Geographical coordinates. Shape and size of the earth. coordinate systems. Heights
Polar coordinate system determined by specifying a certain point O, called the pole, emanating from this point of the ray O.A.(also denoted as Ox), called the polar axis, and a scale for changing lengths. In addition, when specifying a polar coordinate system, it must be determined which rotations around the point O are considered positive (in drawings, counterclockwise turns are usually considered positive).
So, let’s select a certain point on the plane (figure above) O(pole) and some ray emerging from it Ox. In addition, we indicate the scale unit. Polar coordinates of a point M two numbers ρ and φ are called, the first of which (the polar radius ρ) is equal to the distance of the point M from the pole O, and the second (polar angle φ, which is also called amplitude) is the angle by which the beam must be rotated counterclockwise Ox before aligning with the beam OM.
Full stop M with polar coordinates ρ and φ are designated by the symbol M(ρ, φ) .
Relationship between polar coordinates and Cartesian coordinates
Let's install relationship between the polar coordinates of a point and its Cartesian coordinates . We will assume that the origin of the Cartesian rectangular coordinate system is at the pole, and the positive semi-axis of the abscissa coincides with the polar axis. Let the point M has Cartesian coordinates x And y and polar coordinates ρ and φ. Then
x= ρ cos φ)
y= ρ sin φ) .
Polar coordinates ρ and φ of a point M are determined by its Cartesian coordinates as follows:
In order to find the value of the angle φ, you need to use the signs x And y, determine the quadrant in which the point is located M, and, in addition, take advantage of the fact that the tangent of the angle φ is equal to .
The above formulas are called formulas for the transition from Cartesian to polar coordinates.
Problems about points in the polar coordinate system
Example 1.
A(3; π /4) ;
B(2; -π /2) ;
C(3; -π /3) .
Find the polar coordinates of points symmetrical to these points about the polar axis.
Solution. With symmetry, the length of the beam does not change. Consequently, the first coordinate - the length of the ray - for a point symmetrical relative to the polar axis will be the same as for the given point. As can be seen from the figure at the beginning of the lesson, when constructing a point symmetrical relative to the polar axis, this point must be rotated around the polar axis by the same angle φ. Consequently, in the polar coordinate system, the second coordinate of the symmetrical point will be the angle for the original point, taken with the opposite sign, that is, -φ. So, the polar coordinates of a point symmetrical to a given one relative to the polar axis will differ only in the second coordinate, and this coordinate will have the opposite sign. The polar coordinates of the required symmetrical points will be as follows:
A"(3; -π /4) ;
B"(2; π /2) ;
C"(3; π /3) .
Example 2. In the polar coordinate system, points are given on the plane
A(1; π /4) ;
B(5; π /2) ;
C(2; -π /3) .
Find the polar coordinates of points symmetrical to these points relative to the pole.
Solution. With symmetry, the length of the beam does not change. Consequently, the first coordinate - the length of the ray - for a point symmetrical relative to the pole will be the same as for the given point. A point symmetrical relative to the pole is obtained by rotating the starting point 180 degrees counterclockwise, that is, by the angle π . Consequently, the second coordinate of a point symmetrical to the given one relative to the pole is calculated as φ + π (if the result is a numerator greater than the denominator, then subtract one full revolution from the resulting number, that is, 2 π ). We obtain the following coordinates of points symmetrical to the data relative to the pole:
A"(1; 3π /4) ;
B"(5; -π /2) ;
C"(2; 2π /3) .
Example 3. The pole of the polar coordinate system coincides with the origin of Cartesian rectangular coordinates, and the polar axis coincides with the positive semi-axis of the abscissa. Points are given in the polar coordinate system
A(6; π /2) ;
B(5; 0) ;
C(2; π /4) .
Find the Cartesian coordinates of these points.
Solution. We use formulas for the transition from polar coordinates to Cartesian:
x= ρ cos φ)
y= ρ sin φ) .
We obtain the following Cartesian coordinates of these points:
A(0; 6) ;
B(5; 0) ;
C"(√2; √2) .
Example 4. The pole of the polar coordinate system coincides with the origin of Cartesian rectangular coordinates, and the polar axis coincides with the positive semi-axis of the abscissa. Points are given in the Cartesian rectangular coordinate system
A(0; 5) ;
B(-3; 0) ;
C(√3; 1) .
Find the polar coordinates of these points.
Coordinate systems used in topography: geographic, flat rectangular, polar and bipolar coordinates, their essence and use
Coordinates are called angular and linear quantities (numbers) that determine the position of a point on any surface or in space.
In topography, coordinate systems are used that make it possible to most simply and unambiguously determine the position of points earth's surface both from the results of direct measurements on the ground and using maps. Such systems include geographic, flat rectangular, polar and bipolar coordinates.
Geographical coordinates(Fig.1) - angular values: latitude (Y) and longitude (L), which determine the position of an object on the earth’s surface relative to the origin of coordinates - the point of intersection of the prime (Greenwich) meridian with the equator. On a map, the geographic grid is indicated by a scale on all sides of the map frame. The western and eastern sides of the frame are meridians, and the northern and southern sides are parallels. In the corners of the map sheet the geographical coordinates of the points of intersection of the sides of the frame are written.
Rice. 1. System of geographical coordinates on the earth's surface
In the geographic coordinate system, the position of any point on the earth's surface relative to the origin of coordinates is determined in angular measure. In our country and in most other countries, the point of intersection of the prime (Greenwich) meridian with the equator is taken as the beginning. Being thus uniform for our entire planet, the system of geographical coordinates is convenient for solving problems by determining mutual position objects located at significant distances from each other.
Therefore, in military affairs this system is used mainly for conducting calculations related to the use of combat weapons. long range, for example, ballistic missiles, aviation, etc.
Plane rectangular coordinates(Fig. 2) - linear quantities that determine the position of an object on a plane relative to the accepted origin of coordinates - the intersection of two mutually perpendicular lines ( coordinate axes X and Y).
In topography, each 6-degree zone has its own system of rectangular coordinates. The X axis is the axial meridian of the zone, the Y axis is the equator, and the point of intersection of the axial meridian with the equator is the origin of coordinates.
Rice. 2. System of flat rectangular coordinates on maps
The plane rectangular coordinate system is zonal; it is established for each six-degree zone into which the Earth’s surface is divided when depicting it on maps in the Gaussian projection, and is intended to indicate the position of images of points of the earth’s surface on a plane (map) in this projection.
The origin of coordinates in a zone is the point of intersection of the axial meridian with the equator, relative to which the position of all other points in the zone is determined in a linear measure. The origin of the zone and its coordinate axes occupy a strictly defined position on the earth's surface. Therefore, the system of flat rectangular coordinates of each zone is connected both with the coordinate systems of all other zones, and with the system of geographical coordinates.
The use of linear quantities to determine the position of points makes the system of flat rectangular coordinates very convenient for carrying out calculations both when working on the ground and on a map. Therefore, this system is most widely used among the troops. Rectangular coordinates indicate the position of terrain points, their battle formations and targets, and with their help determine the relative position of objects within one coordinate zone or in adjacent areas of two zones.
Polar and bipolar coordinate systems are local systems. In military practice, they are used to determine the position of some points relative to others in relatively small areas of the terrain, for example, when designating targets, marking landmarks and targets, drawing up terrain diagrams, etc. These systems can be associated with systems of rectangular and geographic coordinates.
If we introduce a coordinate system on a plane or in three-dimensional space, we will be able to describe geometric figures and their properties using equations and inequalities, that is, we will be able to use algebra methods. Therefore, the concept of a coordinate system is very important.
In this article we will show how a rectangular Cartesian coordinate system is defined on a plane and in three-dimensional space and find out how the coordinates of points are determined. For clarity, we provide graphic illustrations.
Page navigation.
Rectangular Cartesian coordinate system on a plane.
Let us introduce a rectangular coordinate system on the plane.
To do this, draw two mutually perpendicular lines on the plane and select on each of them positive direction, indicating it with an arrow, and select on each of them scale(unit of length). Let us denote the point of intersection of these lines by the letter O and consider it starting point. So we got rectangular coordinate system on surface.
Each of the straight lines with a selected origin O, direction and scale is called coordinate line or coordinate axis.
A rectangular coordinate system on a plane is usually denoted by Oxy, where Ox and Oy are its coordinate axes. The Ox axis is called x-axis, and the Oy axis – y-axis.
Now let's agree on the image of a rectangular coordinate system on a plane.
Typically, the unit of measurement of length on the Ox and Oy axes is chosen to be the same and is laid off from the origin on each coordinate axis in the positive direction (marked with a dash on the coordinate axes and the unit is written next to it), the abscissa axis is directed to the right, and the ordinate axis is directed upward. All other options for the direction of the coordinate axes are reduced to the voiced one (Ox axis - to the right, Oy axis - up) by rotating the coordinate system at a certain angle relative to the origin and looking at it from the other side of the plane (if necessary).
The rectangular coordinate system is often called Cartesian, since it was first introduced on the plane by Rene Descartes. Even more commonly, a rectangular coordinate system is called a rectangular Cartesian coordinate system, putting it all together.
Rectangular coordinate system in three-dimensional space.
The rectangular coordinate system Oxyz is set in a similar way in three-dimensional Euclidean space, only not two, but three mutually perpendicular lines are taken. In other words, a coordinate axis Oz is added to the coordinate axes Ox and Oy, which is called axis applicate.
Depending on the direction of the coordinate axes, right and left rectangular coordinate systems in three-dimensional space are distinguished.
If viewed from the positive direction of the Oz axis and the shortest rotation from the positive direction of the Ox axis to the positive direction of the Oy axis occurs counterclockwise, then the coordinate system is called right.
If viewed from the positive direction of the Oz axis and the shortest rotation from the positive direction of the Ox axis to the positive direction of the Oy axis occurs clockwise, then the coordinate system is called left.
Coordinates of a point in a Cartesian coordinate system on a plane.
First, consider the coordinate line Ox and take some point M on it.
Each real number corresponds to a single point M on this coordinate line. For example, a point located on a coordinate line at a distance from the origin in the positive direction corresponds to the number , and the number -3 corresponds to a point located at a distance of 3 from the origin in the negative direction. The number 0 corresponds to the starting point.
On the other hand, each point M on the coordinate line Ox corresponds to a real number. This real number is zero if point M coincides with the origin (point O). This real number is positive and equal to the length of the segment OM on a given scale if point M is removed from the origin in the positive direction. This real number is negative and equal to the length of the segment OM with a minus sign if point M is removed from the origin in the negative direction.
The number is called coordinate points M on the coordinate line.
Now consider a plane with the introduced rectangular Cartesian coordinate system. Let us mark an arbitrary point M on this plane.
Let be the projection of point M onto the line Ox, and let be the projection of point M onto the coordinate line Oy (if necessary, see the article). That is, if through the point M we draw lines perpendicular to the coordinate axes Ox and Oy, then the points of intersection of these lines with the lines Ox and Oy are points and, respectively.
Let the number correspond to a point on the Ox coordinate axis, and the number to a point on the Oy axis.
Each point M of the plane in a given rectangular Cartesian coordinate system corresponds to a unique ordered pair of real numbers, called coordinates of point M on surface. The coordinate is called abscissa of point M, A - ordinate of point M.
The converse statement is also true: each ordered pair of real numbers corresponds to a point M on the plane in a given coordinate system.
Coordinates of a point in a rectangular coordinate system in three-dimensional space.
Let us show how the coordinates of point M are determined in a rectangular coordinate system defined in three-dimensional space.
Let and be the projections of point M onto the coordinate axes Ox, Oy and Oz, respectively. Let these points on the coordinate axes Ox, Oy and Oz correspond to real numbers and.
Projections of the point M onto the coordinate axes can also be obtained by constructing planes perpendicular to the lines Ox, Oy and Oz and passing through the point M. These planes will intersect the coordinate lines Ox, Oy and Oz at points and, respectively.
Each point in three-dimensional space in a given Cartesian coordinate system corresponds to an ordered triple of real numbers, called coordinates of point M, the numbers are called abscissa, ordinate And applicate points M respectively. The converse statement is also true: each ordered triple of real numbers in a given rectangular coordinate system corresponds to a point M in three-dimensional space.
Bibliography.
- Atanasyan L.S., Butuzov V.F., Kadomtsev S.B., Poznyak E.G., Yudina I.I. Geometry. Grades 7 – 9: textbook for general education institutions.
- Atanasyan L.S., Butuzov V.F., Kadomtsev S.B., Kiseleva L.S., Poznyak E.G.. Geometry. Textbook for 10-11 grades of secondary school.
- Mordkovich A.G. Algebra. 7th grade. Part 1: textbook for students of general education institutions.
Determining the position of a point in space
So, the position of a point in space can only be determined in relation to some other points. The point relative to which the position of other points is considered is called reference point . We will also use another name for the reference point - observation point . Usually a reference point (or an observation point) is associated with some coordinate system , which is called reference system. In the selected reference system, the position of EACH point is determined by THREE coordinates.
Right-hand Cartesian (or rectangular) coordinate system
This coordinate system consists of three mutually perpendicular directed lines, also called coordinate axes , intersecting at one point (origin). The origin point is usually denoted by the letter O.
The coordinate axes are named:
1. Abscissa axis – designated as OX;
2. Y axis – denoted as OY;
3. Applicate axis – designated as OZ
Now let us explain why this coordinate system is called right-handed. Let's look at the XOY plane from the positive direction of the OZ axis, for example from point A, as shown in the figure.
Let's assume that we begin to rotate the OX axis around point O. So - the right coordinate system has such a property that if you look at the XOY plane from any point on the positive semi-axis OZ (for us this is point A), then, when turning OX axis by 90 counterclockwise, its positive direction will coincide with the positive direction of the OY axis.
This decision was made in scientific world, we just have to accept it as it is.
So, after we have decided on the reference system (in our case, the right-hand Cartesian coordinate system), the position of any point is described through the values of its coordinates or, in other words, through the values of the projections of this point on the coordinate axes.
It is written like this: A(x, y, z), where x, y, z are the coordinates of point A.
A rectangular coordinate system can be thought of as the lines of intersection of three mutually perpendicular planes.
It should be noted that you can orient a rectangular coordinate system in space in any way you like, and only one condition must be met - the origin of coordinates must coincide with the reference center (or observation point).
Spherical coordinate system
The position of a point in space can be described in another way. Let's assume that we have chosen a region of space in which the reference point O (or observation point) is located, and we also know the distance from the reference point to a certain point A. Let's connect these two points with a straight line OA. This line is called radius vector and is denoted as r. All points that have the same radius vector value lie on a sphere, the center of which is at the reference point (or observation point), and the radius of this sphere is equal, respectively, to the radius vector.
Thus, it becomes obvious to us that knowing the value of the radius vector does not give us an unambiguous answer about the position of the point of interest to us. You need TWO more coordinates, because to unambiguously determine the location of a point, the number of coordinates must be THREE.
Next we will proceed as follows - we will construct two mutually perpendicular planes, which, naturally, will give a line of intersection, and this line will be infinite, because the planes themselves are not limited by anything. Let's set a point on this line and designate it, for example, as point O1. Now let’s combine this point O1 with the center of the sphere – point O and see what happens?
And it turns out a very interesting picture:
· Both one and the other planes will be central planes.
· The intersection of these planes with the surface of the sphere is denoted by big circles
· One of these circles - arbitrarily, we will call EQUATOR, then the other circle will be called MAIN MERIDIAN.
· The line of intersection of two planes will uniquely determine the direction LINES OF THE MAIN MERIDIAN.
We denote the points of intersection of the line of the main meridian with the surface of the sphere as M1 and M2
Through the center of the sphere, point O in the plane of the main meridian, we draw a straight line perpendicular to the line of the main meridian. This straight line is called POLAR AXIS .
The polar axis will intersect the surface of the sphere at two points called POLES OF THE SPHERE. Let's denote these points as P1 and P2.
Determining the coordinates of a point in space
Now we will consider the process of determining the coordinates of a point in space, and also give names to these coordinates. To complete the picture, when determining the position of a point, we indicate the main directions from which the coordinates are counted, as well as the positive direction when counting.
1. Set the position in space of the reference point (or observation point). Let's denote this point with the letter O.
2. Construct a sphere whose radius is equal to the length of the radius vector of point A. (The radius vector of point A is the distance between points O and A). The center of the sphere is located at the reference point O.
3. We set the position in space of the EQUATOR plane, and accordingly the plane of the MAIN MERIDIAN. It should be recalled that these planes are mutually perpendicular and are central.
4. The intersection of these planes with the surface of the sphere determines for us the position of the circle of the equator, the circle of the main meridian, as well as the direction of the line of the main meridian and the polar axis.
5. Determine the position of the poles of the polar axis and the poles of the main meridian line. (The poles of the polar axis are the points of intersection of the polar axis with the surface of the sphere. The poles of the line of the main meridian are the points of intersection of the line of the main meridian with the surface of the sphere).
6. Through point A and the polar axis we construct a plane, which we will call the plane of the meridian of point A. When this plane intersects with the surface of the sphere, a large circle will be obtained, which we will call the MERIDIAN of point A.
7. The meridian of point A will intersect the circle of the EQUATOR at some point, which we will designate as E1
8. The position of point E1 on the equatorial circle is determined by the length of the arc enclosed between points M1 and E1. The countdown is COUNTERclockwise. The arc of the equatorial circle enclosed between points M1 and E1 is called the LONGITUDE of point A. Longitude is denoted by the letter .
Let's sum up the intermediate results. At the moment, we know TWO of THREE coordinates that describe the position of point A in space - this is the radius vector (r) and longitude (). Now we will determine the third coordinate. This coordinate is determined by the position of point A on its meridian. But the position of the starting point from which the counting takes place is not clearly defined: we can start counting both from the pole of the sphere (point P1) and from point E1, that is, from the point of intersection of the meridian lines of point A and the equator (or in other words - from the equator line).
In the first case, the position of point A on the meridian is called POLAR DISTANCE (denoted as R) and is determined by the length of the arc enclosed between point P1 (or the pole point of the sphere) and point A. The counting is carried out along the meridian line from point P1 to point A.
In the second case, when the countdown is from the equator line, the position of point A on the meridian line is called LATITUDE (denoted as and is determined by the length of the arc enclosed between point E1 and point A.
Now we can finally say that the position of point A in a spherical coordinate system is determined by:
· sphere radius length (r),
length of the arc of longitude (),
arc length of polar distance (p)
In this case, the coordinates of point A will be written as follows: A(r, , p)
If we use a different reference system, then the position of point A in the spherical coordinate system is determined through:
· sphere radius length (r),
length of the arc of longitude (),
· arc length of latitude ()
In this case, the coordinates of point A will be written as follows: A(r, , )
Methods for measuring arcs
The question arises - how do we measure these arcs? The simplest and most natural way is to directly measure the lengths of the arcs with a flexible ruler, and this is possible if the size of the sphere is comparable to the size of a person. But what to do if this condition is not met?
In this case, we will resort to measuring the RELATIVE arc length. We will take the circumference as a standard, part which is the arc we are interested in. How can I do that?
Coordinate system- a way to specify points in space using numbers. The number of numbers needed to uniquely determine any point in space determines its dimension. A mandatory element of the coordinate system is origin- the point from which distances are counted. Another required element is the unit of length, which allows you to measure distances. All points of one-dimensional space can be specified with a selected origin using one number. For two-dimensional space, two numbers are needed, for three-dimensional space, three. These numbers are called coordinates.
1. History
The development of coordinate systems in the history of mankind is associated with both mathematical problems and practical problems in the art of navigation, based on cartography and astronomy. Known system coordinates, rectangular, was proposed by Rene Descartes in the year. The concept of a polar coordinate system in European mathematics developed around these times, but the first ideas about it existed in Ancient Greece, in medieval Arab mathematicians who developed methods for calculating the direction of the Kaaba.
The emergence of the concept of coordinate systems led to the development of new sections of geometry: analytical, projective, descriptive.
2. Cartesian coordinate system
The most common coordinate system in mathematics is the Cartesian coordinate system, named after René Descartes. The Cartesian coordinate system is specified by the origin and three vectors that determine the direction of the coordinate axes. Each point in space is specified by numbers that correspond to the distance from this point to coordinate planes.
The coordinates of the Cartesian system on a hollow are usually denoted by In space.
Various Cartesian coordinate systems are interconnected by affine transformations: displacement and rotations.
3. Curvilinear coordinate systems
Based on the Cartesian coordinate system, it is possible to define a curvilinear coordinate system, that is, for example, for a three-dimensional space of numbers associated with Cartesian coordinates:
,where all functions are single-valued and continuously differentiated, and the Jacobian is:
. An example of a curvilinear coordinate system on a plane is the polar coordinate system, in which the position of a point is specified by two numbers: the distance between the point and the origin, and the angle between the ray that connects the origin to the point and the selected axis. Cartesian and polar coordinates of a point are related to each other by the formulas:
, ,For three-dimensional space, cylindrical and spherical coordinate systems are popular. Thus, the position of an aircraft in space can be specified by three numbers: altitude, distance to the point on the surface of the Earth over which it flies, and the angle between the direction towards the aircraft and the direction to the north. This task corresponds to a cylindrical coordinate system. Alternatively, the position of the aircraft can be specified by the distance to it and two angles: polar and azimuthal. This task corresponds to a spherical coordinate system.
The variety of coordinate systems is not limited to those listed. There are many curvilinear coordinate systems that are convenient for use when solving one or another mathematical problem.
3.1. Properties
Each of the equations specifies coordinate plane. The intersection of two coordinate planes with different i sets coordinate line. Each point in space is defined by the intersection of three coordinate planes.
Important characteristics of curvilinear coordinate systems are the length of the arc element and the volume element in them. These quantities are used in integration. The length of the arc element is given by the quadratic form:
,They are components of the metric tensor.
The volume element is equal in the curvilinear coordinate system
. The square of the Jacobian is equal to the determinant of the metric tensor:
. The coordinate system is called right, if they touch the coordinate lines and are directed in the direction of growth of the corresponding coordinates, they form a right-hand triple of vectors.
When describing vectors in a curvilinear coordinate system, it is convenient to use a local basis defined at each point.
4. In geography
6. In physics
To describe the movement of physical bodies, physics uses the concept
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