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's 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 an intersection line, 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 designate 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. On this 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 natural way- This is to directly measure the lengths of the arcs with a flexible ruler, and this is possible if the dimensions of the sphere are comparable to the dimensions 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?
A rectangular (other names are flat, two-dimensional) coordinate system, named after the French scientist Descartes (1596-1650) “Cartesian coordinate system on the plane,” is formed by the intersection on the plane at right angles (perpendicular) of two numerical axes so that the positive semi-axis of one one is directed to the right (x-axis, or abscissa axis), and the second is directed upward (y-axis, or ordinate axis).
The intersection point of the axes coincides with the 0 point of each of them and is called the origin of coordinates.
For each of the axes, an arbitrary scale (a single length segment) is selected. Each point on the plane corresponds to one pair of numbers, called the coordinates of this point on the plane. Conversely, any ordered pair of numbers corresponds to one point on the plane for which these numbers are coordinates.
The first coordinate of a point is called the abscissa of that point, and the second coordinate is called the ordinate.
The entire coordinate plane is divided into 4 quadrants (quarters). Quadrants are located from the first to the fourth counterclockwise (see figure).
To determine the coordinates of a point, you need to find its distance to the abscissa and ordinate axis. Since the distance (shortest) is determined by the perpendicular, then from the point two perpendiculars (auxiliary lines on the coordinate plane) are lowered onto the axis so that the point of their intersection is the location given point in the coordinate plane. The points of intersection of perpendiculars with axes are called projections of the point on the coordinate axes.
The first quadrant is limited by the positive semi-axes of abscissa and ordinate. Therefore, the coordinates of the points in this quarter of the plane will be positive
(signs "+" and
For example, point M (2; 4) in the figure above.
The second quadrant is limited by the negative x-axis and the positive y-axis. Consequently, the coordinates of points along the abscissa axis will be negative (sign “-”), and along the ordinate axis they will be positive (sign “+”).
For example, point C (-4; 1) in the figure above.
The third quadrant is limited by the negative x-axis and the negative y-axis. Consequently, the coordinates of the points along the abscissa and ordinate axis will be negative (signs “-” and “-”).
For example, point D (-6; -2) in the figure above.
The fourth quadrant is limited by the positive x-axis and the negative y-axis. Consequently, the coordinates of the points along the abscissa axis will be positive (the “+” sign). and along the ordinate axis - negative (sign “-”).
For example, point R (3; -3) in the figure above.
Constructing a point using its specified coordinates
we will find the first coordinate of the point on the x-axis and draw an auxiliary line through it - a perpendicular;
we find the second coordinate of the point on the ordinate axis and draw an auxiliary line through it - a perpendicular;
the point of intersection of two perpendiculars (auxiliary lines) will correspond to the point with the given coordinates.
A rectangular coordinate system on a plane is formed by two mutually perpendicular coordinate axes X’X and Y’Y. The coordinate axes intersect at point O, which is called the origin, a positive direction is selected on each axis. The positive direction of the axes (in a right-handed coordinate system) is chosen so that when the X'X axis is rotated counterclockwise by 90°, its positive direction coincides with the positive direction of the Y'Y axis. The four angles (I, II, III, IV) formed by the coordinate axes X'X and Y'Y are called coordinate angles (see Fig. 1).
The position of point A on the plane is determined by two coordinates x and y. The x coordinate is equal to the length of the segment OB, the y coordinate is equal to the length of the segment OC in the selected units of measurement. Segments OB and OC are defined by lines drawn from point A parallel to the Y'Y and X'X axes, respectively. The x coordinate is called the abscissa of point A, the y coordinate is called the ordinate of point A. It is written as follows: A(x, y).
If point A lies in coordinate angle I, then point A has a positive abscissa and ordinate. If point A lies in coordinate angle II, then point A has a negative abscissa and a positive ordinate. If point A lies in coordinate angle III, then point A has a negative abscissa and ordinate. If point A lies in coordinate angle IV, then point A has a positive abscissa and a negative ordinate.
Rectangular coordinate system in space is formed by three mutually perpendicular coordinate axes OX, OY and OZ. The coordinate axes intersect at point O, which is called the origin, on each axis a positive direction is selected, indicated by arrows, and a unit of measurement for the segments on the axes. The units of measurement are the same for all axes. OX - abscissa axis, OY - ordinate axis, OZ - applicate axis. The positive direction of the axes is chosen so that when the OX axis is rotated counterclockwise by 90°, its positive direction coincides with the positive direction of the OY axis, if this rotation is observed from the positive direction of the OZ axis. Such a coordinate system is called right-handed. If thumb right hand take the X direction as the X direction, the index one as the Y direction, and the middle one as the Z direction, then a right-handed coordinate system is formed. Similar fingers of the left hand form the left coordinate system. It is impossible to combine the right and left coordinate systems so that the corresponding axes coincide (see Fig. 2).
The position of point A in space is determined by three coordinates x, y and z. The x coordinate is equal to the length of the segment OB, the y coordinate is the length of the segment OC, the z coordinate is the length of the segment OD in the selected units of measurement. The segments OB, OC and OD are defined by planes drawn from point A parallel to the planes YOZ, XOZ and XOY, respectively. The x coordinate is called the abscissa of point A, the y coordinate is called the ordinate of point A, the z coordinate is called the applicate of point A. It is written as follows: A(a, b, c).
Orty
A rectangular coordinate system (of any dimension) is also described by a set of unit vectors aligned with the coordinate axes. The number of unit vectors is equal to the dimension of the coordinate system and they are all perpendicular to each other.
In the three-dimensional case, such unit vectors are usually denoted i j k or e x e y e z. In this case, in the case of a right-handed coordinate system, the following formulas with the vector product of vectors are valid:
- [i j]=k ;
- [j k]=i ;
- [k i]=j .
Story
The rectangular coordinate system was first introduced by Rene Descartes in his work “Discourse on Method” in 1637. Therefore, the rectangular coordinate system is also called - Cartesian coordinate system. The coordinate method of describing geometric objects marked the beginning of analytical geometry. Pierre Fermat also contributed to the development of the coordinate method, but his works were first published after his death. Descartes and Fermat used the coordinate method only on the plane.
The coordinate method for three-dimensional space was first used by Leonhard Euler already in the 18th century.
see also
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See what “Cartesian coordinate system” is in other dictionaries:
CARTESIAN COORDINATE SYSTEM, a rectilinear coordinate system on a plane or in space (usually with mutually perpendicular axes and equal scales along the axes). Named after R. Descartes (see DESCARTES Rene). Descartes first introduced... encyclopedic Dictionary
CARTESIAN COORDINATE SYSTEM- a rectangular coordinate system on a plane or in space, in which the scales along the axes are the same and the coordinate axes are mutually perpendicular. D. s. K. is denoted by the letters x:, y for a point on a plane or x, y, z for a point in space. (Cm.… …
CARTESIAN COORDINATE SYSTEM, a system introduced by Rene DESCARTES, in which the position of a point is determined by the distance from it to mutually intersecting lines (axes). In the simplest version of the system, the axes (denoted x and y) are perpendicular.... ... Scientific and technical encyclopedic dictionary
Cartesian coordinate system
A rectilinear coordinate system (See Coordinates) on a plane or in space (usually with equal scales along the axes). R. Descartes himself in “Geometry” (1637) used only a coordinate system on a plane (in general, oblique). Often… … Great Soviet Encyclopedia
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 specific point is called the coordinates of this point. In... ... Wikipedia
cartesian system- Dekarto koordinačių sistema statusas T sritis fizika atitikmenys: engl. Cartesian system; Cartesian system of coordinates vok. cartesisches Koordinatensystem, n; kartesisches Koordinatensystem, n rus. Cartesian system, f; Cartesian system... ... Fizikos terminų žodynas
COORDINATE SYSTEM- a set of conditions that determine the position of a point on a straight line, on a plane, in space. There are various linear shapes: Cartesian, oblique, cylindrical, spherical, curvilinear, etc. Linear and angular values, determining the position... ... Big Polytechnic Encyclopedia
Orthonormal rectilinear coordinate system in Euclidean space. D.p.s. on a plane is specified by two mutually perpendicular straight coordinate axes, on each of which a positive direction is chosen and a segment of the unit ... Mathematical Encyclopedia
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. Very easily and directly summarized for... ... Wikipedia
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Construction of a Cartesian rectangular coordinate system
on surface
A Cartesian rectangular coordinate system in a plane is formed by two mutually perpendicular coordinate axes OX 1 And OX 2 , which intersect at the point O, called the origin of coordinates (Fig. 1). On each axis, a positive direction is selected, indicated by arrows, and a unit of measurement for the segments on the axes. The units are usually the same for all axes (which is not mandatory). IN right-sided coordinate system, the positive direction of the axes is chosen so that when the axis is directed OX 2 up, axis OX 1 looked to the right. OX 1 -- abscissa axis, OX 2 -- ordinate axis. Four corners (I, II, III, IV) formed by the coordinate axes OX 1 And OX 2 , are called coordinate angles or quadrants.
Dot B A to the coordinate axis OX 1 ;
Dot C - orthographic projection points A to the coordinate axis OX 2 ;
Construction of a Cartesian rectangular coordinate system in space
The Cartesian rectangular coordinate system in space is formed by three mutually perpendicular coordinate axes OX, OY And OZ. The coordinate axes intersect at the point O, which is called the origin of coordinates, on each axis a positive direction is selected, indicated by arrows, and a unit of measurement for the segments on the axes. The units are usually the same for all axes (which is not mandatory). OX-- abscissa axis, OY-- ordinate axis, OZ-- applicator axis.
If the thumb of the right hand is taken as the direction X, index - for direction Y and the middle one is for the direction Z, then it is formed right coordinate system. Similar fingers of the left hand form the left coordinate system. In other words, the positive direction of the axes is chosen so that when the axis rotates OX counterclockwise by 90° its positive direction coincides with the positive direction of the axis OY, if this rotation is observed from the positive direction of the axis OZ. It is impossible to combine the right and left coordinate systems so that the corresponding axes coincide (Fig. 2). Dot F- orthogonal projection of a point A to the coordinate plane OXY; Dot E- orthogonal projection of a point A to the coordinate plane OYZ; Dot G- orthogonal projection of a point A to the coordinate plane OX Z ;
Layout representation of Cartesian rectangular coordinate system in space shown in Figures 3, 4 and 5.
Determining the coordinates of a point in the Cartesian rectangular coordinate system
The main question of any coordinate system is the question of determining the coordinates of a point located in its plane or space.
Determining the coordinates of a point on a plane Cartesian coordinate system
Point position A on the plane is determined by two coordinates - x And y (Fig. 5). Coordinate x equal to the length of the segment O.B., coordinate y -- length of the segment O.C. in selected units of measurement. Segments O.B. And O.C. are determined by lines drawn from the point A parallel to the axes OY And OX respectively. Coordinate x called abscissa (lat. abscissa- segment), coordinate y -- ordinate (lat. ordinates- located in order) points A. Write it like this:
If the point A lies in the coordinate angle I, then it has positive abscissa and ordinate. If the point A lies in coordinate angle II, then there is a negative abscissa and a positive ordinate. If the point A lies in coordinate angle III, then it has negative abscissa and ordinate. If the point A lies in the coordinate angle IV, then there is a positive abscissa and a negative ordinate.
This is how coordinates are determined in the Cartesian coordinate system on a plane.
