Rectangular space cartesian coordinate system

Spatial Rectangular Coordinate System Transformations

Linear mapping transformations

Reducing a general quadratic form to a canonical one

Curvilinear coordinates

Understanding Curvilinear Coordinate Systems

Curvilinear coordinates on a surface

Polar coordinate systems and their generalizations

Spatial system of polar coordinates

Cylindrical coordinate system

Spherical coordinate system

Polar coordinates on the surface

Chapter 3. COORDINATE SYSTEMS USED IN GEODESY

General classification of coordinate systems used in geodesy

Earth Geodetic Coordinate Systems

Polar coordinate systems in geodesy

Curvilinear ellipsoidal systems of geodetic coordinates

Determination of ellipsoidal geodetic coordinates with a separate method for determining the horizontal and vertical positions of points on the earth's surface

Convert Spatial Polar Geodetic Coordinates to Ellipsoidal Geodetic Coordinates

Conversion of reference systems of geodetic coordinates to general terrestrial ones and vice versa

Spatial rectangular coordinate systems

Relationship of spatial rectangular coordinates with ellipsoidal geodetic coordinates

Converting spatial rectangular reference coordinates to general terrestrial coordinates and vice versa

Topocentric coordinate systems in geodesy

Relationship of the spatial topocentric horizontal geodesic SC with the spatial polar spherical coordinates

Converting topocentric horizontal geodetic coordinates to spatial rectangular coordinates X, Y, Z

Plane rectangular coordinate systems in geodesy

Relationship of planar rectangular Gauss - Kruger coordinates with ellipsoidal geodetic coordinates

Converting planar rectangular Gauss-Kruger coordinates from one zone to another

Conversion of plane rectangular coordinates of points of local geodetic constructions to other systems of plane rectangular coordinates

Chapter 4. COORDINATE SYSTEMS USED IN GEODESIC ASTRONOMY AND SPACE GEODESY

Spherical astronomy coordinate systems

Reference systems in space geodesy

Stellar (celestial) inertial geocentric equatorial coordinates

Greenwich Terrestrial Geocentric System of Spatial Rectangular Coordinates

Topocentric coordinate systems

Chapter 5. COORDINATIZATION OF THE ENVIRONMENTAL SPACE AT THE BEGINNING OF THE XXI CENTURY IN RUSSIA

Systems of state geodetic coordinates at the beginning of the XXI century.

Construction of the State Geodetic Network

BIBLIOGRAPHY

APPENDIX 1. SOLUTION OF A DIRECT GEODETIC PROBLEM IN SPACE

APPENDIX 2. SOLUTION OF THE INVERSE GEODETIC PROBLEM IN SPACE

APPENDIX 3. CONVERSION OF GEODESIC COORDINATES B, L, H INTO SPATIAL RECTANGULAR X, Y, Z

APPENDIX 4. CONVERSION OF SPATIAL RECTANGULAR COORDINATES X, Y, Z INTO GEODESIC B, L, H

APPENDIX 5. CONVERSION OF SPATIAL RECTANGULAR COORDINATES X, Y, Z CK-42 INTO COORDINATES OF THE PZ-90 SYSTEM

APPENDIX 6. CONVERSION OF THE REFERENCE SYSTEM OF GEODETIC COORDINATES B, L, H INTO THE SYSTEM OF GEODESIC COORDINATES PZ-90 B0, L0, H0

APPENDIX 7. CONVERSION OF SPATIAL POLAR COORDINATES OF SYSTEM S, ZG, A INTO TOPOCENTRIC HORIZONTAL GEODESIC COORDINATES XT, YT, ZT

APPENDIX 8. CONVERSION OF TOPOCENTRIC HORIZONTAL GEODETIC COORDINATES ХТ, УТ, ZТ INTO POLAR SPATIAL COORDINATES - S, ZГ, A

APPENDIX 9. TRANSFORMATION OF TOPOCENTRIC HORIZONTAL GEODESIC COORDINATES XT, YT, ZT INTO SPATIAL RECTANGULAR COORDINATES X, Y, Z

APPENDIX 10. TRANSFORMATION OF ELLIPSOIDAL GEODESIC COORDINATES B, L INTO PLANE RECTANGULAR GAUSS - KRUGER COORDINATES X, Y

APPENDIX 11. CONVERSION OF PLANE RECTANGULAR GAUSSIAN - KRUGER COORDINATES X, Y INTO ELLIPSOIDAL GEODETIC COORDINATES B, L

(a 11 - λ1) (a 22 - λ1) - a 12 a 21 = 0;

λ 12 - (a 11 + a 22) λ 1 + (a 11a 22 - a 12 a 21) = 0.

The discriminant of these quadratic equations is ³ 0, that is,

D = (a 11 + a 22) 2 - 4 (a 11a 22 - a 12 a 21) = (a 11 - a 22) 2 + 4a 122 ³ 0.

Equations (2.56), (2.57) are called characteristic equations

matrices, and the roots of these equations are eigenvalues the matrices A. Substitute the eigenvalues ​​found from (2.57) into (2.39), we obtain

canonical equation.

A quadratic form is given in the form: F (x x) = 5x 2				2x 2.

Find the canonical form of this equation.

Since here a 11 = 5; a 21 = 2; a 22 = 2, then the characteristic equation (2.56) for a given quadratic form will have the form

5 - λ 2

2 2 - λ 1

Equating the determinant of this matrix equation to zero

(5 - λ) (2 - λ) - 4 = λ2 - 7λ + 6 = 0

and solving this quadratic equation, we get λ1 = 6; λ2 = 1.

And then the canonical form of this quadratic form will have the form

F (x 1, x 2) = 6 x 1 2 + x 2 2.

2.3. Curvilinear coordinates

2.3.1. Understanding Curvilinear Coordinate Systems

The class of curvilinear coordinates, in comparison with the class of rectilinear coordinates, is extensive and much more diverse and, from an analytical point of view, is the most universal, since it expands the capabilities of the method of rectilinear coordinates. The use of curvilinear coordinates can sometimes greatly simplify the solution of many problems, especially problems solved directly on the surface of revolution. So, for example, when solving a problem on a surface of revolution associated with finding a certain function, it is possible, in the domain of this function on a given surface, to choose a system of curvilinear coordinates that will allow this function to be endowed with a new property - to be constant in a given coordinate system that cannot always be done using rectilinear coordinate systems.

A system of curvilinear coordinates, given in a certain region of three-dimensional Euclidean space, puts in correspondence to each point of this space an ordered triple of real numbers - φ, λ, r (curvilinear coordinates of a point).

If the curvilinear coordinate system is located directly on some surface (surface of revolution), then in this case, in accordance with each point of the surface, already two real numbers are put - φ, λ, which uniquely determine the position of the point on this surface.

There must be a mathematical relationship between the system of curvilinear coordinates φ, λ, r and a rectilinear Cartesian CS (X, Y, Z). Indeed, let the curvilinear coordinate system be given in a certain region of space. Each point of this space corresponds to a unique triple of curvilinear coordinates - φ, λ, r. On the other hand, the only triple of rectilinear Cartesian coordinates - X, Y, Z - corresponds to the same point.Then it can be argued that in general form

ϕ = ϕ (X, Y, Z);

λ = λ (,); (2.58)

X Y Z

r = r (X, Y, Z).

There is both a direct (2.58) and an inverse mathematical relationship between these CSs.

From the analysis of formulas (2.58) it follows that for a constant value of one of the spatial curvilinear coordinates φ, λ, r, for example,

ϕ = ϕ (X, Y, Z) = const,

and variable values ​​of the other two (λ, r), we get in general a surface, which is called coordinate. Coordinate surfaces corresponding to the same coordinate do not intersect with each other. However, two coordinate surfaces corresponding to different coordinates intersect and give a coordinate line corresponding to the third coordinate.

2.3.2. Curvilinear coordinates on a surface

For geodesy, surface curvilinear coordinates are of greatest interest.

Let the equation of the surface in the form of a function of Cartesian coordinates in

implicitly has the form
F (X, Y, Z) = 0.

By directing the unit vectors i, j, l along the coordinate axes (Fig. 2.11), the surface equation can be written in vector form

r = X i + Y j + Z l. (2.60)

We introduce two new independent variables φ and λ such that the functions

satisfy equation (2.59). Equalities (2.61) are parametric equations of the surface.

λ1 = const

					λ2 = const
					λ3 = const
					φ3 = const
					φ2 = const
					φ1 = const

Fig. 2.11. Curved surface coordinate system

Each pair of numbers φ and λ corresponds to a certain (unique) point on the surface, and these variables can be taken as the coordinates of points on the surface.

If we give φ different constant values ​​φ = φ1, φ = φ2,…, then we get a family of curves on the surface corresponding to these constants. Similarly, giving constant values ​​for λ, we will have

second family of curves. Thus, a network of coordinate lines φ = const and λ = const is formed on the surface. Coordinate lines in general

are curved lines. Therefore, the numbers φ, λ are called

curvilinear coordinates points on the surface.

Curvilinear coordinates can be both linear and angular values. The simplest example of a curvilinear coordinate system, in which one coordinate is a linear quantity and the other is an angular quantity, can be polar coordinates on a plane.

The choice of curved coordinates does not have to precede the formation of coordinate lines. In some cases, it is more expedient to establish a grid of coordinate lines that is most convenient for solving certain problems on the surface, and then select for these lines such parameters (coordinates) that would have a constant value for each coordinate line.

A well-defined grid of coordinate lines corresponds to a certain system of parameters, but for each given family of coordinate lines, many other parameters can be selected, which are continuous and single-valued functions of this parameter. In the general case, the angles between the coordinate lines of the φ = const family and the lines of the λ = const family can have different values.

We will consider only orthogonal curvilinear coordinate systems in which each coordinate line φ = const intersects any other coordinate line λ = const at a right angle.

When solving many problems on the surface, especially problems associated with calculating the curvilinear coordinates of points on the surface, it is necessary to have differential equations for changing the curvilinear coordinates φ and λ depending on the change in the length S of the surface curve.

The connection between the differentials dS, dφ, dλ can be established by introducing a new variable α, i.e., the angle

α dS

φ = const

λ = const

λ + d λ = const

positive direction of the line λ = const to positive

directions of this curve (Fig. 2.12). This angle, as it were, establishes the direction (orientation) of the line in

a given point on the surface. Then (without output):

Fig. 2.12. The geometry of the connection of the differential of an arc of a curve on a surface with changes (differentials) of curvilinear

coordinates

					∂X		2 ∂ Y 2
E = (rϕ)
						∂ϕ		∂ϕ
G = (						∂X		∂ Y 2
						∂λ		∂λ

+ ∂ Z 2;

∂ϕ

+ ∂ Z 2. ∂λ

		cosα
		sinα

IN geodesy angle α corresponds to the geodetic azimuth: α = BUT.

2.3.3. Polar coordinate systems and their generalizations

2.3.4. Spatial system of polar coordinates

To specify the spatial system of polar coordinates, you must first select a plane (hereinafter we will call it the main one). On this plane, some point O is chosen

measurements

segments

space then

position

any point in space will

unequivocally

be defined

quantities: r, φ, λ, where r -

polar

distance in a straight line from a pole

O to point Q (Fig. 2.13); λ -

polar angle - the angle between

polar

Fig. 2.13. Spatial system

orthogonal

projection

polar radius to the main

polar coordinates and its modifications

plane

changes

(polar radius) and its

0 ≤ λ < 2π); φ – угол между

vector

projection

OQ0 on

the main

plane, assumed to be positive (0 ≤ φ ≤ π / 2) for points in the positive half-space and negative (-π / 2 ≤ φ ≤ 0) for points in the negative half-space.

Any spatial polar CS can be easily associated (transformed) with a spatial Cartesian rectangular CS.

If the scale and origin of coordinates in the spatial rectangular system are taken as the scale and the origin of the polar system, the polar axis OR is the abscissa semiaxis OX, the line OZ drawn from the pole O perpendicular to the main plane in the positive direction of the polar system is the semiaxis OZ of the rectangular Cartesian system, and for the semiaxis - OU, take the axis into which the abscissa axis goes when it is rotated by an angle π / 2 in the positive direction in the main plane of the polar system, then from Fig. 2.13

Formulas (2.64) allow expressing X, Y, Z in terms of r, φ, λ and vice versa

On any surface, you can set a coordinate system, defining the position of a point on it again with two numbers. To do this, in some way we will cover the entire surface with two families of lines so that one, and only one, line from each family passes through each point (perhaps with a few exceptions). Now you just need to provide the lines of each family with numerical marks according to some firm rule that allows you to find the required family line by the numerical mark (Fig. 22).

Point coordinates M surfaces are numbers u, v, Where u- numerical mark of the line of the first family passing through M, and v- marking the lines of the second family. We will continue to write: M (u; v), the numbers and, v are called curvilinear coordinates of the point M. What has been said will become quite clear if we turn to the sphere for an example. It can all be covered with meridians (first family); each of them corresponds to a numerical mark, namely the value of longitude u(or c). All parallels form a second family; each of them has a numerical mark - latitude v(or and). Only one meridian and one parallel passes through each point of the sphere (excluding the poles).

As another example, consider the lateral surface of a straight round cylinder of height H, radius a(fig. 23). For the first family we will take the system of its generators, we will take one of them as the initial one. To each generator we assign a mark u, equal to the length of the arc on the circumference of the base between the initial generatrix and the given one (we will count the arc, for example, counterclockwise). For the second family we will take the system of horizontal sections of the surface; numerically marked v we will consider the height at which the section is drawn above the base. With proper choice of axes x, y, z in space we will have for any point M (x; y; z) of our surface:

(Here the arguments for cosine and sine are not in degrees, but in radians.) These equations can be viewed as parametric equations for the surface of a cylinder.

Problem 9. What curve should a piece of sheet metal be cut to make a downspout elbow so that, after proper bending, a cylinder of radius is obtained but, truncated by a plane at an angle of 45 ° to the plane of the base?

Decision. Let's use the parametric equations of the cylinder surface:

Draw the cutting plane through the axis Oh, her equation z = y. Combining it with the equations just written, we get the equation

lines of intersection in curvilinear coordinates. After unfolding the surface onto a plane, curvilinear coordinates and and v will turn into Cartesian coordinates.

So, a piece of tin should be outlined on top along a sinusoid

Here u and v already Cartesian coordinates on the plane (Fig. 24).

As in the case of a sphere and a cylindrical surface, and in the general case, the specification of the surface by parametric equations entails the establishment of a curvilinear coordinate system on the surface. Indeed, the expression for Cartesian coordinates x, y, z arbitrary point M (x; y; z) surface in two parameters u, v(this is generally written as follows: x= q ( u; v), y = c (u; v), z = u (u; v), q, w, u are functions of two arguments) makes it possible, knowing a pair of numbers u, v, find the corresponding coordinates x, y, z, and hence the position of the point M on a surface; the numbers u, v serve as its coordinates. By giving one of them a constant value, for example u=u 0, we get the expression x, y, z through one parameter v, i.e. the parametric equation of the curve. This is the coordinate line of one family, its equation u = u 0. Likewise the line v = v 0 is the coordinate line of another family.

coordinate cartesian radius vector

On surface.

Local properties of curved coordinates

When considering curvilinear coordinates in this section, we will assume that we are considering a three-dimensional space (n = 3) equipped with Cartesian coordinates x, y, z. The case of other dimensions differs only in the number of coordinates.

In the case of a Euclidean space, the metric tensor, also called the square of the differential of the arc, in these coordinates will have the form corresponding to the unit matrix:

$dS\; ^\; 2\; =\; \backslash \; mathbf\; (dx)\; ^\; 2\; +\; \backslash \; mathbf\; (dy)\; ^\; 2\; +\; \backslash \; mathbf\; (dz)\; ^\; 2.$

General case

Let be $q\_1$, $q\_2$, $q\_3$- some curvilinear coordinates, which we will consider as given smooth functions of x, y, z. To make three functions $q\_1$, $q\_2$, $q\_3$ served as coordinates in a certain region of space, the existence of an inverse mapping is necessary:

$\backslash \; left\; \backslash \; (\backslash \; begin\; (matrix)\; x\; =\; \backslash \; varphi\_1\; \backslash \; left\; (q\_1,\; \backslash ;\; q\_2,\; \backslash ;\; q\_3\; \backslash \; right);\; \backslash \backslash \; y\; =\; \backslash \; varphi\_2\; \backslash \; left\; (q\_1,\; \backslash ;\; q\_2,\; \backslash ;\; q\_3\; \backslash \; right)\; ;\; \backslash \backslash \; z\; =\; \backslash \; varphi\_3\; \backslash \; left\; (q\_1,\; \backslash ;\; q\_2,\; \backslash ;\; q\_3\; \backslash \; right),\; \backslash \; end\; (matrix)\; \backslash \; right.$

Where $\backslash \; varphi\_1,\; \backslash ;\; \backslash \; varphi\_2,\; \backslash ;\; \backslash \; varphi\_3$- functions defined in a certain area of ​​sets $\backslash \; left\; (q\_1,\; \backslash ;\; q\_2,\; \backslash ;\; q\_3\; \backslash \; right)$ coordinates.

Local basis and tensor analysis

In tensor calculus, local basis vectors can be introduced: $\backslash \; mathbf\; (R\_j)\; =\; \backslash \; frac\; (d\; \backslash \; mathbf\; r)\; (dy\; ^\; j)\; =\; \backslash \; frac\; (dx\; ^\; i)\; (dy\; ^\; j)\; \backslash \; mathbf\; e\_i\; =\; Q\; ^\; i\_j\; \backslash \; mathbf\; e\_i$ where $\backslash \; mathbf\; e\_i$- unit vectors of the Cartesian coordinate system, $Q\; ^\; i\_j$- Jacobi matrix, $x\; ^\; i$ coordinates in the Cartesian system, $y\; ^\; i$- introduced curvilinear coordinates.
It is not difficult to see that curvilinear coordinates, generally speaking, change from point to point.
Let us indicate the formulas for the relationship between curvilinear and Cartesian coordinates:
$\backslash \; mathbf\; R\_i\; =\; Q\; ^\; j\_i\; \backslash \; mathbf\; e\_j$
$\backslash \; mathbf\; e\_i\; =\; P\; ^\; j\_i\; \backslash \; mathbf\; R\_j$ Where $P\; ^\; j\_i\; Q\; ^\; i\_j\; =\; E$, where E is the identity matrix.
The product of two vectors of the local basis forms a metric matrix:
$\backslash \; mathbf\; R\_i\; \backslash \; mathbf\; R\_j\; =\; Q\; ^\; n\_i\; Q\; ^\; m\_j\; d\_\; (nm)\; =\; g\_\; (ij)$
$\backslash \; mathbf\; R\; ^\; i\; \backslash \; mathbf\; R\; ^\; j\; =\; P\; ^\; i\_n\; P\; ^\; j\_m\; d\; ^\; (nm)\; =\; g\; ^\; (ij)$
$g\_\; (ij)\; g\; ^\; (jk)\; =\; g\; ^\; (jk)\; g\_\; (ij)\; =\; d\_i\; ^\; k$ where $d\_\; (ij),\; d\; ^\; (ij),\; d\; ^\; i\_j$ contravariant, covariant and mixed Kronecker symbols
Thus, any tensor field $\backslash \; mathbf\; T$ rank n can be expanded in a local polyadic basis:
$\backslash \; mathbf\; T\; =\; T\; ^\; (i\_1\; ...\; i\_n)\; \backslash \; mathbf\; e\_i\; \backslash \; otimes\; ...\; \backslash \; otimes\; \backslash \; mathbf\; e\_n\; =\; T\; ^\; (i\_1\; ...\; i\_n)\; P\; ^\; (j\_1)\; \_\; (i\_1)\; ...\; P\; ^\; (j\_n)\; \_\; (i\_n)\; \backslash \; mathbf\; R\_\; (j\_1)\; \backslash \; otimes\; ...\; \backslash \; otimes\; \backslash \; mathbf\; R\_\; (j\_n)$
For example, in the case of a first rank tensor field (vector):
$\backslash \; mathbf\; v\; =\; v\; ^\; i\; \backslash \; mathbf\; e\_i\; =\; v\; ^\; i\; P\; ^\; j\_i\; \backslash \; mathbf\; R\_j$

Orthogonal Curvilinear Coordinates

In Euclidean space, the use of orthogonal curvilinear coordinates is of particular importance, since the formulas related to length and angles look simpler in orthogonal coordinates than in the general case. This is due to the fact that the metric matrix in systems with an orthonormal basis will be diagonal, which will greatly simplify the calculations.
An example of such systems is a spherical system in $\backslash \; mathbb\; (R)\; ^\; 2$

Lamé odds

Let us write the differential of the arc in curvilinear coordinates in the form (using the Einstein summation rule):

$dS\; ^\; 2\; =\; \backslash \; left\; (\backslash \; frac\; (\backslash \; partial\; \backslash \; varphi\_1)\; (\backslash \; partial\; q\_i)\; \backslash \; mathbf\; (dq)\; \_i\; \backslash \; right)\; ^\; 2\; +$

\ left (\ frac (\ partial \ varphi_2) (\ partial q_i) \ mathbf (dq) _i \ right) ^ 2 + \ left (\ frac (\ partial \ varphi_3) (\ partial q_i) \ mathbf (dq) _i \ right) ^ 2, ~ i = 1,2,3

Taking into account the orthogonality of coordinate systems ( $\backslash \; mathbf\; (dq)\; \_i\; \backslash \; cdot\; \backslash \; mathbf\; (dq)\; \_j\; =\; 0$ at $i\; \backslash \; ne\; j$) this expression can be rewritten as

$dS\; ^\; 2\; =\; H\_1\; ^\; 2dq\_1\; ^\; 2\; +\; H\_2\; ^\; 2dq\_2\; ^\; 2\; +\; H\_3\; ^\; 2dq\_3\; ^\; 2,$

$H\_i\; =\; \backslash \; sqrt\; (\backslash \; left\; (\backslash \; frac\; (\backslash \; partial\; \backslash \; varphi\_1)\; (\backslash \; partial\; q\_i)\; \backslash \; right)\; ^\; 2\; +\; \backslash \; left\; (\backslash \; frac\; (\backslash \; partial\; \backslash \; varphi\_2)\; (\backslash \; partial\; q\_i)\; \backslash \; right)\; ^\; 2\; +\; \backslash \; left\; (\backslash \; frac\; (\backslash \; partial\; \backslash \; varphi\_3)\; (\backslash \; partial\; q\_i)\; \backslash \; right)\; ^\; 2);\; \backslash \; i\; =\; 1,\; \backslash ;\; 2,\; \backslash ;\; 3$

Positive values $H\_i\; \backslash$ that depend on a point in space are called Lamé coefficients or scale factors. Lamé coefficients show how many units of length are contained in the unit of coordinates of a given point and are used to transform vectors when moving from one coordinate system to another.

Riemannian metric tensor written in coordinates $(q\_i)$, is a diagonal matrix, on the diagonal of which are the squares of the Lamé coefficients:

Examples of

Polar coordinates ( n=2)

Polar coordinates on a plane include the distance r to the pole (origin) and the direction (angle) φ.

Relationship of polar coordinates with Cartesian ones:

$\backslash \; left\; \backslash \; (\backslash \; begin\; (matrix)\; x\; =\; r\; \backslash \; cos\; (\backslash \; varphi);\; \backslash \backslash \; y\; =\; r\; \backslash \; sin\; (\backslash \; varphi).\; \backslash \; end\; (matrix)\; \backslash \; right.$

Lame coefficients:

$\backslash \; begin\; (matrix)\; H\_r\; =\; 1;\; \backslash \backslash \; H\_\; \backslash \; varphi\; =\; r.\; \backslash \; end\; (matrix)$

Arc differential:

$dS\; ^\; 2\; \backslash \; =\; \backslash \; dr\; ^\; 2\; \backslash \; +\; \backslash \; r\; ^\; 2d\; \backslash \; varphi\; ^\; 2.$

At the origin, the function φ is undefined. If the φ coordinate is considered not a number, but an angle (a point on the unit circle), then the polar coordinates form a coordinate system in the area obtained from the entire plane by removing the origin point. If, nevertheless, φ is considered a number, then in the designated area it will be multivalued, and the construction of a coordinate system strictly in the mathematical sense is possible only in a simply connected area that does not include the origin of coordinates, for example, on a plane without a ray.

Cylindrical coordinates ( n=3)

Cylindrical coordinates are a trivial generalization of polar coordinates to the case of three-dimensional space by adding a third coordinate z. Relationship of cylindrical coordinates with Cartesian ones:

$\backslash \; left\; \backslash \; (\backslash \; begin\; (matrix)\; x\; =\; r\; \backslash \; cos\; (\backslash \; varphi);\; \backslash \backslash \; y\; =\; r\; \backslash \; sin\; (\backslash \; varphi).\; \backslash \backslash \; z\; =\; z.\; \backslash \; end\; (matrix)\; \backslash \; right.$

Lame coefficients:

$\backslash \; begin\; (matrix)\; H\_r\; =\; 1;\; \backslash \backslash \; H\_\; \backslash \; varphi\; =\; r;\; \backslash \backslash \; H\_z\; =\; 1.\; \backslash \; end\; (matrix)$

Arc differential:

$dS\; ^\; 2\; \backslash \; =\; \backslash \; dr\; ^\; 2\; \backslash \; +\; \backslash \; r\; ^\; 2d\; \backslash \; varphi\; ^\; 2\; +\; dz\; ^\; 2.$

Spherical coordinates ( n=3)

Spherical coordinates are associated with latitude and longitude coordinates on a unit sphere. Relationship of spherical coordinates with Cartesian ones:

$\backslash \; left\; \backslash \; (\backslash \; begin\; (matrix)\; x\; =\; r\; \backslash \; sin\; (\backslash \; theta)\; \backslash \; cos\; (\backslash \; varphi);\; \backslash \backslash \; y\; =\; r\; \backslash \; sin\; (\backslash \; theta)\; \backslash \; sin\; (\backslash \; varphi);\; \backslash \backslash \; z\; =\; r\; \backslash \; cos\; (\backslash \; theta).\; \backslash \; end\; (matrix)\; \backslash \; right.$

Lame coefficients:

$\backslash \; begin\; (matrix)\; H\_r\; =\; 1;\; \backslash \backslash \; H\_\; \backslash \; theta\; =\; r;\; \backslash \backslash \; H\_\; \backslash \; varphi\; =\; r\; \backslash \; sin\; (\backslash \; theta).\; \backslash \; end\; (matrix)$

Arc differential:

$dS\; ^\; 2\; \backslash \; =\; \backslash \; dr\; ^\; 2\; \backslash \; +\; \backslash \; r\; ^\; 2d\; \backslash \; theta\; ^\; 2\; +\; r\; ^\; 2\; \backslash \; sin\; ^\; 2\; (\backslash \; theta)\; d\; \backslash \; varphi\; ^\; 2.$

Spherical coordinates, like cylindrical ones, do not work on the z-axis (x = 0, y = 0), since the φ coordinate is not defined there.

Various exotic coordinates on the plane ( n= 2) and their generalizations

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Literature

• Korn G., Korn T. Handbook of mathematics (for scientists and engineers). - M .: Nauka, 1974 .-- 832 p.

Coordinate name Coordinate system types 2D coordinates 3D coordinates $n$-dimensional coordinates Physical coordinates Related definitions

Excerpt Characterizing Curvilinear Coordinate System

“If he could attack us, he would do it today,” he said.
“So you think he’s powerless,” said Langeron.
“A lot, if he has 40 thousand troops,” Weyrother answered with the smile of a doctor, to whom the medicine wants to indicate the remedy.
“In that case, he is going to his death, waiting for our attack,” Langeron said with a thin ironic smile, looking back at the nearest Miloradovich for confirmation.
But Miloradovich, obviously, at that moment thought least of all about what the generals were arguing about.
- Ma foi, [By God,] - he said, - tomorrow we will see everything on the battlefield.
Weyrother grinned again with that smile, which said that it was funny and strange for him to meet objections from Russian generals and to prove something in which not only he himself was too sure, but of which the emperors sovereigns were sure of them.
“The enemy has put out the lights, and there is a continuous noise in his camp,” he said. - What does it mean? - Either he leaves, which is one thing we should be afraid of, or he changes his position (he chuckled). But even if he took a position in Turas, he only saves us a lot of trouble, and all orders, down to the smallest details, remain the same.
- In what way? .. - said Prince Andrey, who had long been waiting for an opportunity to express his doubts.
Kutuzov woke up, cleared his throat heavily and looked around at the generals.
“Gentlemen, the disposition for tomorrow, even today (because it's already the first hour), cannot be changed,” he said. “You heard her, and we will all fulfill our duty. And before the battle, there is nothing more important ... (he paused) how to get a good night's sleep.
He pretended to stand up. The generals bowed and left. It was past midnight. Prince Andrew went out.

The council of war, at which Prince Andrei failed to express his opinion, as he hoped, left him with a vague and disturbing impression. Who was right: Dolgorukov with Weyrother or Kutuzov with Lanzheron and others, who did not approve of the plan of attack, he did not know. “But was it really impossible for Kutuzov to directly express his thoughts to the sovereign? Can't it be done otherwise? Is it possible for court and personal considerations to risk tens of thousands of my, my life? " he thought.
“Yes, very likely they will kill tomorrow,” he thought. And suddenly, at the thought of death, a whole series of memories, the most distant and most soulful, arose in his imagination; he recalled the last farewell to his father and wife; he recalled the early days of his love for her! He remembered her pregnancy, and he felt sorry for both her and himself, and in a nervous, softened and agitated state he left the hut in which he stood with Nesvitsky, and began to walk in front of the house.
The night was hazy, and moonlight mysteriously shone through the fog. “Yes, tomorrow, tomorrow! He thought. - Tomorrow, maybe everything will be over for me, all these memories will no longer be, all these memories will no longer have any meaning for me. Tomorrow, maybe, maybe even tomorrow, I have a presentiment of it, for the first time I will have to finally show everything that I can do. " And he imagined a battle, the loss of it, the concentration of the battle on one point and the confusion of all the commanding persons. And now that happy moment, that Toulon, which he had been waiting for so long, finally appears to him. He firmly and clearly speaks his opinion to Kutuzov, Weyrother, and the emperors. Everyone is amazed at the fidelity of his reasoning, but no one undertakes to fulfill it, and so he takes a regiment, a division, makes a condition that no one interferes in his orders, and leads his division to the decisive point and one wins. And death and suffering? says another voice. But Prince Andrey does not answer this voice and continues his successes. The disposition of the next battle is made by him alone. He carries the title of duty officer in the army under Kutuzov, but he does everything alone. The next battle is won by him alone. Kutuzov is replaced, he is appointed ... Well, and then? another voice speaks again, and then, if you have not been wounded, killed or deceived ten times before; well, and then what? “Well, then,” Prince Andrey answers himself, “I don’t know what will happen next, I don’t want and cannot know: but if I want this, I want fame, I want to be known to people, I want to be loved by them, then it is not my fault that I want this, that this alone I want, for this alone I live. Yes, for this one! I will never tell anyone this, but my God! what am I to do if I love nothing but glory, human love. Death, wounds, loss of family, nothing scares me. And no matter how dear and dear to me many people - father, sister, wife - are the people most dear to me - but, no matter how terrible and unnatural it seems, I will give them all now for a minute of glory, triumph over people, for love of to myself people whom I do not know and will not know, for the love of these people, ”he thought, listening to the dialect in the courtyard of Kutuzov. In the courtyard of Kutuzov one could hear the voices of orderlies who were packing; one voice, probably the coachman, teasing the old Kutuzov cook, whom Prince Andrey knew, and whose name was Titus, said: "Titus, and Titus?"
- Well, - answered the old man.
“Titus, go thresh,” said the joker.
- Ugh, well, to hell, - a voice was heard, covered with laughter of orderlies and servants.
"And yet, I love and treasure only the triumph over all of them, I treasure this mysterious power and glory that rushes above me in this fog!"

Rostov that night was with a platoon in a flanker chain, in front of Bagration's detachment. His hussars were scattered in chains in pairs; he himself rode on horseback along this line of the chain, trying to overcome the dream that irresistibly drove him. Behind it could be seen a vast expanse of fires of our army, indistinctly burning in the fog; ahead of him was hazy darkness. No matter how much Rostov peered into this misty distance, he did not see anything: now it was turning gray, now it was as if something were blackening; then flashed like lights, where the enemy should be; then he thought that it was only glittering in his eyes. His eyes were closed, and in his imagination he imagined the sovereign, then Denisov, then Moscow memories, and he again hastily opened his eyes and close in front of him he saw the head and ears of the horse on which he was sitting, sometimes the black figures of hussars when he was six paces away ran into them, and in the distance the same foggy darkness. "From what? it is very possible, ”thought Rostov,“ that the sovereign, upon meeting me, would give an assignment, as he would to any officer: he would say: 'Go ahead and find out what is there.' Many told how, quite by chance, he recognized some officer and brought him closer to him. What if he brought me closer to him! Oh, how I would protect him, how I would tell him the whole truth, how I would expose his deceivers, ”and Rostov, in order to vividly imagine his love and devotion to the sovereign, imagined an enemy or deceiver of the German, whom he enjoyed not only killed, but hit on the cheeks in the eyes of the sovereign. Suddenly, a distant cry awakened Rostov. He shuddered and opened his eyes.
"Where I am? Yes, in the chain: the slogan and the password are the tongue, Olmutz. What a shame that our squadron will be in reserves tomorrow ... - he thought. - I'll ask for the case. This may be the only time to see the sovereign. Yes, now it's not long before the shift. I will drive around again and, as soon as I return, I will go to the general and ask him. " He straightened himself on the saddle and moved the horse to go round his hussars once more. It seemed to him that it was brighter. On the left side there was a gentle, illuminated slope and the opposite, black hillock, which seemed steep like a wall. On this hillock there was a white spot, which Rostov could not understand in any way: was it a clearing in the forest, illuminated by a month, or the remaining snow, or white houses? It even seemed to him that something was stirring over this white spot. “The snow must be a stain; the spot is une tache, ”thought Rostov. "So much for you ..."

Corresponding to such a vector space. In this article, the first definition will be taken as a starting point.

N (\ displaystyle n)-dimensional Euclidean space is denoted E n, (\ displaystyle \ mathbb (E) ^ (n),) the notation is also often used (if it is clear from the context that the space has a Euclidean structure).

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✪ 04 - Linear Algebra. Euclidean space

✪ Non-Euclidean geometry. Part one.

✪ Non-Euclidean geometry. Part two

✪ 01 - Linear algebra. Linear (vector) space

✪ 8. Euclidean spaces

Subtitles

Formal definition

To define a Euclidean space, it is easiest to take the concept of the scalar product as the main one. Euclidean vector space is defined as a finite-dimensional vector space over the field of real numbers, on the vectors of which a real-valued function is given (⋅, ⋅), (\ displaystyle (\ cdot, \ cdot),) having the following three properties:

An example of Euclidean space - coordinate space R n, (\ displaystyle \ mathbb (R) ^ (n),) consisting of all possible tuples of real numbers (x 1, x 2,…, x n), (\ displaystyle (x_ (1), x_ (2), \ ldots, x_ (n)),) dot product in which is defined by the formula (x, y) = ∑ i = 1 n x i y i = x 1 y 1 + x 2 y 2 + ⋯ + x n y n. (\ displaystyle (x, y) = \ sum _ (i = 1) ^ (n) x_ (i) y_ (i) = x_ (1) y_ (1) + x_ (2) y_ (2) + \ cdots + x_ (n) y_ (n).)

Lengths and angles

The scalar product given on the Euclidean space is sufficient to introduce the geometric concepts of length and angle. Vector length u (\ displaystyle u) defined as (u, u) (\ displaystyle (\ sqrt ((u, u)))) and denoted | u | ... (\ displaystyle | u |.) The positive definiteness of the dot product guarantees that the length of a nonzero vector is nonzero, and bilinearity implies that | a u | = | a | | u | , (\ displaystyle | au | = | a || u |,) that is, the lengths of proportional vectors are proportional.

Angle between vectors u (\ displaystyle u) and v (\ displaystyle v) is determined by the formula φ = arccos ⁡ ((x, y) | x | | y |). (\ displaystyle \ varphi = \ arccos \ left ((\ frac ((x, y)) (| x || y |)) \ right).) It follows from the cosine theorem that for a two-dimensional Euclidean space ( euclidean plane) this definition of the angle coincides with the usual one. Orthogonal vectors, as in three-dimensional space, can be defined as vectors, the angle between which is equal to π 2. (\ displaystyle (\ frac (\ pi) (2)).)

The Cauchy - Bunyakovsky - Schwarz inequality and the triangle inequality

In the definition of the angle given above, there is one space left: in order to arccos ⁡ ((x, y) | x | | y |) (\ displaystyle \ arccos \ left ((\ frac ((x, y)) (| x || y |)) \ right)) was determined, it is necessary that the inequality | (x, y) | x | | y | | ⩽ 1. (\ displaystyle \ left | (\ frac ((x, y)) (| x || y |)) \ right | \ leqslant 1.) This inequality really holds in an arbitrary Euclidean space, it is called the Cauchy - Bunyakovsky - Schwartz inequality. This inequality, in turn, implies the triangle inequality: | u + v | ⩽ | u | + | v | ... (\ displaystyle | u + v | \ leqslant | u | + | v |.) The triangle inequality, together with the length properties listed above, means that the length of the vector is the norm on the Euclidean vector space, and the function d (x, y) = | x - y | (\ displaystyle d (x, y) = | x-y |) defines the structure of a metric space on the Euclidean space (this function is called the Euclidean metric). In particular, the distance between elements (points) x (\ displaystyle x) and y (\ displaystyle y) coordinate space R n (\ displaystyle \ mathbb (R) ^ (n)) is given by the formula d (x, y) = ‖ x - y ‖ = ∑ i = 1 n (x i - y i) 2. (\ displaystyle d (\ mathbf (x), \ mathbf (y)) = \ | \ mathbf (x) - \ mathbf (y) \ | = (\ sqrt (\ sum _ (i = 1) ^ (n) (x_ (i) -y_ (i)) ^ (2))).)

Algebraic properties

Orthonormal bases

Conjugate spaces and operators

Any vector x (\ displaystyle x) Euclidean space defines a linear functional x ∗ (\ displaystyle x ^ (*)) on this space, defined as x ∗ (y) = (x, y). (\ displaystyle x ^ (*) (y) = (x, y).) This mapping is an isomorphism between Euclidean space and

Until now, wanting to know the position of a point on a plane, or in space, we have used the Cartesian coordinate system. So, for example, we determined the position of a point in space using three coordinates. These coordinates were the abscissa, ordinate and applicate of a variable point in space. However, it is clear that specifying the abscissa, ordinate, and applicate of a point is not the only way to determine the position of a point in space. This can be done in another way, for example, using curvilinear coordinates.

Let, according to some well-defined rule, each point M space uniquely corresponds to some triplet of numbers ( q 1 , q 2 , q 3), and different triples of numbers correspond to different points. Then they say that a coordinate system is given in space; the numbers q 1 , q 2 , q 3, which correspond to the point M are called the coordinates (or curvilinear coordinates) of this point.

Depending on the rule according to which the triple of numbers ( q 1 , q 2 , q 3) is put in correspondence to a point in space, they speak of a particular coordinate system.

If you want to note that in a given coordinate system, the position of point M is determined by the numbers q 1 , q 2 , q 3, then it is written as follows M(q 1 , q 2 , q 3).

Example 1. Let some fixed point be marked in space ABOUT(origin), and three mutually perpendicular axes with a scale selected on them are drawn through it. (Axis Ox, Oy, Оz). Three of numbers x, y, z put in line with the dot M, such that the projections of its radius vector OM on the axis Ox, Oy, Оz will be equal respectively x, y, z... This way of establishing a relationship between triples of numbers ( x, y, z) and points M leads us to the well-known Cartesian coordinate system.

It is easy to see that in the case of a Cartesian coordinate system, not only a certain point in space corresponds to each triple of numbers, but also vice versa, a certain triple of coordinates corresponds to each point in space.

Example 2. Let the coordinate axes be drawn in space again Ox, Oy, Оz passing through a fixed point ABOUT(origin).

Consider a triple of numbers r, j, z where r³0; £ 0 j£ 2 p, –¥<z<¥, и поставим в соответствие этой тройке чисел точку M, such that its applicate is z, and its projection onto the plane Oxy has polar coordinates r and j(see fig. 4.1). It is clear that here for every triple of numbers r, j, z corresponds to a certain point M and back, every point M a certain triple of numbers answers r, j, z... The only exceptions are points lying on the axis. Оz: in this case r and z are uniquely determined, and the angle j any meaning can be assigned. The numbers r, j, z are called the cylindrical coordinates of the point M.

It is easy to establish a relationship between cylindrical and Cartesian coordinates:

x = r× cos j; y = r× sin j; z = z.

And back ; ; z = z.

Example 3. Let's introduce a spherical coordinate system. Let's set three numbers r, q, j characterizing the position of the point M in space as follows: r- distance from the origin to the point M(length of the radius vector), q Оz and radius vector OM(point latitude M) j- the angle between the positive direction of the axis Ox and the projection of the radius vector onto the plane Oxy(point longitude M). (See Figure 4.2).

It is clear that in this case, not only each point M matches a certain triple of numbers r, q, j where r³ 0, 0 £ q £ p, 0£ j£ 2 p, but vice versa, each such triplet of numbers corresponds to a certain point in space (again, with the exception of Оz where this unambiguity is violated).

It is easy to find the relationship between spherical and Cartesian coordinates:

x = r sin q cos j; y = r sin q sin j; z = r cos q.

Let's return to an arbitrary coordinate system ( Oq 1 , Oq 2 , Oq 3). We will assume that not only each point in space corresponds to a certain triple of numbers ( q 1 , q 2 , q 3), but vice versa, each triple of numbers corresponds to a certain point in space. Let's introduce the concept of coordinate surfaces and coordinate lines.

Definition... The set of those points for which the coordinate q 1 is constant, called the coordinate surface q one . Coordinate surfaces are defined similarly q 2, and q 3 (see fig. 4.3).

Obviously, if point M has coordinates FROM 1 , FROM 2 , FROM 3 then at this point the coordinate surfaces intersect q 1 =C 1 ; q 2 =C 2 ; q 3 =C 3 .

Definition... The set of those points along which only the coordinate changes q 1 (and the other two coordinates q 2 and q 3 remain constant) is called the coordinate line q 1 .

Obviously, any coordinate line q 1 is the line of intersection of the coordinate planes q 2 and q 3 .

Coordinate lines are defined similarly q 2 and q 3 .

Example 1. Coordinate surfaces (by coordinate x) in the Cartesian coordinate system are all planes x= const. (They are parallel to the plane Oyz). Coordinate surfaces are defined similarly by coordinates y and z.

Coordinate x-line is a straight line parallel to the axis Ox... Coordinate y-line ( z-line) - straight, parallel to the axis OU(axes Оz).

Example 2. Coordinate surfaces in a cylindrical system are: any plane parallel to the plane Oxy(coordinate surface z= const), the surface of a circular cylinder, the axis of which is directed along the axis Оz(coordinate surface r= const) and the half-plane bounded by the axis Оz(coordinate surface j= const) (see Fig.4.4).

The name cylindrical coordinate system is explained by the fact that there are cylindrical surfaces among its coordinate surfaces.

The coordinate lines in this system are z-line - straight, parallel to the axis Оz; j-line - a circle lying in a horizontal plane centered on an axis Оz; and r-line - a ray emerging from an arbitrary point on the axis Оz parallel to the plane Oxy.

Fig. 4.5

Since there are spheres among the coordinate surfaces, this coordinate system is called spherical.

The coordinate lines here are as follows: r-line - a ray outgoing from the origin, q-line - a semicircle centered at the origin, connecting two points on the axis Оz; j-line - a circle lying in the horizontal plane, centered on the axis Оz.

In all the examples discussed above, the coordinate lines passing through any point M, are orthogonal to each other. This does not happen in every coordinate system. However, we restrict ourselves to studying only those coordinate systems for which this is the case; such coordinate systems are called orthogonal.

Definition... Coordinate system ( Oq 1 , Oq 2 , Oq 3) is called orthogonal if at each point M coordinate lines passing through this point intersect at right angles.

Consider now some point M and draw unit vectors tangent at this point to the corresponding coordinate lines and directed towards the increasing direction of the corresponding coordinate. If these vectors form a right triple at each point, then we are given a right coordinate system. So, for example, the Cartesian coordinate system x, y, z(with the usual arrangement of the axes) is right. Also right-handed cylindrical coordinates r, j, z(but precisely with this order of coordinates; if we change the order of the coordinates, taking, for example, r, z, j, we no longer get the right system).

The spherical coordinate system is also right-handed (if we establish such an order of r, q, j).

Note that in the Cartesian coordinate system, the direction of the unit vector does not depend on at what point M we draw this vector; the same is true for vectors. We observe something else in curvilinear coordinate systems: for example, in a cylindrical coordinate system, the vectors at the point M and at some other point M 1 no longer have to be parallel to each other. The same applies to the vector (at different points it has, generally speaking, different directions).

Thus, the triple orthogonal unit vectors in a curvilinear coordinate system depend on the position of the point M in which these vectors are considered. A triplet of unit orthogonal vectors is called a moving frame, and the vectors themselves are called unit vectors (or simply orts).