Chapter I. Vectors on the plane and in space
§ 13. Transition from one rectangular Cartesian coordinate system to another
We offer you to consider this topic in two versions.
1) Based on the textbook by I.I. Privalov "Analytical Geometry" (textbook for higher technical educational institutions, 1966)
I.I. Privalov "Analytical geometry"
§ 1. The problem of coordinate transformation.
The position of a point on the plane is determined by two coordinates relative to some coordinate system. The coordinates of the point will change if we choose a different coordinate system.
The task of transforming coordinates is to to, 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 that relate the coordinates of an arbitrary point in two systems, and the coefficients of these formulas will include constant values that determine the mutual position of the systems.
Let two Cartesian coordinate systems be given hoy And XO 1Y(Fig. 68).
Position of the new system XO 1Y relative to the old system hoy will be determined if the coordinates are known A And b new beginning O 1 according to the old system and the angle α between axles Oh And About 1 X. Denote by X And at coordinates 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 expressed in terms of the new X and Y. The resulting transformation formulas must obviously include the constants a, b And α .
We will obtain the solution of 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, while the origin of coordinates remains unchanged ( a = b = 0).
§ 2. Transfer of the origin.
Let two systems of Cartesian coordinates with different origins be given O And O 1 and the same directions of the axes (Fig. 69).
Denote by A 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 Oh, as well as the point About 1 per axle Oh, we get on the axis Oh three dots Oh, a And R. Segment values OA, AR And OR are related by the following relation:
| OA| + | AR | = | OR |. (1)
Noticing that | | OA| = A , | OR | = X , | AR | = | O 1 R 1 | = X, we rewrite equality (1) in the form:
A + X = x or x = X + A . (2)
Similarly, projecting M and About 1 on the y-axis, we get:
y = Y + b (3)
So, the old coordinate is equal to the new one plus the coordinate of the new origin according to the old system.
From formulas (2) and (3), the new coordinates can be expressed in terms of the old ones:
X = x - a , (2")
Y = y-b . (3")
§ 3. Rotation of coordinate axes.
Let two Cartesian coordinate systems with the same origin be given ABOUT and different directions of the axes (Fig. 70).
Let α is the angle between the axes Oh And OH. Denote by x, y And X, Y coordinates of an arbitrary point M, respectively, in the old and new systems:
X = | OR | , at = | RM | ,
X= | OR 1 |, Y= | R 1 M |.
Consider a broken line OR 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 (Chapter I, § 8), we have:
OR 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 (Chapter I, § 8); therefore, equality (4) will be written as follows:
etc OR 1+ pr R 1 M+ pr MP= | OR | (4")
Since the projection of a directed segment is equal to its value multiplied by the cosine of the angle between the projection axis and the axis on which the segment lies (Chapter I, § 8), then
etc OR 1 = X cos α
etc R 1 M = Y cos (90° + α ) = - Y sin α ,
pr MP= 0.
Hence equality (4") gives us:
x = X cos α - Y sin α . (5)
Similarly, projecting the same broken line onto the axis OU, we get an expression for at. Indeed, we have:
etc OR 1+ pr R 1 M+ pr MP= pr OR = 0.
Noticing that
etc OR 1 = X cos ( α - 90°) = X sin α ,
etc R 1 M = Y cos α ,
pr MP = - y ,
will have:
X sin α + Y cos α - y = 0,
y = X sin 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.
From fig. 71 we have:
X = OP = OM cos ( α + φ ) = OM cos α cos φ - OM sin α sin φ ,
at = PM = OM sin ( α + φ ) = OM sin α cos φ + OM cos α sin φ .
Since (Ch. I, § 11) OM cos φ = X, OM sin φ =Y, That
x = X cos α - Y sin α , (5)
y = X sin α + Y cos α . (6)
§ 4. General case.
Let two Cartesian coordinate systems with different origins and different directions of the axes be given (Fig. 72).
Denote by A 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 through X And Y, we introduce an auxiliary coordinate system x 1 O 1 y 1 , whose beginning we place at the new beginning ABOUT 1 , and take the directions of the axes to coincide with the directions of the old axes. Let 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 = y 1 +b .
X 1 = X cos α - Y sin α , y 1 = X sin 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 - Y sin α + a
y = X sin α + Y cos α + b (I)
Formulas (I) contain, as a special case, the formulas of §§ 2 and 3. Thus, for α = 0 formulas (I) turn into
x = X + A , y = Y + b ,
and at a = b = 0 we have:
x = X cos α - Y sin α , y = X sin 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.
We 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 the old X And at also formulas of the first degree with respect to X And y.
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 one or another coordinate system is not limited by anything and is determined in each particular case only by considerations of convenience. Often the same set has to be considered in different coordinate systems. One and the same point in different systems obviously has 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 in the transition from one coordinate system to another.
Let two rectangular coordinate systems be given on the plane: O, i, j and about", i",j" (Fig. 41).
The first system with origin at point O and basis vectors i And j we agree to call the old one, the second - with the beginning at the point O" and the basis vectors i" And j" - new.
We will consider the position of the new system relative to the old one to be known: let the point O" in the old system have coordinates ( a;b ), a vector i" forms with vector i corner α . Corner α counting in the opposite direction of the clockwise movement.
Consider an arbitrary point M. Denote its coordinates in the old system through ( x;y ), in the new one - through ( x"; y" ). Our task is to establish the relationship between the old and new coordinates of the point M.
Connect in pairs the points O and O", O" and M, O and M. According to the triangle rule, we obtain
OM > = OO" > + O"M > . (1)
Let's decompose the vectors OM> and OO"> by basis vectors i And j , and the vector O"M> 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 basis vectors i" And j" expanded over the old basis vectors i And j in the following way:
i" = cos α i + sin α j ,
j" = cos ( π / 2 + α ) i + sin ( π / 2 + α ) j = - 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 = a + X" cos α
- at" sin α
, |
Formulas (3) give the desired 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 sufficient to solve the system of equations (3) with respect to the unknowns X" And at".
So, the coordinates of the points when moving the origin to the point ( A; b ) and rotate the axes by 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, assuming in formulas (3) α = 0, we get
Formulas (5) are called rotation formulas.
Task 1. Let the coordinates of the new beginning in the old system be (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)
Task 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.
And According to formulas (4), we obtain
|
-
2= a + 5 |
where A = - 7, b = - 2.
Answer. (-7; -2).
Task 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 = 45°.
By formulas (5) we find
Task 4. The 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 = 30°.
By formulas (3) we have
Solving this system of equations for X" And at", we find: X" = 4, at" = -2.
Answer. A(4;-2).
Task 5. Given the equation of a straight line at = 2X - 6. Find the equation of the same line in the new coordinate system, which is obtained from the old system by rotating the axes by an angle α = 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
Coordinates
- these are quantities that determine the position of any point on the surface or in space in the accepted coordinate system. The coordinate system sets the initial (original) points, lines or planes for reading the required quantities - the origin of the coordinates and the units of their calculation. In topography and geodesy, systems of geographic, rectangular, polar and bipolar coordinates have received the greatest application.
Geographic coordinates (Fig. 2.8) are used to determine the position of points on the Earth's surface on an ellipsoid (ball). In this coordinate system, the initial meridian plane and the equatorial plane are the initial ones. A meridian is a line of section of an ellipsoid by a plane passing through a given point and the axis of rotation of the Earth.
A parallel is a line of section of an ellipsoid by a plane passing through a given point and perpendicular to the earth's axis. The parallel whose plane passes through the center of the ellipsoid is called the equator. Through each point lying on the surface of the globe, only one meridian and only one parallel can be drawn.
Geographical coordinates
are angular quantities: longitude l and latitude j.
Geographic longitude l is the dihedral angle enclosed between the plane of the given meridian (passing through point B) and the plane of the initial meridian. For the initial (zero) meridian, the meridian passing through the center of the main hall of the Greenwich Observatory within the city of London was taken. For point B, longitude is determined by the angle l = WCD. Longitudes are counted from the prime meridian in both directions - east and west. In this regard, we distinguish between western and eastern longitudes, which vary from 0° to 180°.
Geographic latitude
j is the angle formed by the plane of the equator and the plumb line passing through the given point. If the Earth is taken as a ball, then for point B (Fig. 2.8) latitude j is determined by the angle DCB. The latitudes measured from the equator to the north are called northern, and to the south - southern, they vary from 0 ° at the equator to 90 ° at the poles.
Geographic coordinates may be derived from astronomical observations or geodetic measurements. In the first case, they are called astronomical, and in the second - geodetic (L - longitude, B - latitude). In astronomical observations, the projection of points onto the reference surface is carried out by plumb lines, in geodetic measurements - by normals. Therefore, the values of astronomical and geodetic coordinates differ by the amount of deviation of the plumb line.
The use of different reference ellipsoids by different states leads to differences in the coordinates of the same points calculated relative to different initial surfaces. In practice, this is expressed in the general displacement of the cartographic image relative to the meridians and parallels on maps of large and medium scales.
Rectangular coordinates
linear quantities are called - the abscissa and the ordinate, which determine the position of a point on the plane relative to the original directions.
(Fig. 2.9)
In geodesy and topography, the right-hand system of rectangular coordinates is adopted. This distinguishes it from the left coordinate system used in mathematics. The initial directions are two mutually perpendicular lines with the origin at the point of their intersection O.
The XX straight line (abscissa axis) is aligned with the direction of the meridian passing through the origin, or with the direction parallel to some meridian. The straight line YY (y-axis) passes through the point O perpendicular to the x-axis. In such a system, the position of a point on a plane is determined by the shortest distance to it from the coordinate axes. The position of point A is determined by the length of the perpendiculars Xa and Ya. The segment Xa is called the abscissa of the point A, and Yа is the ordinate of this point. Rectangular coordinates are usually expressed in meters. The abscissa and ordinate axes divide the terrain at point O into four quarters (Fig. 2.9). The name of the quarters is determined by the accepted designations of the countries of the world. Quarters are numbered clockwise: I - SV; II - SE; III - SW; IV - NW.
In table. 2.3 shows the signs of abscissas X and ordinates Y for points located in different quarters and their names are given.
Table 2.3
The abscissas of points located up from the origin are considered positive, and down from it - negative, the ordinates of points located to the right - positive, to the left - negative. The system of flat rectangular coordinates is used in limited areas of the earth's surface, which can be taken as flat.
Coordinates, the origin of which is any point in the terrain, are called polar. In this coordinate system, orientation angles are measured. On a horizontal plane (Fig. 2.10), through an arbitrarily chosen point O, called the pole, a straight line OX is drawn - the polar axis.
Then the position of any point, for example, M will be determined by the radius - the vector r1 and the direction angle a1, and the point N - respectively r2 and a2. Angles a1 and a2 are measured from the polar axis clockwise to the radius vector. The polar axis can be located arbitrarily or combined with the direction of any meridian passing through the pole O.
The bipolar coordinate system (Fig. 2.11) represents two selected fixed poles O1 and O2, connected by a straight line - the polar axis. This coordinate system allows you to determine the position of the point M relative to the polar axis on the plane using two angles b1 and b2, two radius vectors r1 and r2, or combinations thereof. If the rectangular coordinates of the points O1 and O2 are known, then the position of the point M can be calculated analytically. 
Rice. 2.11
Rice. 2.12
Heights of points on the earth's surface. To determine the position of the points of the physical surface of the Earth, it is not enough to know only the planned coordinates X, Y or l, j, a third coordinate is needed - the height of the point H. The height of the point H (Fig. 2.12) is the distance along the vertical direction from a given point (A´; B´ ´) to the accepted main level surface MN. The numerical value of the height of a point is called elevation. The heights measured from the main level surface MN are called absolute heights (AA´; BB´´), and those determined relative to an arbitrarily chosen level surface are called conditional heights (В´В´´). The height difference between two points or the distance along the vertical direction between level surfaces passing through any two points on the Earth is called the relative height (В´В´´) or the excess of these points h.
In the Republic of Belarus, the Baltic system of heights of 1977 was adopted. The heights are counted from the level surface, which coincides with the average water level in the Gulf of Finland, from the zero of the Kronstadt footstock.
Here's another
For determining point positions in geodesy use spatial rectangular, geodetic, and planar rectangular coordinates.
Spatial rectangular coordinates. The origin of the coordinate system is located in the center O earth ellipsoid(Fig. 2.2).
Axis Z directed along the axis of rotation of the ellipsoid to the north. Axis X lies at the intersection of the equatorial plane with the initial - Greenwich meridian. Axis Y directed perpendicular to the axes Z And X to the East.
Geodetic coordinates. The geodetic coordinates of a point are its latitude, longitude and height (Fig. 2.2).
Geodetic latitude points M called the angle IN, formed by the normal to the surface of the ellipsoid passing through the given point, and the plane of the equator.
Latitude is measured from the equator north and south from 0° to 90° and is called north or south. North latitude is considered positive, and south latitude is negative.
Sectional planes of an ellipsoid passing through an axis oz, are called geodetic meridians.
Geodetic longitude points M called a dihedral angle L, formed by the planes of the initial (Greenwich) geodesic meridian and the geodesic meridian of the given point.
Longitudes are measured from the prime meridian within the range from 0° to 360° east, or from 0° to 180° east (positive) and from 0° to 180° west (negative).
Geodetic height points M is her height H above the surface of the earth's ellipsoid.
Geodetic coordinates with spatial rectangular coordinates are related by the formulas
X=(N+H)cos B cos L,
Y=(N+H)cos B sin L,
Z=[(1- e 2)N+H] sin B,
Where e is the first eccentricity of the meridian ellipse and N-radius of curvature of the first vertical. N=a/(1 - e 2 sin 2 B) 1/2 .
Geodetic and spatial rectangular coordinates of points are determined using satellite measurements, as well as by linking them with geodetic measurements to points with known coordinates.
Note that along with with geodesics there are also astronomical latitude and longitude. Astronomical latitude j is the angle made by the plumb line at the given point with the equatorial plane. Astronomical longitude l is the angle between the planes of the Greenwich meridian and the astronomical meridian passing through the plumb line at a given point. Astronomical coordinates are determined on the ground from astronomical observations.
Astronomical coordinates differ from geodesics because the directions of the plumb lines do not coincide with the directions of the normals to the surface of the ellipsoid. The angle between the direction of the normal to the surface of the ellipsoid and the plumb line at a given point on the earth's surface is called plumb line.
A generalization of geodetic and astronomical coordinates is the term - geographical coordinates.
Planar rectangular coordinates. To solve the problems of engineering geodesy from spatial and geodetic coordinates, they switch to simpler - flat coordinates, which make it possible to depict the terrain on a plane and determine the position of points with two coordinates X And at.
Since the convex surface of the Earth it is impossible to depict on a plane without distortion, the introduction of flat coordinates is possible only in limited areas where the distortions are so small that they can be neglected. In Russia, a system of rectangular coordinates is adopted, the basis of which is an equiangular transverse-cylindrical Gaussian projection. The surface of an ellipsoid is depicted on a plane in parts called zones. The zones are spherical bicagons bounded by meridians and extending from the north pole to the south (Fig. 2.3). The size of the zone in longitude is 6°. The central meridian of each zone is called the axial meridian. The zones are numbered from Greenwich to the east.
The longitude of the axial meridian of the zone with the number N is equal to:
l 0 \u003d 6 ° × N - 3 °.
Axial meridian of the zone and equator are depicted on a plane by straight lines (Fig. 2.4). The axial meridian is taken as the abscissa axis x, and the equator - for the y-axis y. Their intersection (point O) serves as the origin of the given zone.
To avoid negative ordinate values, the intersection coordinates are taken equal to x 0 = 0, y 0 = 500 km, which is equivalent to an axis shift X west for 500 km.
So that by the rectangular coordinates of a point it is possible to judge in which zone it is located, to the ordinate y on the left, the number of the coordinate zone is assigned.
Let, for example, the coordinates of the point A look like:
x A= 6 276 427 m
y A= 12 428 566 m
These coordinates indicate at what point A located at a distance of 6276427 m from the equator, in the western part ( y < 500 км) 12-ой координатной зоны, на расстоянии 500000 - 428566 = 71434 м от осевого меридиана.
For spatial rectangular, geodetic and flat rectangular coordinates in Russia, a unified coordinate system SK-95 has been adopted, fixed on the ground by points of the state geodetic network and built on satellite and ground-based measurements as of the epoch of 1995.
Local systems of rectangular coordinates. During the construction of various objects, local (conditional) coordinate systems are often used, in which the directions of the axes and the origin of coordinates are assigned based on the convenience of their use during the construction and subsequent operation of the object.
So, when shooting railway station axis at are directed along the axis of the main railway track in the direction of increasing picketage, and the axis X- along the axis of the passenger station building.
During construction Bridge Crossing Axis X usually combined with the axis of the bridge, and the axis y goes in a perpendicular direction.
During construction large industrial and civil facilities axis x And y directed parallel to the axes of the buildings under construction.
To solve most problems in applied sciences, it is necessary to know the location of an object or point, which is determined using one of the accepted coordinate systems. In addition, there are elevation systems that also determine the altitude location of a point on
What are coordinates
Coordinates are numeric or literal values that can be used to determine the location of a point on the terrain. As a consequence, a coordinate system is a set of values of the same type that have the same principle for finding a point or object.
Finding the location of a point is required to solve many practical problems. In a science such as geodesy, determining the location of a point in a given space is the main goal, on the achievement of which all subsequent work is based.
Most coordinate systems, as a rule, define the location of a point on a plane limited by only two axes. In order to determine the position of a point in three-dimensional space, a system of heights is also used. With its help, you can find out exact location desired object.
Briefly about coordinate systems used in geodesy
Coordinate systems determine the location of a point on a territory by giving it three values. The principles of their calculation are different for each coordinate system.
The main spatial coordinate systems used in geodesy:
- Geodetic.
- Geographic.
- Polar.
- Rectangular.
- Zonal Gauss-Kruger coordinates.
All systems have their own starting point, values for the location of the object and scope.
Geodetic coordinates
The main figure used to read geodetic coordinates is the earth's ellipsoid.
An ellipsoid is a three-dimensional compressed figure that best represents the figure of the globe. Due to the fact that the globe is a mathematically incorrect figure, it is the ellipsoid that is used instead to determine geodetic coordinates. This facilitates the implementation of many calculations to determine the position of the body on the surface.
Geodetic coordinates are defined by three values: geodetic latitude, longitude, and altitude.
- Geodetic latitude is an angle whose beginning lies on the plane of the equator, and the end lies at the perpendicular drawn to the desired point.
- Geodetic longitude is the angle that is measured from the zero meridian to the meridian on which the desired point is located.
- Geodetic height - the value of the normal drawn to the surface of the ellipsoid of the Earth's rotation from a given point.
Geographical coordinates
To solve high-precision problems of higher geodesy, it is necessary to distinguish between geodetic and geographical coordinates. In the system used in engineering geodesy, such differences, due to the small space covered by the work, as a rule, do not.
An ellipsoid is used as a reference plane to determine geodetic coordinates, and a geoid is used to determine geographical coordinates. The geoid is a mathematically incorrect figure, closer to the actual figure of the Earth. For its leveled surface, they take that which is continued under sea level in its calm state.
The geographic coordinate system used in geodesy describes the position of a point in space with three values. longitude coincides with the geodesic, since the reference point will also be called Greenwich. It passes through the observatory of the same name in the city of London. determined from the equator drawn on the surface of the geoid.
Height in the local coordinate system used in geodesy is measured from sea level in its calm state. On the territory of Russia and the countries of the former Union, the mark from which the heights are determined is the Kronstadt footstock. It is located at the level of the Baltic Sea.
Polar coordinates
The polar coordinate system used in geodesy has other nuances of the product of measurements. It is used in small areas of terrain to determine the relative location of a point. The reference point can be any object marked as a source. Thus, using polar coordinates, it is impossible to determine the unambiguous location of a point on the territory of the globe.
Polar coordinates are defined by two quantities: angle and distance. The angle is measured from the north direction of the meridian to a given point, determining its position in space. But one angle will not be enough, so a radius vector is introduced - the distance from the standing point to the desired object. With these two options, you can determine the location of the point in the local system.
As a rule, this coordinate system is used for engineering work carried out on a small area of area.
Rectangular coordinates
The rectangular coordinate system used in geodesy is also used in small areas of the terrain. The main element of the system is the coordinate axis from which the reference is made. The coordinates of a point are found as the length of perpendiculars drawn from the abscissa and ordinate axes to the desired point.
The north direction of the x-axis and the east of the y-axis are considered positive, and the south and west are negative. Depending on the signs and quarters, the location of a point in space is determined.
Gauss-Kruger coordinates
The Gauss-Kruger coordinate zonal system is similar to the rectangular one. The difference is that it can be applied to the entire territory of the globe, and not just to small areas.
The rectangular coordinates of the Gauss-Kruger zones, in fact, are the projection of the globe onto a plane. It arose for practical purposes to depict large areas of the Earth on paper. Transferring distortions are considered insignificant.
According to this system, the globe is divided by longitude into six-degree zones with the axial meridian in the middle. The equator is in the center along a horizontal line. As a result, there are 60 such zones.
Each of the sixty zones has its own system of rectangular coordinates, measured along the ordinate axis from X, and along the abscissa - from the area of the earth's equator Y. To unambiguously determine the location on the territory of the entire globe, the zone number is put in front of the X and Y values.
The values of the x-axis in Russia are usually positive, while the values of y can be negative. In order to avoid the minus sign in the values of the abscissa axis, the axial meridian of each zone is conditionally moved 500 meters to the west. Then all coordinates become positive.
The coordinate system was proposed by Gauss as possible and calculated mathematically by Krüger in the middle of the twentieth century. Since then, it has been used in geodesy as one of the main ones.
Height system
The systems of coordinates and heights used in geodesy are used to accurately determine the position of a point on the Earth. Absolute heights are measured from sea level or other surface taken as the original. In addition, there are relative heights. The latter are counted as an excess from the desired point to any other. It is convenient to use them for working in the local coordinate system in order to simplify the subsequent processing of the results.
Application of coordinate systems in geodesy
In addition to the above, there are other coordinate systems used in geodesy. Each of them has its own advantages and disadvantages. There are also their own areas of work for which this or that method of determining the location is relevant.
It is the purpose of the work that determines which coordinate systems used in geodesy are best used. For work in small areas, it is convenient to use rectangular and polar coordinate systems, and for solving large-scale problems, systems are needed that allow covering the entire territory of the earth's surface.
Origin
Origin(reference point) in Euclidean space - a singular point, usually denoted by the letter ABOUT, which is used as a reference point for all other points. In Euclidean geometry, the origin of coordinates can be chosen arbitrarily at any convenient point.
A vector drawn from the origin to another point is called a radius vector.
Cartesian coordinate system
The origin of coordinates divides each of the axes into two beams - a positive semi-axis and a negative semi-axis.
In particular, the origin can be entered on the number axis. In this sense, we can talk about the origin of coordinates for various extensive quantities (time, temperature, etc.)
Polar coordinate systems
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See what the "Origin of coordinates" is in other dictionaries:
origin- Zero point (point of intersection of the axes) in a flat coordinate system used in graphic systems that work with two-dimensional images. The point coordinate is set by the distance from the origin (center) of coordinates along the horizontal X axis (abscissa) ... ...
origin- koordinačių pradžia statusas T sritis automatika atitikmenys: engl. origin of coordinates vok. Koordinatenanfangspunkt, m; Koordinatenursprung, m rus. origin, npranc. origine de cordonnées, f … Automatikos terminų žodynas
origin (plotter)- — [E.S. Alekseev, A.A. Myachev. English Russian explanatory dictionary of computer systems engineering. Moscow 1993] Topics information Technology in general EN plot origin … Technical Translator's Handbook
- (origin) The point on the graph that represents zero for any measurement. A chart can have more than one reference point. A two-factor square diagram (box diagram), for example, is constructed in such a way that the total available volumes of any factors ... Economic dictionary
directional resistance relay with characteristic not passing through the origin- - [V.A. Semenov. English Russian dictionary of relay protection] Topics relay protection EN offset mho distance relay ... Technical Translator's Handbook
characteristic of a directional resistance relay in the form of a circle passing through the origin- — [Ya.N. Luginsky, M.S. Fezi Zhilinskaya, Yu.S. Kabirov. English Russian Dictionary of Electrical Engineering and Power Industry, Moscow, 1999] Electrical engineering topics, basic concepts EN mho characteristic ... Technical Translator's Handbook
reference point- The position on the display screen from which all coordinate systems start. Usually located in the upper left corner of the screen. Topics information technology in general EN origin … Technical Translator's Handbook
Rectangular coordinate system is a rectilinear coordinate system with mutually perpendicular axes on a plane or in space. The simplest and therefore most commonly used coordinate system. It is very easily and directly generalized for ... ... Wikipedia
A point has three Cartesian and three spherical coordinates. It is convenient to define a spherical coordinate system by referring to q ... Wikipedia
A set of definitions that implements the coordinate method, that is, a way to determine the position of a point or body using numbers or other symbols. The set of numbers that determine the position of a particular point is called the coordinates of this point. In ... ... Wikipedia
Books
- Spring, Stefania Danilova, Poet Stefania Danilova was born on August 16, 1994 in St. Petersburg, and is unconditionally in love with this city. Ambidextrous, child prodigy, polyglot, who created the first adult poem at the age of three.… Category: Modern Russian poetry Series: Runet Star Publisher: AST,
- Providence, Rogatko Sergey Alexandrovich, The new novel "Promysl" by the writer Sergei Rogatko, who professes a realistic beginning in Russian literature and confirmed this in his famous novel "Layman", is written in the genre of a parable, "... Category: