientation has the form
A matrix is \u200b\u200bcalled orthogonal if the sum of the squares of the elements located in each column is equal to 1, and the sum of the products of the corresponding elements of different columns is equal to zero, i.e. if
Thus, the matrix (2) of the transition from one rectangular coordinate system to another rectangular system with the same orientation is orthogonal. Note also that the determinant of this matrix is \u200b\u200b+1:
Conversely, if an orthogonal matrix (3) with determinant equal to +1 is given, and a Cartesian rectangular coordinate system is introduced on the plane hoy, then by virtue of relations (4) the vectors and are unit and mutually perpendicular, therefore, the coordinates of the vector in the system hoy are equal and, where is the angle from the vector to the vector, and since the vector is unit and we obtain from the vector by turning by, then either, or.
The second possibility is excluded, since if we had, then we were given that.
Hence, and the matrix AND has the form
those. is the transition matrix from one rectangular coordinate system hoy to another rectangular system having the same orientation, and the angle.
2. Transition from one Cartesian rectangular coordinate system on the plane to another Cartesian rectangular coordinate system with opposite orientation and with the same origin.
Let two rectangular Cartesian coordinate systems be introduced on the plane hoy and with a common origin ABOUT, but having opposite orientation we denote the angle from the axis Oh to the axis through (the orientation of the plane is set by the system hoy).
Since the angle from the axis Oh before the vector is equal, then the coordinates of the vector are:
Now the angle from the vector to the vector is (Fig. 146), so the angle from the axis Oh to the vector is equal (according to the Shales theorem for angles) and therefore the coordinates of the vector are:
And formulas (3) § 97 take the form
Transition matrix
orthogonal, but its determinant is –1. (7)
Conversely, any orthogonal matrix with determinant equal to –1 specifies the transformation of one rectangular coordinate system on the plane into another rectangular system with the same origin, but opposite orientation. So, if two Cartesian rectangular coordinate systems hoyand have a common origin, then
where x, at - coordinates of any point in the system hoy; and are the coordinates of the same point in the system, and
orthogonal matrix.
Conversely if
is an arbitrary orthogonal matrix, then the relations
the transformation of a Cartesian rectangular coordinate system to a Cartesian rectangular coordinate system is expressed the system with the same origin; - coordinates in the system hoy a unit vector giving the positive direction of the axis; - coordinates in the system hoy a unit vector giving the positive direction of the axis.
coordinate systems hoyand have the same orientation, and in the case - the opposite.
3. General transformation of one Cartesian rectangular coordinate system on the plane to another rectangular system.
Based on points 1) and 2) of this section, as well as on the basis of § 96, we conclude that if rectangular coordinate systems are introduced on the plane hoyand, then the coordinates x and at arbitrary point M plane in the system hoy with coordinates of the same point M in the system are related by the relations - coordinates of the origin of the coordinate system in the system hoy.
Note that the old and new coordinates x, at and, vectors under the general transformation of the Cartesian rectangular coordinate system are related by the relations
in case the systems hoy and have the same orientation and relations
if these systems have opposite orientation, or in the form
orthogonal matrix. Transformations (10) and (11) are called orthogonal.
Chapter I. Vectors on the plane and in space
§ 13. Transition from one rectangular Cartesian coordinate system to another
We suggest you consider this topic in two versions.
1) According to the textbook I. I. Privalov "Analytical geometry" (textbook for higher technical educational institutions in 1966)
II Privalov "Analytical geometry"
§ 1. The problem of transformation of coordinates.
The position of a point on a plane is determined by two coordinates relative to a certain coordinate system. The coordinates of the point will change if we choose a different coordinate system.
The coordinate transformation task is so that, knowing the coordinates of a point in one coordinate system, find its coordinates in another system.
This problem will be solved if we establish formulas connecting the coordinates of an arbitrary point in two systems, and the coefficients of these formulas include constant values \u200b\u200bthat determine the mutual position of the systems.
Let two Cartesian coordinate systems be given hoy and XO 1 Y (fig. 68).
Position of the new system XO 1 Y relatively old system hoy will be determined if coordinates are known and and b a new beginning O 1 according to the old system and angle α between axles Oh and About 1 X... Let us denote by x and atcoordinates of an arbitrary point M relative to the old system, through X and Y-coordinates of the same point relative to the new system. Our task is to make the old coordinates x and at express in terms of new X and Y. Obviously, the resulting transformation formulas should include the constants a, b and α .
We will obtain a solution to this general problem by considering two special cases.
1. The origin of coordinates changes, while the directions of the axes remain unchanged ( α = 0).
2. The directions of the axes change, but the origin of coordinates remains unchanged ( a \u003d b = 0).
§ 2. Transfer of the origin of coordinates.
Let there be given two systems of Cartesian coordinates with different origins O and O 1 and the same directions of the axes (fig. 69).
Let us denote by and and b coordinates of a new beginning About 1 in the old system and through x, y and X, Y-coordinates of an arbitrary point M, respectively, in the old and new systems. Projecting point M on the axis About 1 X and Ohand also point About 1 per axis Oh, we get on the axis Oh three dots Oh, ah and R... Segment values OA, AR and OR are related by the following relationship:
| OA| + | AR | = | OR |. (1)
Noticing that | OA| = and , | OR | = x , | AR | = | О 1 Р 1 | = X , we rewrite equality (1) as:
and + X = x or x = X + and . (2)
Similarly, projecting M and About 1 on the ordinate axis, we get:
y = Y + b (3)
So, the old coordinate is equal to the new one plus the old coordinate of the new origin.
From formulas (2) and (3), the new coordinates can be expressed through the old ones:
X = x - a , (2")
Y = y - b . (3")
§ 3. Rotation of the coordinate axes.
Let two Cartesian coordinate systems with the same origin be given ABOUT and different directions of the axes (Fig. 70).
Let be α there is an angle between the axes Oh and OH... Let us denote by x, y and X, Y coordinates of an arbitrary point M in the old and new systems, respectively:
x = | OR | , at = | PM | ,
X= | OP 1 |, Y= | P 1 M |.
Consider a broken line OP 1 MP and take its projection onto the axis Oh... Noticing that the projection of the broken line is equal to the projection of the closing segment (Ch. I, § 8) we have:
OP 1 MP = | OR |. (4)
On the other hand, the projection of a broken line is equal to the sum of the projections of its links (Ch. I, § 8); therefore, equality (4) will be written as follows:
etc OP 1 + pr P 1 M+ np MP= | OR | (4")
Since the projection of a directed segment is equal to its magnitude multiplied by the cosine of the angle between the projection axis and the axis on which the segment lies (Chapter I, § 8), then
etc OP 1 = X cos α
etc P 1 M = Y cos (90 ° + α ) = - Ysin α ,
np MP= 0.
Hence equality (4 ") gives us:
x = X cos α - Ysin α . (5)
Similarly, projecting the same polyline onto the axis OU, we get an expression for at... Indeed, we have:
etc OP 1 + pr P 1 M+ np MP\u003d pr OR = 0.
Noticing that
etc OP 1 = X cos ( α - 90 °) \u003d Xsin α ,
etc P 1 M = Y cos α ,
np MP = - y ,
will have:
Xsin α + Y cos α - y = 0,
y = Xsin α + Y cos α . (6)
From formulas (5) and (6) we obtain new coordinates X and Y expressed through old x and at if we solve equations (5) and (6) with respect to X and Y.
Comment. Formulas (5) and (6) can be obtained differently.
Fig. 71 we have:
x \u003d OP \u003d ОМ cos ( α + φ ) \u003d ОМ cos α cos φ - ОМ sin α sin φ ,
at \u003d РМ \u003d ОМ sin ( α + φ ) \u003d ОМ sin α cos φ + ОМ cos α sin φ .
Since (Ch. I, § 11) OM cos φ = X, ОМ sin φ =Ythen
x = X cos α - Ysin α , (5)
y = Xsin α + Y cos α . (6)
§ 4. General case.
Let there be given two Cartesian coordinate systems with different origins and different directions of the axes (Fig. 72).
Let us denote by and and b coordinates of a new beginning ABOUT, according to the old system, through α -the angle of rotation of the coordinate axes and, finally, through x, y and X, Y- coordinates of an arbitrary point M, respectively, according to the old and new systems.
To express x and at across X and Y, we introduce an auxiliary coordinate system x 1 O 1 y 1, the beginning of which we will place at the new beginning ABOUT 1, and the directions of the axes are taken to coincide with the directions of the old axes. Let be x 1 and y 1 denote the coordinates of the point M relative to this auxiliary system. Passing from the old coordinate system to the auxiliary one, we have (§ 2):
x = x 1 + a , y \u003d y 1 + b .
x 1 = X cos α - Ysin α , y 1 = Xsin α + Y cos α .
Replacing x 1 and y 1 in the previous formulas by their expressions from the last formulas, we finally find:
x = X cos α - Ysin α + a
y = Xsin α + Y cos α + b (I)
Formulas (I) contain, as a special case, the formulas of §§ 2 and 3. Thus, for α \u003d 0 formulas (I) turn into
x = X + and , y = Y + b ,
and at a \u003d b \u003d 0 we have:
x = X cos α - Ysin α , y = Xsin α + Y cos α .
From formulas (I), we obtain new coordinates X and Y expressed through old x and at if equations (I) are solvable with respect to X and Y.
Note a very important property of formulas (I): they are linear with respect to X and Y, i.e. of the form:
x = AX + BY + C, y = A 1 X + B 1 Y + C 1 .
It is easy to check that the new coordinates X and Y expressed through old x and at also formulas of the first degree with respect to x and at.
G.N. Yakovlev "Geometry"
§ 13. Transition from one rectangular Cartesian coordinate system to another
By choosing a rectangular Cartesian coordinate system, a one-to-one correspondence is established between the points of the plane and ordered pairs of real numbers. This means that each point of the plane corresponds to a single pair of numbers, and each ordered pair of real numbers corresponds to a single point.
The choice of this or that coordinate system is not limited by anything and is determined in each specific case only by considerations of convenience. Often one and the same set has to be considered in different coordinate systems. One and the same point in different systems has, obviously, different coordinates. A set of points (in particular, a circle, a parabola, a straight line) in different coordinate systems is given by different equations.
Let us find out how the coordinates of the points of the plane are transformed when passing from one coordinate system to another.
Let two rectangular coordinate systems be given on the plane: О, i, j and about", i ", j" (fig. 41).
The first system with origin at point O and basis vectors i and j let's agree to call the old one, the second - with the beginning at point O "and the basis vectors i " and j " - new.
The position of the new system relative to the old one will be considered known: let point O "in the old system have coordinates ( a; b ), a vector i " forms with vector i angle α ... Angle α counting in the opposite direction to the clockwise movement.
Consider an arbitrary point M. We denote its coordinates in the old system by ( x; y ), in the new one - through ( x "; y" ). Our task is to establish the relationship between the old and new coordinates of point M.
We connect in pairs the points O and O ", O" and M, O and M. By the triangle rule, we obtain
OM > = OO " > + O "M > . (1)
Let us expand vectors OM \u003e and OO " \u003e by basis vectors i and j and the vector O "M \u003e by basis vectors i " and j " :
OM > = x i + y j , OO " > = a i + b j , O "M > = x " i "+ y" j "
Now equality (1) can be written as follows:
x i + y j = (a i + b j ) + (x " i "+ y" j "). (2)
New base vectors i " and j " are decomposed by old basis vectors i and j in the following way:
i " \u003d cos α i + sin α j ,
j " \u003d cos ( π / 2 + α ) i + sin ( π / 2 + α ) j \u003d - sin α i + cos α j .
Substituting the found expressions for i " and j " into formula (2), we obtain the vector equality
x i + y j = a i + b j + x "(cos α i + sin α j ) + at "(- sin α i + cos α j )
equivalent to two numerical equalities:
|
x \u003d a + x "cos α
- at " sin α
, |
Formulas (3) give the sought expressions for the old coordinates x and at points through its new coordinates x " and at "... In order to find expressions for the new coordinates in terms of the old ones, it is enough to solve the system of equations (3) with respect to the unknowns x " and at ".
So, the coordinates of points when moving the origin to a point ( and; b ) and turning the axes at an angle α are transformed by formulas (3).
If only the origin of coordinates changes, and the directions of the axes remain the same, then, setting in formulas (3) α \u003d 0, we get
Formulas (5) are called rotation formulas.
Objective 1. Let the coordinates of the new beginning in the old system (2; 3), and the coordinates of point A in the old system (4; -1). Find the coordinates of point A in the new system, if the directions of the axes remain the same.
By formulas (4), we have
Answer. A (2; -4)
Objective 2. Let the coordinates of the point P in the old system (-2; 1), and in the new system, the directions of the axes of which are the same, the coordinates of this point (5; 3). Find the coordinates of the new beginning in the old system.
A By formulas (4) we obtain
|
-
2 \u003d a + 5 |
from where and = - 7, b = - 2.
Answer. (-7; -2).
Objective 3.Point A coordinates in the new system (4; 2). Find the coordinates of this point in the old system, if the origin remains the same, and the coordinate axes of the old system are rotated by an angle α \u003d 45 °.
By formulas (5) we find
Problem 4.Coordinates of point A in the old system (2 √3; - √3). Find the coordinates of this point in the new system, if the origin of the old system is moved to the point (-1; -2), and the axes are rotated by an angle α \u003d 30 °.
By formulas (3), we have
Having solved this system of equations for x " and at ", we find: x " = 4, at " = -2.
Answer. A (4; -2).
Objective 5. The equation of the straight line at = 2x - 6. Find the equation of the same straight line in the new coordinate system, which is obtained from the old system by rotating the axes by an angle α \u003d 45 °.
The rotation formulas in this case have the form
Replacing the straight line in the equation at = 2x - 6 old variables x and at new, we get the equation
√ 2 / 2 (x "+ y") = 2 √ 2 / 2 (x "- y") - 6 ,
which after simplifications takes the form y " = x " / 3 - 2√2