An ordered system of two or three intersecting axes perpendicular to each other with a common origin (origin) and a common unit of length is called rectangular Cartesian coordinate system .
General Cartesian coordinate system (affine coordinate system) may also include not necessarily perpendicular axes. In honor of the French mathematician Rene Descartes (1596-1662), such a coordinate system is named in which a common unit of length is counted on all axes and the axes are straight.
Rectangular Cartesian coordinate system on the plane has two axes rectangular Cartesian coordinate system in space - three axes. Each point on a plane or in space is determined by an ordered set of coordinates - numbers in accordance with the unit length of the coordinate system.
Note that, as follows from the definition, there is a Cartesian coordinate system on a straight line, that is, in one dimension. The introduction of Cartesian coordinates on a straight line is one of the ways in which any point on a straight line is assigned a well-defined real number, that is, a coordinate.
The method of coordinates, which arose in the works of René Descartes, marked a revolutionary restructuring of all mathematics. It became possible to interpret algebraic equations (or inequalities) in the form of geometric images (graphs) and, conversely, to search for a solution to geometric problems using analytical formulas, systems of equations. Yes, inequality z < 3 геометрически означает полупространство, лежащее ниже плоскости, параллельной координатной плоскости xOy and located above this plane by 3 units.
With the help of the Cartesian coordinate system, the belonging of a point to a given curve corresponds to the fact that the numbers x and y satisfy some equation. So, the coordinates of a point of a circle centered at a given point ( a; b) satisfy the equation (x - a)² + ( y - b)² = R² .
Rectangular Cartesian coordinate system on the plane
Two perpendicular axes on a plane with a common origin and the same scale unit form Cartesian coordinate system on the plane . One of these axes is called the axis Ox, or x-axis , the other - the axis Oy, or y-axis . These axes are also called coordinate axes. Denote by Mx and My respectively the projection of an arbitrary point M on axle Ox and Oy. How to get projections? Pass through the dot M Ox. This line intersects the axis Ox at the point Mx. Pass through the dot M straight line perpendicular to the axis Oy. This line intersects the axis Oy at the point My. This is shown in the figure below.
x and y points M we will call respectively the magnitudes of the directed segments OMx and OMy. The values of these directional segments are calculated respectively as x = x0 - 0 and y = y0 - 0 . Cartesian coordinates x and y points M abscissa and ordinate . The fact that the dot M has coordinates x and y, is denoted as follows: M(x, y) .
The coordinate axes divide the plane into four quadrant , whose numbering is shown in the figure below. It also indicates the arrangement of signs for the coordinates of points, depending on their location in one or another quadrant.
In addition to Cartesian rectangular coordinates in the plane, the polar coordinate system is also often considered. About the method of transition from one coordinate system to another - in the lesson polar coordinate system .
Rectangular Cartesian coordinate system in space
Cartesian coordinates in space are introduced in complete analogy with Cartesian coordinates on a plane.
Three mutually perpendicular axes in space (coordinate axes) with a common origin O and the same scale unit form Cartesian rectangular coordinate system in space .
One of these axes is called the axis Ox, or x-axis , the other - the axis Oy, or y-axis , third - axis Oz, or applicate axis . Let be Mx, My Mz- projections of an arbitrary point M spaces on the axis Ox , Oy and Oz respectively.
Pass through the dot M OxOx at the point Mx. Pass through the dot M plane perpendicular to the axis Oy. This plane intersects the axis Oy at the point My. Pass through the dot M plane perpendicular to the axis Oz. This plane intersects the axis Oz at the point Mz.
Cartesian rectangular coordinates x , y and z points M we will call respectively the magnitudes of the directed segments OMx, OMy and OMz. The values of these directional segments are calculated respectively as x = x0 - 0 , y = y0 - 0 and z = z0 - 0 .
Cartesian coordinates x , y and z points M are named accordingly abscissa , ordinate and applique .
Taken in pairs, the coordinate axes are located in the coordinate planes xOy , yOz and zOx .
Problems about points in the Cartesian coordinate system
Example 1
A(2; -3) ;
B(3; -1) ;
C(-5; 1) .
Find the coordinates of the projections of these points on the x-axis.
Decision. As follows from the theoretical part of this lesson, the projection of a point onto the x-axis is located on the x-axis itself, that is, the axis Ox, and therefore has an abscissa equal to the abscissa of the point itself, and an ordinate (coordinate on the axis Oy, which the x-axis intersects at point 0), equal to zero. So we get the following coordinates of these points on the x-axis:
Ax(2;0);
Bx(3;0);
Cx(-5;0).
Example 2 Points are given in the Cartesian coordinate system on the plane
A(-3; 2) ;
B(-5; 1) ;
C(3; -2) .
Find the coordinates of the projections of these points on the y-axis.
Decision. As follows from the theoretical part of this lesson, the projection of a point onto the y-axis is located on the y-axis itself, that is, the axis Oy, and therefore has an ordinate equal to the ordinate of the point itself, and an abscissa (the coordinate on the axis Ox, which the y-axis intersects at point 0), equal to zero. So we get the following coordinates of these points on the y-axis:
Ay(0; 2);
By (0; 1);
Cy(0;-2).
Example 3 Points are given in the Cartesian coordinate system on the plane
A(2; 3) ;
B(-3; 2) ;
C(-1; -1) .
Ox .
Ox Ox Ox, will have the same abscissa as the given point, and the ordinate equal in absolute value to the ordinate of the given point, and opposite in sign to it. So we get the following coordinates of points symmetrical to these points about the axis Ox :
A"(2; -3) ;
B"(-3; -2) ;
C"(-1; 1) .
Solve problems on the Cartesian coordinate system yourself, and then look at the solutions
Example 4 Determine in which quadrants (quarters, figure with quadrants - at the end of the paragraph "Rectangular Cartesian coordinate system on the plane") the point can be located M(x; y) , if
1) xy > 0 ;
2) xy < 0 ;
3) x − y = 0 ;
4) x + y = 0 ;
5) x + y > 0 ;
6) x + y < 0 ;
7) x − y > 0 ;
8) x − y < 0 .
Example 5 Points are given in the Cartesian coordinate system on the plane
A(-2; 5) ;
B(3; -5) ;
C(a; b) .
Find the coordinates of points symmetrical to these points about the axis Oy .
We continue to solve problems together
Example 6 Points are given in the Cartesian coordinate system on the plane
A(-1; 2) ;
B(3; -1) ;
C(-2; -2) .
Find the coordinates of points symmetrical to these points about the axis Oy .
Decision. Rotate 180 degrees around the axis Oy directed line segment from an axis Oy up to this point. In the figure, where the quadrants of the plane are indicated, we see that the point symmetrical to the given one with respect to the axis Oy, will have the same ordinate as the given point, and an abscissa equal in absolute value to the abscissa of the given point, and opposite in sign to it. So we get the following coordinates of points symmetrical to these points about the axis Oy :
A"(1; 2) ;
B"(-3; -1) ;
C"(2; -2) .
Example 7 Points are given in the Cartesian coordinate system on the plane
A(3; 3) ;
B(2; -4) ;
C(-2; 1) .
Find the coordinates of points that are symmetrical to these points with respect to the origin.
Decision. We rotate 180 degrees around the origin of the directed segment going from the origin to the given point. In the figure, where the quadrants of the plane are indicated, we see that a point symmetrical to a given one with respect to the origin of coordinates will have an abscissa and an ordinate equal in absolute value to the abscissa and ordinate of the given point, but opposite in sign to them. So we get the following coordinates of points symmetrical to these points with respect to the origin:
A"(-3; -3) ;
B"(-2; 4) ;
C(2; -1) .
Example 8
A(4; 3; 5) ;
B(-3; 2; 1) ;
C(2; -3; 0) .
Find the coordinates of the projections of these points:
1) on a plane Oxy ;
2) to the plane Oxz ;
3) to the plane Oyz ;
4) on the abscissa axis;
5) on the y-axis;
6) on the applique axis.
1) Projection of a point onto a plane Oxy located on this plane itself, and therefore has an abscissa and ordinate equal to the abscissa and ordinate of the given point, and an applicate equal to zero. So we get the following coordinates of the projections of these points on Oxy :
Axy(4;3;0);
Bxy (-3; 2; 0);
Cxy(2;-3;0).
2) Projection of a point onto a plane Oxz located on this plane itself, and therefore has an abscissa and applicate equal to the abscissa and applicate of the given point, and an ordinate equal to zero. So we get the following coordinates of the projections of these points on Oxz :
Axz (4; 0; 5);
Bxz (-3; 0; 1);
Cxz(2;0;0).
3) Projection of a point onto a plane Oyz located on this plane itself, and therefore has an ordinate and an applicate equal to the ordinate and applicate of a given point, and an abscissa equal to zero. So we get the following coordinates of the projections of these points on Oyz :
Ayz (0; 3; 5);
Byz (0; 2; 1);
Cyz(0;-3;0).
4) As follows from the theoretical part of this lesson, the projection of a point onto the x-axis is located on the x-axis itself, that is, the axis Ox, and therefore has an abscissa equal to the abscissa of the point itself, and the ordinate and applicate of the projection are equal to zero (since the ordinate and applicate axes intersect the abscissa at point 0). We get the following coordinates of the projections of these points on the x-axis:
Ax(4;0;0);
Bx(-3;0;0);
Cx(2;0;0).
5) The projection of a point on the y-axis is located on the y-axis itself, that is, the axis Oy, and therefore has an ordinate equal to the ordinate of the point itself, and the abscissa and applicate of the projection are equal to zero (since the abscissa and applicate axes intersect the ordinate axis at point 0). We get the following coordinates of the projections of these points on the y-axis:
Ay(0;3;0);
By(0;2;0);
Cy(0;-3;0).
6) The projection of a point on the applicate axis is located on the applicate axis itself, that is, the axis Oz, and therefore has an applicate equal to the applicate of the point itself, and the abscissa and ordinate of the projection are equal to zero (since the abscissa and ordinate axes intersect the applicate axis at point 0). We get the following coordinates of the projections of these points on the applicate axis:
Az(0; 0; 5);
Bz(0;0;1);
Cz(0; 0; 0).
Example 9 Points are given in the Cartesian coordinate system in space
A(2; 3; 1) ;
B(5; -3; 2) ;
C(-3; 2; -1) .
Find the coordinates of points that are symmetrical to these points with respect to:
1) plane Oxy ;
2) plane Oxz ;
3) plane Oyz ;
4) abscissa axis;
5) y-axis;
6) applique axis;
7) the origin of coordinates.
1) "Advance" the point on the other side of the axis Oxy Oxy, will have an abscissa and an ordinate equal to the abscissa and ordinate of the given point, and an applicate equal in magnitude to the applicate of the given point, but opposite in sign to it. So, we get the following coordinates of points symmetrical to the data with respect to the plane Oxy :
A"(2; 3; -1) ;
B"(5; -3; -2) ;
C"(-3; 2; 1) .
2) "Advance" the point on the other side of the axis Oxz for the same distance. According to the figure displaying the coordinate space, we see that the point symmetrical to the given one with respect to the axis Oxz, will have an abscissa and applicate equal to the abscissa and applicate of the given point, and an ordinate equal in magnitude to the ordinate of the given point, but opposite in sign to it. So, we get the following coordinates of points symmetrical to the data with respect to the plane Oxz :
A"(2; -3; 1) ;
B"(5; 3; 2) ;
C"(-3; -2; -1) .
3) "Advance" the point on the other side of the axis Oyz for the same distance. According to the figure displaying the coordinate space, we see that the point symmetrical to the given one with respect to the axis Oyz, will have an ordinate and an applicate equal to the ordinate and an applicate of the given point, and an abscissa equal in magnitude to the abscissa of the given point, but opposite in sign to it. So, we get the following coordinates of points symmetrical to the data with respect to the plane Oyz :
A"(-2; 3; 1) ;
B"(-5; -3; 2) ;
C"(3; 2; -1) .
By analogy with symmetric points on the plane and points in space symmetric to data with respect to planes, we note that in the case of symmetry about some axis of the Cartesian coordinate system in space, the coordinate on the axis about which the symmetry is set will retain its sign, and the coordinates on the other two axes will be the same in absolute value as the coordinates of the given point, but opposite in sign.
4) The abscissa will retain its sign, while the ordinate and applicate will change signs. So, we get the following coordinates of points symmetrical to the data about the x-axis:
A"(2; -3; -1) ;
B"(5; 3; -2) ;
C"(-3; -2; 1) .
5) The ordinate will retain its sign, while the abscissa and applicate will change signs. So, we get the following coordinates of points symmetrical to the data about the y-axis:
A"(-2; 3; -1) ;
B"(-5; -3; -2) ;
C"(3; 2; 1) .
6) The applicate will retain its sign, and the abscissa and ordinate will change signs. So, we get the following coordinates of points symmetrical to the data about the applicate axis:
A"(-2; -3; 1) ;
B"(-5; 3; 2) ;
C"(3; -2; -1) .
7) By analogy with symmetry in the case of points on a plane, in the case of symmetry about the origin of coordinates, all coordinates of a point symmetrical to a given one will be equal in absolute value to the coordinates of a given point, but opposite in sign to them. So, we get the following coordinates of points that are symmetrical to the data with respect to the origin.
Definition 1 . Numeric axis ( number line, coordinate line) Ox is called a straight line on which the point O is chosen reference point (origin of coordinates)(fig.1), direction
O → x
listed as positive direction and a segment is marked, the length of which is taken as unit of length.
Definition 2 . The segment, the length of which is taken as a unit of length, is called scale.
Each point of the numerical axis has a coordinate , which is a real number. The coordinate of the point O is equal to zero. The coordinate of an arbitrary point A lying on the ray Ox is equal to the length of the segment OA . The coordinate of an arbitrary point A of the numerical axis, not lying on the ray Ox , is negative, and in absolute value it is equal to the length of the segment OA .
Definition 3 . Rectangular Cartesian coordinate system Oxy on the plane call the two mutually perpendicular numerical axes Ox and Oy with the same scale and common origin at the point O, moreover, such that the rotation from the ray Ox through an angle of 90 ° to the ray Oy is carried out in the direction anti-clockwise(Fig. 2).
Remark . The rectangular Cartesian coordinate system Oxy shown in Figure 2 is called right coordinate system, Unlike left coordinate systems, in which the rotation of the beam Ox at an angle of 90° to the beam Oy is carried out in a clockwise direction. In this guide, we consider only right coordinate systems without mentioning it in particular.
If we introduce some system of rectangular Cartesian coordinates Oxy on the plane, then each point of the plane will acquire two coordinates – abscissa and ordinate, which are calculated as follows. Let A be an arbitrary point of the plane. Let us drop perpendiculars from point A AA 1 and AA 2 to the lines Ox and Oy, respectively (Fig. 3).
Definition 4 . The abscissa of point A is the coordinate of the point A 1 on the numerical axis Ox, the ordinate of point A is the coordinate of the point A 2 on the numeric axis Oy .
Designation . Coordinates (abscissa and ordinate) of a point A in the rectangular Cartesian coordinate system Oxy (Fig. 4) is usually denoted A(x;y) or A = (x; y).
Remark . Point O, called origin, has coordinates O(0 ; 0) .
Definition 5 . In the rectangular Cartesian coordinate system Oxy, the Ox numerical axis is called the abscissa axis, and the Oy numerical axis is called the ordinate axis (Fig. 5).
Definition 6 . Each rectangular Cartesian coordinate system divides the plane into 4 quarters ( quadrants), the numbering of which is shown in Figure 5.
Definition 7 . A plane on which a rectangular Cartesian coordinate system is given is called coordinate plane.
Remark . The abscissa axis is given on the coordinate plane by the equation y= 0 , the y-axis is given on the coordinate plane by the equation x = 0.
Statement 1 . Distance between two points coordinate plane
A 1 (x 1 ;y 1) and A 2 (x 2 ;y 2)
calculated according to the formula
Proof . Consider Figure 6.
| |A 1 A 2 | 2 = = (x 2 -x 1) 2 + (y 2 -y 1) 2 . | (1) |
Hence,
Q.E.D.
Equation of a circle on the coordinate plane
Consider on the coordinate plane Oxy (Fig. 7) a circle of radius R centered at the point A 0 (x 0 ;y 0) .
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 of coordinates, a positive direction is chosen on each axis. The positive direction of the axes (in the 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 X'X and Y'Y coordinate axes 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 OB segment, the y-coordinate is the length of the OC segment in the selected units. 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. They write it like this: A (x, y).
If point A lies in coordinate angle I, then point A has 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 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 the point O, which is called the origin of coordinates, on each axis the positive direction indicated by the arrows is chosen, and the unit of measurement of the segments on the axes. The units of measure 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. If the thumb of the right hand is taken as the X direction, the index finger as the Y direction, and the middle finger as the Z direction, then a right coordinate system is formed. Similar fingers of the left hand form the left coordinate system. The right and left coordinate systems cannot be combined 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 equal to the length of the segment OC, the z coordinate is the length of the segment OD in the selected units. 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. They write it like this: A (a, b, c).
Horts
A rectangular coordinate system (of any dimension) is also described by a set of orts , co-directed with the coordinate axes. The number of orts is equal to the dimension of the coordinate system, and they are all perpendicular to each other.
In the three-dimensional case, such vectors are usually denoted i j k or e x e y e z . In this case, in the case of the right coordinate system, the following formulas with the vector product of vectors are valid:
- [i j]=k ;
- [j k]=i ;
- [k i]=j .
Story
René Descartes was the first to introduce a rectangular coordinate system in his Discourse on the Method in 1637. Therefore, the rectangular coordinate system is also called - Cartesian coordinate system. The coordinate method for describing geometric objects laid the foundation for analytical geometry. Pierre Fermat also contributed to the development of the coordinate method, but his work was 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 applied by Leonhard Euler already in the 18th century.
see also
Links
Wikimedia Foundation. 2010 .
- Cartesian coordinate system
- Cartesian degree
See what "Cartesian coordinates" are in other dictionaries:
CARTSTIAN COORDINATES- (Cartesian coordinate system) a coordinate system on a plane or in space, usually with mutually perpendicular axes and the same scale along the axes, rectangular Cartesian coordinates. Named after R. Descartes ... Big Encyclopedic Dictionary
Cartesian coordinates- A coordinate system consisting of two perpendicular axes. The position of a point in such a system is formed using two numbers that determine the distance from the center of coordinates along each of the axes. Information topics ... ... Technical Translator's Handbook
Cartesian coordinates- (Cartesian coordinate system), a coordinate system on a plane or in space, usually with mutually perpendicular axes and the same scale along the axes, rectangular Cartesian coordinates. Named after R. Descartes ... encyclopedic Dictionary
Cartesian coordinates- Dekarto koordinatės statusas T sritis Standartizacija ir metrologija apibrėžtis Tiesinė plokštumos arba erdvės koordinačių sistema. Joje ašių masteliai paprastai būna lygūs. atitikmenys: engl. Cartesian coordinates vok. kartesische Koordinaten, f … Penkiakalbis aiskinamasis metrologijos terminų žodynas
Cartesian coordinates- Dekarto koordinatės statusas T sritis fizika atitikmenys: engl. Cartesian coordinates; grid coordinates vok. kartesische Koordinaten, f rus. Cartesian coordinates, f pranc. coordonnées cartésiennes, f … Fizikos terminų žodynas
CARTSTIAN COORDINATES- a method for determining the position of points on a plane by their distances to two fixed perpendicular straight axes. This concept is already seen in Archimedes and the Appologia of Perga more than two thousand years ago, and even among the ancient Egyptians. For the first time this…… Mathematical Encyclopedia
CARTSTIAN COORDINATES- Cartesian coordinate system [named after the French. philosopher and mathematician R. Descartes (R. Descartes; 1596 1650)], a coordinate system on a plane or in space, usually with mutually perpendicular axes and the same scale along the axes, rectangular D ... Big encyclopedic polytechnic dictionary
CARTSTIAN COORDINATES- (Cartesian coordinate system), a coordinate system on a plane or in space, usually with mutually perpendicular axes and the same scale along the axes, rectangular D. to. Named after R. Descartes ... Natural science. encyclopedic Dictionary
CARTSTIAN COORDINATES- The system of location of any point found bones relative to two axes that intersect at right angles. Developed by René Descartes, this system became the basis for standard methods for graphical representation of data. Horizontal line… … Explanatory Dictionary of Psychology
Coordinates- Coordinates. On the plane (left) and in space (right). COORDINATES (from the Latin co together and ordinatus ordered), numbers that determine the position of a point on a line, plane, surface, in space. Coordinates are distances... Illustrated Encyclopedic Dictionary
Instruction
Write down mathematical operations in text form and enter them in the search query field on the main page of the Google site, if you cannot use a calculator, but have Internet access. This search engine has a built-in multifunctional calculator, which is much easier to use than any other. There is no interface with buttons - all data must be entered in text form in a single field. For example, if known coordinates extreme points segment in the three-dimensional coordinate system A(51.34 17.2 13.02) and A(-11.82 7.46 33.5), then coordinates middle point segment C((51.34-11.82)/2 (17.2+7.46)/2 (13.02+33.5)/2). Entering (51.34-11.82) / 2 in the search query field, then (17.2 + 7.46) / 2 and (13.02 + 33.5) / 2, you can use Google to get coordinates C (19.76 12.33 23.26).
The standard circle equation allows you to find out several important information about this figure, for example, the coordinates of its center, the length of the radius. In some problems, on the contrary, it is required to make an equation for the given parameters.
Instruction
Determine if you have information about the circle, based on the task given to you. Remember that the end goal is to determine the coordinates of the center as well as the diameter. All your actions should be aimed at achieving this particular result.
Use the data on the presence of intersection points with coordinate lines or other lines. Please note that if the circle passes through the abscissa axis, the second one will have coordinate 0, and if through the ordinate axis, then the first one. These coordinates will allow you to find the coordinates of the center of the circle and also calculate the radius.
Don't forget about the basic properties of secants and tangents. In particular, the most useful theorem is that at the point of contact, the radius and the tangent form a right angle. But note that you may be asked to prove all the theorems used in the course.
Solve the most common types to learn how to immediately see how to use certain data for a circle equation. So, in addition to the already indicated problems with directly given coordinates and those under which information is given about the presence of intersection points, to compile the equation of a circle, you can use knowledge about the center of the circle, the length of the chord and on which this chord lies.
To solve, build an isosceles triangle, the base of which will be the given chord, and equal sides will be the radii. Make up, from which you can easily find the necessary data. To do this, it is enough to use the formula for finding the length of a segment in a plane.
Related videos
A circle is understood as a figure that consists of a set of points in a plane equidistant from its center. Distance from center to points circles called the radius.
Polar coordinates
The number is called polar radius dots or first polar coordinate. Distance cannot be negative, so the polar radius of any point is . The first polar coordinate is also denoted by the Greek letter ("rho"), but I'm used to the Latin version, and in the future I will use it.
The number is called polar angle given point or second polar coordinate. The polar angle is standardly changed within (the so-called principal values of the angle). However, it is quite acceptable to use the range, and in some cases there is a direct need to consider all angle values from zero to "plus infinity". I recommend, by the way, to get used to the radian measure of the angle, since it is not considered comme il faut to operate with degrees in higher mathematics.
The couple is called polar coordinates points . Of easy to find and their specific meanings. The tangent of an acute angle of a right triangle is the ratio of the opposite leg to the adjacent leg: therefore, the angle itself: 
. According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the legs:, therefore, the polar radius:
Thus, 
.
One penguin is good, but a flock is better: 
Negatively oriented corners 
just in case, I marked with arrows, suddenly one of the readers did not know about this orientation yet. If desired, you can “screw” 1 turn to each of them (rad. or 360 degrees) and get, by the way, comfortable table values:
But the disadvantage of these "traditionally" oriented corners is that they are too far (more than 180 degrees) "twisted" counterclockwise. I foresee the question: “why the lack and why do we need any negative angles at all?” In mathematics, the shortest and most rational paths are valued. Well, from the point of view of physics, the direction of rotation is often of fundamental importance - each of us tried to open the door by pulling the handle in the wrong direction =)
The order and technique of constructing points in polar coordinates
Beautiful pictures are beautiful, but building in a polar coordinate system is a rather painstaking task. Difficulties do not arise with points whose polar angles are 
, in our example these are the points 
; values that are multiples of 45 degrees also do not cause much trouble: . But how to correctly and competently build, say, a point?
You will need a checkered piece of paper, a pencil and the following drawing tools: ruler, compass, protractor. In extreme cases, you can get by with one ruler, or even ... without it at all! Read on and you will get one more proof that this country is invincible =)
Example 1
Construct a point in the polar coordinate system.
First of all, you need to find out the degree measure of the angle. If the angle is unfamiliar or you have doubts, then it is always better to use table or the general formula for converting radians to degrees. So our angle is (or ).
Let's draw a polar coordinate system (see the beginning of the lesson) and pick up a protractor. It will not be difficult for owners of a round instrument to mark 240 degrees, but with a high probability you will have a semi-circular version of the device on your hands. The problem of the complete absence of a protractor in the presence of a printer and scissors solved by needlework.
There are two ways: turn the sheet over and mark 120 degrees, or “screw” half a turn and consider the opposite angle. Let's choose the adult method and make a mark of 60 degrees: 
Either a midget protractor, or a giant cage =) However, in order to measure the angle, the scale is not important.
We draw with a pencil a thin straight line passing through the pole and the mark made: 
We figured out the angle, the next step is the polar radius. We take a compass and by ruler we set its solution to 3 units, most often, these are, of course, centimeters: 
Now we carefully place the needle on the pole, and with a rotational movement we make a small notch (red). The desired point is built: 
You can do without a compass by attaching a ruler directly to the constructed line and measuring 3 centimeters. But, as we will see later, in tasks for construction in the polar coordinate system a typical situation is when you need to mark two or more points with the same polar radius, so it is more efficient to harden the metal. In particular, in our drawing, by turning the leg of the compass by 180 degrees, it is easy to make a second notch and build a point symmetrical with respect to the pole. On it, let's work out the material of the next paragraph:
The relationship of rectangular and polar coordinate systems
Obviously join to the polar coordinate system of the "normal" coordinate grid and draw a point on the drawing: 
This connection is always useful to keep in mind when drawing in polar coordinates. Although, willy-nilly, it suggests itself without too much hint.
Let's establish the relationship between polar and Cartesian coordinates using the example of a specific point. Consider a right-angled triangle, in which the hypotenuse is equal to the polar radius: , and the legs are the "x" and "game" coordinates of the point in the Cartesian coordinate system: 
.
The sine of an acute angle is the ratio of the opposite leg to the hypotenuse: 
The cosine of an acute angle is the ratio of the adjacent leg to the hypotenuse: 
At the same time, they repeated the definitions of sine, cosine (and a little earlier tangent) from the program of the 9th grade of a comprehensive school.
Please add working formulas to your reference book that express the Cartesian coordinates of a point in terms of its polar coordinates - we will have to deal with them more than once, and next time right now =)
Let's find the coordinates of a point in a rectangular coordinate system: 
Thus:
The resulting formulas open up another loophole in the construction problem, when you can do without a protractor at all: first we find the Cartesian coordinates of the point (of course, on the draft), then we mentally find the right place on the drawing and mark this point. At the final stage, we draw a thin straight line that passes through the constructed point and the pole. As a result, it turns out that the angle was allegedly measured by a protractor.
It's funny that absolutely desperate students can even do without a ruler, using instead the smooth edge of a textbook, notebook or gradebook - after all, the manufacturers of notebooks took care of the metric, 1 cell = 5 millimeters.
All this reminded me of a well-known anecdote in which resourceful pilots plotted a course along the Belomor pack \u003d) Although, jokes are jokes, and the anecdote is not so far from reality, I remember that on one of the domestic flights across the Russian Federation, all navigation devices failed in the liner, and the crew successfully landed the board using an ordinary glass of water, which showed the angle of inclination of the aircraft relative to the ground. And the airstrip - here it is, visible from the windshield.
Using the Pythagorean theorem cited at the beginning of the lesson, it is easy to obtain inverse formulas: , therefore:
The angle "phi" itself is standardly expressed through the arc tangent - exactly the same as complex number argument with all its quirks.
It is also advisable to place the second group of formulas in your reference luggage.
After a detailed analysis of flights with individual points, let's move on to the natural continuation of the topic:
Line equation in polar coordinates
Essentially, the equation of a line in a polar coordinate system is polar radius function of polar angle (argument). In this case, the polar angle is taken into account in radians(!) and continuously takes values from to (sometimes it should be considered ad infinitum, or in a number of problems for convenience from to ). Each value of the angle "phi", which is included in domain function, corresponds to a single value of the polar radius.
The polar function can be compared to a kind of radar - when a beam of light emanating from the pole rotates counterclockwise and “detects” (draws) a line.
A common example of a polar curve is Archimedean spiral. The following figure shows her first turn– when the polar radius following the polar angle takes values from 0 to : 
Further, crossing the polar axis at the point , the spiral will continue to unwind, infinitely far away from the pole. But such cases are quite rare in practice; a more typical situation, when on all subsequent revolutions we “walk along the same line”, which is obtained in the range .
In the first example, we also encounter the concept domains polar function: since the polar radius is non-negative, negative angles cannot be considered here.
! Note : in some cases it is customary to use generalized polar coordinates, where the radius can be negative, and we will briefly study this approach a little later
In addition to the Archimedes spiral, there are many other well-known curves, but, as they say, you won’t be full of art, so I picked up examples that are very common in real practical tasks.
First, the simplest equations and the simplest lines:
An equation of the form specifies the outgoing from the pole Ray. Indeed, think about it if the value of the angle always(whatever "er" is) constantly, then what is the line?
Note : in the generalized polar coordinate system, this equation defines a straight line passing through the pole
The equation of the form determines ... guess the first time - if for anyone corner "phi" radius remains constant? In fact, this definition circles centered at the pole of radius .
For example, . For clarity, let's find the equation of this line in a rectangular coordinate system. Using the formula obtained in the previous paragraph, we will carry out the replacement:
Let's square both sides:
– circle equation centered at the origin of coordinates of radius 2, which was to be verified.
Since the creation and release of the article on linear dependence and linear independence of vectors I received several letters from site visitors who asked a question in the spirit: “here is a simple and convenient rectangular coordinate system, why do we need some other oblique affine case?”. The answer is simple: mathematics seeks to embrace everything and everyone! In addition, in this or that situation, convenience is important - as you can see, it is much more profitable to work with a circle in polar coordinates due to the extreme simplicity of the equation.
And sometimes a mathematical model anticipates scientific discoveries. So, at one time, the rector of Kazan University N.I. Lobachevsky rigorously proved, through an arbitrary point of the plane it is possible to draw infinite number of lines parallel to the given one. As a result, he was defamed by the entire scientific world, but ... no one could refute this fact. Only after a good century, astronomers found out that light in space propagates along curved trajectories, where the non-Euclidean geometry of Lobachevsky, formally developed by him long before this discovery, begins to work. It is assumed that this is a property of space itself, the curvature of which is invisible to us due to small (by astronomical standards) distances.
Consider more meaningful construction tasks:
Example 2
build a line
Decision: first find domain. Since the polar radius is non-negative, the inequality must hold. You can remember the school rules for solving trigonometric inequalities, but in simple cases like this, I advise a faster and more visual method of solving:
Imagine a cosine plot. If he has not yet managed to be deposited in memory, then find him on the page Graphs of elementary functions. What does inequality tell us? It tells us that the cosine graph should be located not less abscissa axis. And this happens on a segment. And, accordingly, the interval does not fit.
Thus, the domain of our function is: , that is, the graph is located to the right of the pole (according to the terminology of the Cartesian system, in the right half-plane).
In polar coordinates, there is often a vague idea of which line defines this or that equation, so in order to build it, you need to find the points that belong to it - and the more, the better. Usually limited to a dozen or two (or even less). The easiest way, of course, is to take tabular angle values. For greater clarity, I will “fasten” one turn to negative values: 
Due to the parity of the cosine 
the corresponding positive values can be omitted again: 
Let's depict the polar coordinate system and set aside the found points, while it is convenient to set aside the same values of "er" at a time, making paired serifs with a compass according to the technology discussed above: 
In principle, the line is clearly drawn, but to absolutely confirm the guess, let's find its equation in the Cartesian coordinate system. You can apply newly derived formulas 
, but I'll tell you about a trickier trick. We artificially multiply both parts of the equation by "er": and use more compact transition formulas:
Selecting the full square, we bring the equation of the line to a recognizable form: 
– circle equation centered at point , radius 2.
Since, according to the condition, it was simply necessary to complete the construction and that's it, we smoothly connect the found points with a line: 
Ready. It's okay if it turns out a little uneven, you didn't have to know that it was a circle ;-)
Why didn't we consider angle values outside the interval ? The answer is simple: it doesn't make sense. In view of the periodicity of the function, we are waiting for an endless run along the constructed circle.
It is easy to carry out a simple analysis and come to the conclusion that the equation of the form defines a circle of diameter with a center at the point . Figuratively speaking, all such circles "sit" on the polar axis and necessarily pass through the pole. If , then the cheerful company will move to the left - to the continuation of the polar axis (think why).
A similar problem for an independent solution:
Example 3
Draw a line and find its equation in a rectangular coordinate system.
We systematize the procedure for solving the problem:
First of all, we find the domain of the function, for this it is convenient to look at sinusoid to immediately understand where the sine is non-negative.
In the second step, we calculate the polar coordinates of the points using tabular values of angles; analyze whether it is possible to reduce the number of calculations?
In the third step, we set aside the points in the polar coordinate system and carefully connect them with a line.
And, finally, we find the equation of the line in the Cartesian coordinate system.
Sample solution at the end of the lesson.
We detail the general algorithm and technique for constructing in polar coordinates
and significantly speed up in the second part of the lecture, but before that, let's get acquainted with one more common line:
polar rose
Quite right, we are talking about a flower with petals:
Example 4
Plot lines given by equations in polar coordinates
There are two approaches to constructing a polar rose. First, let's go along the knurled track, assuming that the polar radius cannot be negative:
Decision:
a) Find the domain of the function: 
Such a trigonometric inequality is also easy to solve graphically: from the materials of the article Geometric Plot Transformations It is known that if the function argument is doubled, then its graph will shrink to the y-axis by 2 times. Please find the graph of the function in the first example of the specified lesson. Where is this sinusoid located above the x-axis? At intervals 
. Therefore, the corresponding segments satisfy the inequality, and domain our function: 
.
Generally speaking, the solution of the inequalities under consideration is the union of an infinite number of segments, but, again, we are only interested in one period.
Perhaps, some readers will find the analytical method of finding the domain of definition easier, I will conditionally call it “slicing a round pie”. We will cut into equal parts and, first of all, find the boundaries of the first piece. We argue as follows: sine is non-negative, when his argument ranges from 0 to rad. inclusive. In our example: . Dividing all parts of the double inequality by 2, we obtain the required interval:
Now we start sequentially “cut equal pieces of 90 degrees” counterclockwise:
- the segment found, of course, is included in the definition area;
– next interval – not included;
- the next segment - enters;
- and, finally, the interval - is not included.
Just like a chamomile - "loves, does not love, loves, does not love" =) With the difference that this is not fortune-telling. Yes, just some kind of love in Chinese turns out ....
So, 
and the line represents a rose with two identical petals. It is quite possible to draw a drawing schematically, but it is highly desirable to correctly find and mark the tops of the petals. The vertices correspond midpoints of segments of the domain of definition, which in this example have obvious angular coordinates 
. Wherein petal length are: 
Here is the natural result of a caring gardener: 
It should be noted that the length of the petal is easy to immediately see from the equation - since the sine is limited: , then the maximum value of "er" will certainly not exceed two.
b) Let's construct the line given by the equation. Obviously, the length of the petal of this rose is also two, but, first of all, we are interested in the domain of definition. We apply the analytical method of "slicing": sine is non-negative when its argument is in the range from zero to "pi" inclusive, in this case: . We divide all parts of the inequality by 3 and get the first interval:
Next, we begin “cutting the pie into pieces” according to the rad. (60 degrees):
– the segment will enter the definition area;
– interval – will not enter;
- segment - will enter;
– interval – will not enter;
- segment - will enter;
- interval - will not enter.
The process has been successfully completed at the 360 degree mark.
So the scope is: 
.
The actions carried out in whole or in part are easy to carry out mentally.
Construction. If in the previous paragraph everything went well with right angles and 45-degree angles, then here you have to tinker a little. Let's find the tops of the petals. Their length was visible from the very beginning of the task, it remains to calculate the angular coordinates, which are equal to the midpoints of the segments of the domain of definition: 
Please note that between the tops of the petals you must necessarily get equal gaps, in this case 120 degrees.
It is desirable to mark the drawing into 60-degree sectors (delimited by green lines) and draw the directions of the tops of the petals (gray lines). It is convenient to mark the vertices themselves with the help of a compass - once measure the distance of 2 units and apply three notches in the drawn directions at 30, 150 and 270 degrees: 
Ready. I understand that the task is troublesome, but if you want to arrange everything in a smart way, you will have to spend time.
We formulate the general formula: an equation of the form , is a natural number), defines a polar -petal rose whose petal length is .
For example, the equation specifies a quatrefoil with a petal length of 5 units, the equation - a 5-petal rose with a petal length of 3 units. etc.
