On surface.

Local properties of curvilinear 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 arc differential, will in these coordinates have the form corresponding to the identity 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 have three features $q\_1$, $q\_2$, $q\_3$ served as coordinates in some 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 some domain of sets $\backslash left(q\_1,\backslash ;q\_2,\backslash ;q\_3\backslash right)$ coordinates.

Local basis and tensor analysis

In tensor calculus, one can introduce local basis vectors: $\backslash mathbf(R\_j)=\backslash frac(d\backslash mathbf\; r)(dy^j)=\; \backslash frac(dx^i)(dy^j)\; \backslash mathbf\; e\_i=Q^i\_j\; \backslash mathbf\; e\_i$, where $\backslash mathbf\; e\_i$- orts of the Cartesian coordinate system, $Q^i\_j$ is the Jacobian matrix, $x^i$ coordinates in the Cartesian system, $y^i$- input curvilinear coordinates.
It is not difficult to see that the curvilinear coordinates, generally speaking, vary from point to point.
Let us indicate the formulas for the connection 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 local basis vectors 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 symbol
Thus any tensor field $\backslash mathbf\; T$ of rank n can be expanded in a local polyad 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 tensor field of the first rank (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 formulas relating 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 $\backslash mathbb(R)^2$

Lame coefficients

We write the arc differential 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 $H\_i\backslash$, depending on a point in space, are called Lame coefficients or scale factors. The Lame 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 Lame coefficients:

Examples

Polar coordinates ( n=2)

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

Connection of polar coordinates with Cartesian:

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

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 not defined. If the coordinate φ is considered not as a number, but as an angle (a point on a unit circle), then 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:

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

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 related to latitude and longitude coordinates on the unit sphere. Connection of spherical coordinates with Cartesian:

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

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 coordinates, do not work on the z-axis (x=0, y=0) because 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.

Name of coordinates Types of coordinate systems 2D coordinates 3D coordinates $n$-dimensional coordinates Physical coordinates Related definitions

An excerpt characterizing the Curvilinear Coordinate System

“If he could attack us, he would do it today,” he said.
“So you think he is powerless,” said Langeron.
“A lot, if he has 40,000 troops,” Weyrother answered with the smile of a doctor to whom the doctor wants to point out a remedy.
“In that case, he goes 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 was thinking 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 chuckled again with that smile that said that it was ridiculous and strange for him to meet objections from the Russian generals and to prove what not only he himself was too sure of, but what the emperors were sure of.
“The enemy has put out the fires, and there is a continuous noise in his camp,” he said. - What does it mean? “Either he moves away, which is the only thing we should be afraid of, or he changes position (he chuckled). But even if he did take a position in Tyuras, he only saves us a lot of trouble, and the orders, down to the smallest detail, remain the same.
“In what way? ..” said Prince Andrei, 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 is already the first hour), cannot be changed,” he said. “You have heard her, and we will all do our duty. And before the battle, there is nothing more important ... (he paused) how to get a good night's sleep.
He pretended to get up. The generals bowed and retired. It was past midnight. Prince Andrew left.

The military council, at which Prince Andrei failed to express his opinion, as he hoped, left an unclear and disturbing impression on him. Who was right: Dolgorukov with Weyrother or Kutuzov with Langeron 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 differently? Is it really necessary to risk tens of thousands and my, my life because of court and personal considerations? he thought.
"Yes, it's very possible they'll kill you tomorrow," he thought. And suddenly, at this thought of death, a whole series of recollections, the most distant and most sincere, rose in his imagination; he remembered the last farewell to his father and wife; he remembered the first days of his love for her! He remembered her pregnancy, and he felt sorry for both her and himself, and in a nervously 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 misty, and moonlight shone mysteriously through the mist. “Yes, tomorrow, tomorrow! he thought. “Tomorrow, perhaps, everything will be over for me, all these memories will no longer exist, all these memories will no longer have any meaning for me. Tomorrow, maybe, even probably tomorrow, I foresee it, for the first time I will finally have to show everything that I can do. And he imagined the battle, the loss of it, the concentration of the battle on one point and the confusion of all 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 both Kutuzov, and Weyrother, and the emperors. Everyone is amazed at the correctness of his ideas, but no one undertakes to fulfill it, and so he takes a regiment, a division, pronounces a condition that no one should interfere with his orders, and leads his division to a decisive point and alone wins. What about death and suffering? says another voice. But Prince Andrei does not answer this voice and continues his successes. The disposition of the next battle is made by him alone. He bears the rank of army duty officer 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 says again, and then, if you are not wounded, killed or deceived ten times before; well, then what? “Well, and then,” Prince Andrei answers himself, “I don’t know what will happen next, I don’t want and I can’t know: but if I want this, I want glory, I want to be famous people I want to be loved by them, then it's not my fault that I want this, that I want this alone, 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 are - my father, sister, wife - the people dearest to me - but, no matter how terrible and unnatural it seems, I will give them all now for a moment of glory, triumph over people, for love for to myself people whom I do not know and will not know, for the love of these people, ”he thought, listening to the conversation in Kutuzov’s yard. In the yard of Kutuzov, the voices of orderlies packing up were heard; one voice, probably the coachman, teasing the old Kutuzovsky cook, whom Prince Andrei knew, and whose name was Tit, said: “Tit, and Tit?”
“Well,” replied the old man.
“Titus, go thresh,” said the joker.
“Pah, well, to hell with them,” a voice was heard, covered with the laughter of batmen and servants.
“And yet I love and cherish only the triumph over all of them, I cherish this mysterious power and glory, which here rushes over me in this fog!”

Rostov that night was with a platoon in the flanker chain, ahead of Bagration's detachment. His hussars were scattered in pairs in chains; he himself rode along this line of chain, trying to overcome the sleep that irresistibly drooped him. Behind him one could see a huge expanse of fires of our army burning indistinctly in the fog; ahead of him was misty darkness. No matter how much Rostov peered into this foggy distance, he did not see anything: it turned gray, then something seemed to blacken; then flashed like lights, where the enemy should be; then he thought that it was only in his eyes that it glittered. His eyes were closed, and in his imagination he imagined either the sovereign, then Denisov, then Moscow memories, and again he 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, having met me, will give an order, as he would to any officer: he will say: “Go, find out what is there.” They told a lot how, quite by accident, he recognized some officer in such a way 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 the enemy or deceiver of the German, whom he delightedly not only killed, but beat on the cheeks in the eyes of the sovereign. Suddenly a distant cry woke Rostov. He winced and opened his eyes.
"Where I am? Yes, in the chain: the slogan and the password are the drawbar, Olmutz. What a pity that our squadron will be in reserve tomorrow... he thought. - I'll ask to work. This may be the only chance to see the sovereign. Yes, it's not long before the change. I’ll go around again and, when I get back, I’ll go to the general and ask him.” He recovered in the saddle and touched the horse to go around his hussars once more. He thought it was brighter. On the left side one could see a gentle, illuminated slope and the opposite, black hillock, which seemed steep, like a wall. There was a white spot on this hillock, which Rostov could not understand in any way: was it a clearing in the forest, illuminated by the moon, or the remaining snow, or white houses? It even seemed to him that something stirred over this white spot. “The snow must be a stain; the stain is une tache, thought Rostov. “Here you don’t tash ...”

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

point coordinates M surfaces serve as numbers u, v, where u-- numerical mark of the line of the first family passing through M, And v-- marking lines of the second family. We will continue to write: M(u; v), 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 be covered all over by meridians (the 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 corresponds to a numerical mark - latitude v(or and). Through each point of the sphere (excluding the poles) there is only one meridian and one parallel.

As another example, consider the lateral surface of a right circular cylinder of height H, radius a(Fig. 23). For the first family we will take the system of its generators, one of them will be taken as the initial one. We assign a mark to each generatrix 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 take the system of horizontal sections of the surface; numerical mark 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) 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. According to what curve should a piece of tin be cut to make a drainpipe 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?

Solution. Let us use the parametric equations of the cylinder surface:

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

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

So, a piece of tin should be outlined from above along a sinusoid

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

Both in the case of a sphere and a cylindrical surface, and in the general case, the specification of a 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) surfaces through two parameters u, v(This is generally written like this: X\u003d c ( u; v),y= c (u;v), z=u (u;v), ts, sh, u - functions of two arguments) makes it possible, knowing a pair of numbers u, v, find matching coordinates x, y, z, so the position of the point M on the surface; numbers u, v serve as its coordinates. Giving one of them a constant value, like 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 . Just the same line v=v 0 -- coordinate line of another family.

coordinate cartesian radius vector

Rectangular spatial system of Cartesian coordinates

Transformations of spatial rectangular coordinate systems

Linear mapping transformations

Reducing a general quadratic form to a canonical one

Curvilinear coordinates

General information about curvilinear coordinate systems

Curvilinear coordinates on the surface

Polar coordinate systems and their generalizations

Spatial polar coordinate system

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

Terrestrial 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 planned and altitude positions of points on the earth's surface

Converting spatial geodetic polar coordinates to ellipsoidal geodetic coordinates

Converting reference systems of geodetic coordinates to global and vice versa

Spatial rectangular coordinate systems

Relationship of spatial rectangular coordinates with ellipsoidal geodetic coordinates

Converting spatial rectangular reference coordinates to global and vice versa

Topocentric coordinate systems in geodesy

Relationship of spatial topocentric horizontal geodesic CS with spatial polar spherical coordinates

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

Systems of flat rectangular coordinates in geodesy

Relation of planar rectangular Gauss–Krüger coordinates to ellipsoidal geodetic coordinates

Gauss–Kruger planar rectangular coordinate transformation from one zone to another

Recalculation of flat rectangular coordinates of points of local geodetic constructions to other systems of flat rectangular coordinates

Chapter 4

Coordinate systems of spherical astronomy

Reference systems in space geodesy

Stellar (celestial) inertial geocentric equatorial coordinates

Greenwich terrestrial geocentric system of spatial rectangular coordinates

Topocentric coordinate systems

Chapter 5

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

Construction of the State geodetic network

BIBLIOGRAPHY

APPENDIX 1. SOLUTION OF THE DIRECT GEODESIC PROBLEM IN SPACE

APPENDIX 2. SOLUTION OF THE INVERSE GEODESIC PROBLEM IN SPACE

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

APPENDIX 4

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

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

APPENDIX 7. CONVERSION OF THE SPATIAL POLAR COORDINATES OF THE SYSTEM S, ZG, A INTO TOPOCENTRIC HORIZONTAL GEODETIC COORDINATES ХТ, УТ, ZТ

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

APPENDIX 9. CONVERSION OF TOPOCENTRIC HORIZONTAL GEODETIC COORDINATES XT, UT, ZT INTO SPATIAL RECTANGULAR COORDINATES X, Y, Z

APPENDIX 10. CONVERSION OF ELLIPSOIDAL GEODETIC COORDINATES B, L INTO FLAT RECTANGULAR COORDINATES GAUSS - KRUGER X, Y

APPENDIX 11. CONVERSION OF PLANE RECTANGULAR COORDINATES GAUSS - KRUGER 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, i.e.

D \u003d (a 11 + a 22) 2 - 4 (a 11a 22 - a 12 a 21) \u003d (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 own numbers matrices A. We substitute the eigenvalues ​​found from (2.57) into (2.39), we obtain

canonical equation.

Given a quadratic form in the form: F (x x ) = 5x 2				2x2.

Find the canonical form of this equation.

Since here a 11 = 5; and 21 = 2; and 22 = 2, then the characteristic equation (2.56) for the 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 obtain λ1 = 6; λ2 = 1.

And then the canonical form of this quadratic form will look like

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

2.3. Curvilinear coordinates

2.3.1. General information about 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, as it expands the possibilities of the method of rectilinear coordinates. The use of curvilinear coordinates can sometimes greatly simplify the solution of many problems, especially problems that are 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, in the area of ​​specifying this function on a given surface, you can choose such a system of curvilinear coordinates that will allow you to endow this function a new property is to be constant in a given coordinate system, which is not always possible with the use of rectilinear coordinate systems.

The system of curvilinear coordinates, given in some area of ​​three-dimensional Euclidean space, assigns to each point of this space an ordered triple of real numbers - φ, λ, r (curvilinear coordinates of a point).

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

Between the system of curvilinear coordinates φ, λ, r and rectilinear Cartesian CS (X, Y, Z) there must exist mathematical connection. Indeed, let the system of curvilinear coordinates be given in some region of space. Each point of this space corresponds to a single triple of curvilinear coordinates - φ, λ, r. On the other hand, the same point corresponds to the only triple of rectilinear Cartesian coordinates - X, Y, Z. Then it can be argued that in general view

ϕ \u003d 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 SCs.

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

ϕ \u003d ϕ (X, Y, Z) \u003d const,

And variable values ​​of the other two (λ, r ), we get in general a surface, which is called a coordinate one. Coordinate surfaces corresponding to the same coordinate do not intersect. 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 the surface

For geodesy, surface curvilinear coordinates are of the greatest interest.

Let the surface equation be a function of Cartesian coordinates in

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

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

r \u003d 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

Rice. 2.11. Curvilinear Surface Coordinate System

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

If we give φ various constant values ​​φ = φ1 , φ = φ2 , …, then we get a family of curves on the surface corresponding to these constants. Similarly, given 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 quantities. The simplest example of a system of curvilinear coordinates, in which one coordinate is a linear quantity, and the other is an angular quantity, can serve as polar coordinates on a plane.

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

A well-defined network of coordinate lines also corresponds to a certain system of parameters, but for each given family of coordinate lines, many other parameters can be selected that are continuous and single-valued functions of this parameter. In the general case, the angles between the coordinate lines of the family φ = const and the lines of the family λ = const 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 related to the calculation of the curvilinear coordinates of surface points, it is necessary to have differential equations changes in the curvilinear coordinates φ and λ depending on the change in the length S of the surface curve.

The relationship 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

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

given point on the surface. Then (no output):

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

coordinates

					∂X		2 ∂ Y 2
E = (rϕ )
						∂ϕ		∂ϕ
G = (						∂X		∂ U 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 polar coordinate system

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

measurements

segments

space, then

position

any point in space will

definitely

determined

quantities: r, φ, λ, where r is

polar

straight-line distance from the pole

O to point Q (Fig. 2.13); λ -

polar angle is the angle between

polar

Rice. 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

basic

plane, considered positive (0 ≤ φ ≤ π/2) for points of the positive half-space and negative (-π/2 ≤ φ ≤ 0) for points of the negative half-space.

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

If we take the scale and the origin of the polar system as the scale and origin of coordinates in the spatial rectangular system, the polar axis OP - as the semi-axis of the abscissa OX , the line OZ drawn from the pole O perpendicular to the main plane in the positive direction of the polar system - as the semi-axis OZ of the rectangular Cartesian system, and for the semi-axis - OS, take the axis into which the abscissa axis passes when it is rotated through 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 us to express X, Y, Z in terms of r, φ, λ and vice versa

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 triple of numbers ( q 1 , q 2 , q 3), and different points correspond to different triples of numbers. Then we say that a coordinate system is given in space; numbers q 1 , q 2 , q 3 that correspond to the point M, are called coordinates (or curvilinear coordinates) of this point.

Depending on the rule by which the triple of numbers ( q 1 , q 2 , q 3) is put in correspondence with a point in space, they talk about one or another coordinate system.

If you want to note that in a given coordinate system, the position of the 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(the origin), and three mutually perpendicular axes are drawn through it with the scale chosen on them. (axes Ox, Oy, Oz). Three of a kind x, y, z match the dot M, such that the projections of its radius vector OM on axle Ox, Oy, Oz will be equal respectively x, y, z. This way of establishing a relationship between triplets 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 each triple of numbers corresponds to a certain point in space, but vice versa, each point in space corresponds to a certain triple of coordinates.

Example 2. Let the coordinate axes be again drawn in space Ox, Oy, Oz 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 equal to z, and its projection onto the plane Oxy has polar coordinates r And j(see figure 4.1). It is clear that here each triple of numbers r, j, z corresponds to a certain point M and vice versa, each point M answers a certain triple of numbers r, j, z. The exceptions are the points lying on the axis Oz: in this case r And z are uniquely defined, and the corner j any value can be assigned. 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. Set three numbers r, q, j characterizing the position of the point M in space as follows: r is the distance from the origin of coordinates to the point M(length of the radius vector), q Oz and radius vector OM(latitude point M) j is 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 every point M corresponds to a certain triplet of numbers r, q, j, where r³ 0, 0 £ q £ p, 0£ j£2 p, but vice versa, each such triple of numbers corresponds to a certain point in space (again, with the exception of the points of the axis Oz where this uniqueness is violated).

It's 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 us 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 . The coordinate surfaces are defined similarly q 2 , and q 3 (see fig. 4.3).

Obviously, if the point M has coordinates FROM 1 , FROM 2 , FROM 3 then the coordinate surfaces intersect at this point 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 (along the coordinate x) in the Cartesian coordinate system are all planes x= const. (They are parallel to the plane Oyz). The coordinate surfaces are defined similarly by the coordinates y And z.

coordinate x a line is a straight line parallel to the axis Ox. coordinate y-line ( z-line) - a straight line parallel to the axis OU(axes Oz).

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

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

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

Rice. 4.5

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

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

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 confine ourselves to studying only such 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 touching at this point the corresponding coordinate lines and directed in the direction of increasing the corresponding coordinate. If these vectors form a right triple at each point, then we are given a right coordinate system. For example, the Cartesian coordinate system x, y, z(with the usual arrangement of axes) is right. Also right hand cylindrical coordinate system r, j, z(but precisely with this order of coordinates; if you change the order of the coordinates, taking, for example, r, z, j, we no longer obtain a right system).

The spherical coordinate system is also right (if you set such an order r, q, j).

Note that in the Cartesian coordinate system, the direction of the unit vector does not depend on which 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, vectors at a 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 of unit orthogonal vectors in a curvilinear coordinate system depends on the position of the point M, in which these vectors are considered. A triple of unit orthogonal vectors is called a moving frame, and the vectors themselves are called unit orts (or simply orts).

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

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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Subtitles

Formal definition

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

Euclidean space example - 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)),) scalar product in which is determined 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 inner product guarantees that the length of a non-zero vector is non-zero, and it follows from the bilinearity 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)).)

Cauchy-Bunyakovsky-Schwarz inequality and triangle inequality

There is one gap left in the definition of angle given above: in order to arccos ⁡ ((x , y) | x | | y |) (\displaystyle \arccos \left((\frac ((x,y))(|x||y|))\right)) was defined, it is necessary that the inequality | (x, y) | x | | y | | ≤ 1. (\displaystyle \left|(\frac ((x,y))(|x||y|))\right|\leqslant 1.) This inequality is indeed satisfied in an arbitrary Euclidean space, it is called the Cauchy - Bunyakovsky - Schwarz inequality. From this inequality, in turn, follows 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 a vector is a norm on a 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)) 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

Dual 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 the Euclidean space and