Cartesian coordinates of points on the plane. Equation of a circle. Coordinate plane

Understanding the Coordinate Plane

Each object (for example, a house, a place in the auditorium, a point on the map) has its own ordered address (coordinates), which has a numerical or letter designation.

Mathematicians have developed a model that allows you to determine the position of an object and is called coordinate plane.

To construct a coordinate plane, you need to draw $2$ perpendicular straight lines, at the end of which the directions “to the right” and “up” are indicated using arrows. Divisions are applied to the lines, and the point of intersection of the lines is the zero mark for both scales.

Definition 1

The horizontal line is called x-axis and is denoted by x, and the vertical line is called y-axis and is denoted by y.

Two perpendicular x and y axes with divisions make up rectangular, or Cartesian, coordinate system, which was proposed by the French philosopher and mathematician Rene Descartes.

Coordinate plane

Point coordinates

A point on a coordinate plane is defined by two coordinates.

To determine the coordinates of point $A$ on the coordinate plane, you need to draw straight lines through it that will be parallel to the coordinate axes (indicated by a dotted line in the figure). The intersection of the line with the x-axis gives the $x$ coordinate of point $A$, and the intersection with the y-axis gives the y-coordinate of point $A$. When writing the coordinates of a point, the $x$ coordinate is first written, and then the $y$ coordinate.

Point $A$ in the figure has coordinates $(3; 2)$, and point $B (–1; 4)$.

To plot a point on the coordinate plane, proceed in the reverse order.

Constructing a point at specified coordinates

Example 1

On the coordinate plane, construct points $A(2;5)$ and $B(3; –1).$

Solution.

Construction of point $A$:

put the number $2$ on the $x$ axis and draw a perpendicular line;
On the y-axis we plot the number $5$ and draw a straight line perpendicular to the $y$ axis. At the intersection of perpendicular lines we obtain point $A$ with coordinates $(2; 5)$.

Construction of point $B$:

Let us plot the number $3$ on the $x$ axis and draw a straight line perpendicular to the x axis;
On the $y$ axis we plot the number $(–1)$ and draw a straight line perpendicular to the $y$ axis. At the intersection of perpendicular lines we obtain point $B$ with coordinates $(3; –1)$.

Example 2

Construct points on the coordinate plane with given coordinates $C (3; 0)$ and $D(0; 2)$.

Solution.

Construction of point $C$:

put the number $3$ on the $x$ axis;
coordinate $y$ is equal to zero, which means point $C$ will lie on the $x$ axis.

Construction of point $D$:

put the number $2$ on the $y$ axis;
coordinate $x$ is equal to zero, which means point $D$ will lie on the $y$ axis.

Note 1

Therefore, at coordinate $x=0$ the point will lie on the $y$ axis, and at coordinate $y=0$ the point will lie on the $x$ axis.

Example 3

Determine the coordinates of points A, B, C, D.$

Solution.

Let's determine the coordinates of point $A$. To do this, we draw straight lines through this point $2$ that will be parallel to the coordinate axes. The intersection of the line with the x-axis gives the coordinate $x$, the intersection of the line with the y-axis gives the coordinate $y$. Thus, we obtain that the point $A (1; 3).$

Let's determine the coordinates of point $B$. To do this, we draw straight lines through this point $2$ that will be parallel to the coordinate axes. The intersection of the line with the x-axis gives the coordinate $x$, the intersection of the line with the y-axis gives the coordinate $y$. We find that point $B (–2; 4).$

Let's determine the coordinates of point $C$. Because it is located on the $y$ axis, then the $x$ coordinate of this point is zero. The y coordinate is $–2$. Thus, point $C (0; –2)$.

Let's determine the coordinates of point $D$. Because it is on the $x$ axis, then the $y$ coordinate is zero. The $x$ coordinate of this point is $–5$. Thus, point $D (5; 0).$

Example 4

Construct points $E(–3; –2), F(5; 0), G(3; 4), H(0; –4), O(0; 0).$

Solution.

Construction of point $E$:

put the number $(–3)$ on the $x$ axis and draw a perpendicular line;
on the $y$ axis we plot the number $(–2)$ and draw a perpendicular line to the $y$ axis;
at the intersection of perpendicular lines we obtain the point $E (–3; –2).$

Construction of point $F$:

coordinate $y=0$, which means the point lies on the $x$ axis;
Let us plot the number $5$ on the $x$ axis and obtain the point $F(5; 0).$

Construction of point $G$:

put the number $3$ on the $x$ axis and draw a perpendicular line to the $x$ axis;
on the $y$ axis we plot the number $4$ and draw a perpendicular line to the $y$ axis;
at the intersection of perpendicular lines we obtain the point $G(3; 4).$

Construction of point $H$:

coordinate $x=0$, which means the point lies on the $y$ axis;
Let us plot the number $(–4)$ on the $y$ axis and obtain the point $H(0;–4).$

Construction of point $O$:

both coordinates of the point are equal to zero, which means that the point lies simultaneously on both the $y$ axis and the $x$ axis, therefore it is the intersection point of both axes (the origin of coordinates).

§ 1 Coordinate system: definition and method of construction

In this lesson we will get acquainted with the concepts of “coordinate system”, “coordinate plane”, “coordinate axes”, and learn how to construct points on a plane using coordinates.

Let us take a coordinate line x with the origin point O, a positive direction and a unit segment.

Through the origin of coordinates, point O of the coordinate line x, we draw another coordinate line y, perpendicular to x, set the positive direction upward, unit segment same. Thus, we have built a coordinate system.

Let's give a definition:

Two mutually perpendicular coordinate lines intersecting at a point, which is the origin of coordinates of each of them, form a coordinate system.

§ 2 Coordinate axis and coordinate plane

The straight lines that form a coordinate system are called coordinate axes, each of which has its own name: the coordinate line x is the abscissa axis, the coordinate line y is the ordinate axis.

The plane on which the coordinate system is selected is called the coordinate plane.

The described coordinate system is called rectangular. It is often called the Cartesian coordinate system in honor of the French philosopher and mathematician René Descartes.

Each point on the coordinate plane has two coordinates, which can be determined by dropping perpendiculars from the point on the coordinate axis. The coordinates of a point on a plane are a pair of numbers, of which the first number is the abscissa, the second number is the ordinate. The abscissa is perpendicular to the x-axis, the ordinate is perpendicular to the y-axis.

Let's mark point A on the coordinate plane and draw perpendiculars from it to the axes of the coordinate system.

Along the perpendicular to the abscissa axis (x-axis), we determine the abscissa of point A, it is equal to 4, the ordinate of point A - along the perpendicular to the ordinate axis (y-axis) is 3. The coordinates of our point are 4 and 3. A (4;3). Thus, coordinates can be found for any point on the coordinate plane.

§ 3 Construction of a point on a plane

How to construct a point on a plane with given coordinates, i.e. Using the coordinates of a point on the plane, determine its position? In this case, we perform the steps in reverse order. On the coordinate axes we find points corresponding to the given coordinates, through which we draw straight lines perpendicular to the x and y axes. The point of intersection of the perpendiculars will be the desired one, i.e. a point with given coordinates.

Let's complete the task: construct point M (2;-3) on the coordinate plane.

To do this, find a point with coordinate 2 on the x-axis and draw a straight line perpendicular to the x-axis through this point. On the ordinate axis we find a point with coordinate -3, through it we draw a straight line perpendicular to the y axis. The point of intersection of perpendicular lines will be given point M.

Now let's look at a few special cases.

Let us mark points A (0; 2), B (0; -3), C (0; 4) on the coordinate plane.

The abscissas of these points are equal to 0. The figure shows that all points are on the ordinate axis.

Consequently, points whose abscissas are equal to zero lie on the ordinate axis.

Let's swap the coordinates of these points.

The result will be A (2;0), B (-3;0) C (4; 0). In this case, all ordinates are equal to 0 and the points are on the x-axis.

This means that points whose ordinates are equal to zero lie on the abscissa axis.

Let's look at two more cases.

On the coordinate plane, mark the points M (3; 2), N (3; -1), P (3; -4).

It is easy to see that all the abscissas of the points are the same. If these points are connected, you get a straight line parallel to the ordinate axis and perpendicular to the abscissa axis.

The conclusion suggests itself: points that have the same abscissa lie on the same straight line, which is parallel to the ordinate axis and perpendicular to the abscissa axis.

If you swap the coordinates of the points M, N, P, you get M (2; 3), N (-1; 3), P (-4; 3). The ordinates of the points will be the same. In this case, if you connect these points, you get a straight line parallel to the abscissa axis and perpendicular to the ordinate axis.

Thus, points having the same ordinate lie on the same line parallel axis abscissa and perpendicular to the ordinate axis.

In this lesson you became acquainted with the concepts of “coordinate system”, “coordinate plane”, “coordinate axes - abscissa axis and ordinate axis”. We learned how to find the coordinates of a point on a coordinate plane and learned how to construct points on the plane using its coordinates.

List of used literature:

Mathematics. Grade 6: lesson plans for I.I.’s textbook. Zubareva, A.G. Mordkovich // author-compiler L.A. Topilina. – Mnemosyne, 2009.
Mathematics. 6th grade: textbook for students educational institutions. I.I. Zubareva, A.G. Mordkovich. - M.: Mnemosyne, 2013.
Mathematics. 6th grade: textbook for general education institutions/G.V. Dorofeev, I.F. Sharygin, S.B. Suvorov and others/edited by G.V. Dorofeeva, I.F. Sharygina; Russian Academy of Sciences, Russian Academy of Education. - M.: “Enlightenment”, 2010
Handbook of mathematics - http://lyudmilanik.com.ua
Handbook for students in secondary school http://shkolo.ru

Understanding the Coordinate Plane

Each object (for example, a house, a place in the auditorium, a point on the map) has its own ordered address (coordinates), which has a numerical or letter designation.

Mathematicians have developed a model that allows you to determine the position of an object and is called coordinate plane.

Definition 1

The horizontal line is called x-axis and is denoted by x, and the vertical line is called y-axis and is denoted by y.

Two perpendicular x and y axes with divisions make up rectangular, or Cartesian, coordinate system, which was proposed by the French philosopher and mathematician Rene Descartes.

Coordinate plane

Point coordinates

A point on a coordinate plane is defined by two coordinates.

Point $A$ in the figure has coordinates $(3; 2)$, and point $B (–1; 4)$.

To plot a point on the coordinate plane, proceed in the reverse order.

Constructing a point at specified coordinates

Example 1

On the coordinate plane, construct points $A(2;5)$ and $B(3; –1).$

Solution.

Construction of point $A$:

put the number $2$ on the $x$ axis and draw a perpendicular line;
On the y-axis we plot the number $5$ and draw a straight line perpendicular to the $y$ axis. At the intersection of perpendicular lines we obtain point $A$ with coordinates $(2; 5)$.

Construction of point $B$:

Let us plot the number $3$ on the $x$ axis and draw a straight line perpendicular to the x axis;
On the $y$ axis we plot the number $(–1)$ and draw a straight line perpendicular to the $y$ axis. At the intersection of perpendicular lines we obtain point $B$ with coordinates $(3; –1)$.

Example 2

Construct points on the coordinate plane with given coordinates $C (3; 0)$ and $D(0; 2)$.

Solution.

Construction of point $C$:

put the number $3$ on the $x$ axis;
coordinate $y$ is equal to zero, which means point $C$ will lie on the $x$ axis.

Construction of point $D$:

put the number $2$ on the $y$ axis;
coordinate $x$ is equal to zero, which means point $D$ will lie on the $y$ axis.

Note 1

Therefore, at coordinate $x=0$ the point will lie on the $y$ axis, and at coordinate $y=0$ the point will lie on the $x$ axis.

Example 3

Determine the coordinates of points A, B, C, D.$

Solution.

Let's determine the coordinates of point $C$. Because it is located on the $y$ axis, then the $x$ coordinate of this point is zero. The y coordinate is $–2$. Thus, point $C (0; –2)$.

Let's determine the coordinates of point $D$. Because it is on the $x$ axis, then the $y$ coordinate is zero. The $x$ coordinate of this point is $–5$. Thus, point $D (5; 0).$

Example 4

Construct points $E(–3; –2), F(5; 0), G(3; 4), H(0; –4), O(0; 0).$

Solution.

Construction of point $E$:

put the number $(–3)$ on the $x$ axis and draw a perpendicular line;
on the $y$ axis we plot the number $(–2)$ and draw a perpendicular line to the $y$ axis;
at the intersection of perpendicular lines we obtain the point $E (–3; –2).$

Construction of point $F$:

coordinate $y=0$, which means the point lies on the $x$ axis;
Let us plot the number $5$ on the $x$ axis and obtain the point $F(5; 0).$

Construction of point $G$:

put the number $3$ on the $x$ axis and draw a perpendicular line to the $x$ axis;
on the $y$ axis we plot the number $4$ and draw a perpendicular line to the $y$ axis;
at the intersection of perpendicular lines we obtain the point $G(3; 4).$

Construction of point $H$:

coordinate $x=0$, which means the point lies on the $y$ axis;
Let us plot the number $(–4)$ on the $y$ axis and obtain the point $H(0;–4).$

Construction of point $O$:

both coordinates of the point are equal to zero, which means that the point lies simultaneously on both the $y$ axis and the $x$ axis, therefore it is the intersection point of both axes (the origin of coordinates).

Rectangular coordinate system on a plane

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 system coordinates. 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.

Links

Wikimedia Foundation. 2010.

See what “Coordinate plane” is in other dictionaries:

cutting plane- (Pn) Coordinate plane tangent to the cutting edge at the point under consideration and perpendicular to the main plane. [...

In topography, a network of imaginary lines encircling Earth in the latitudinal and meridional directions, with which you can accurately determine the position of any point on earth's surface. Latitudes are measured from the equator - the great circle... ... Geographical encyclopedia

In topography, a network of imaginary lines encircling the globe in the latitudinal and meridional directions, with the help of which you can accurately determine the position of any point on the earth's surface. Latitudes are measured from the equator of the great circle,... ... Collier's Encyclopedia

This term has other meanings, see Phase diagram. Phase plane is a coordinate plane in which any two variables (phase coordinates) are plotted along the coordinate axes, which uniquely determine the state of the system... ... Wikipedia

main cutting plane- (Pτ) Coordinate plane perpendicular to the intersection of the main plane and the cutting plane. [GOST 25762 83] Topics: cutting processing General terms: coordinate plane systems and coordinate planes... Technical Translator's Guide

instrumental main cutting plane- (Pτi) Coordinate plane perpendicular to the line of intersection of the instrumental main plane and the cutting plane. [GOST 25762 83] Topics: cutting processing General terms: coordinate plane systems and coordinate planes... Technical Translator's Guide

tool cutting plane- (Pni) Coordinate plane tangent to the cutting edge at the point under consideration and perpendicular to the instrumental main plane. [GOST 25762 83] Subjects of cutting processing General terms of coordinate plane systems and... ... Technical Translator's Guide

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Introduction

In the speech of adults, you may have heard the following phrase: “Leave me your coordinates.” This expression means that the interlocutor must leave his address or telephone number where he can be found. Those of you who played “sea battle” used the corresponding coordinate system. A similar coordinate system is used in chess. Seats in a cinema auditorium are specified by two numbers: the first number indicates the number of the row, and the second number indicates the number of the seat in this row. The idea of specifying the position of a point on a plane using numbers originated in ancient times. The coordinate system permeates the entire practical life of a person and has a huge practical use. Therefore, we decided to create this project to expand our knowledge on the topic “Coordinate Plane”

Project objectives:

get acquainted with the history of the emergence of a rectangular coordinate system on a plane;

prominent figures involved in this topic;

find interesting historical facts;

perceive coordinates well by ear; carry out constructions clearly and accurately;

prepare a presentation.

Chapter I. Coordinate plane

The idea of specifying the position of a point on a plane using numbers originated in ancient times - primarily among astronomers and geographers when compiling star and geographical maps and calendars.

§1. Origin of coordinates. Coordinate system in geography

200 years BC, the Greek scientist Hipparchus introduced geographical coordinates. He suggested drawing on geographical map parallels and meridians and indicate latitude and longitude with numbers. Using these two numbers, you can accurately determine the position of an island, village, mountain or well in the desert and plot them on a map or globe. Having learned to determine the latitude and longitude of a ship’s location in the open world, sailors were able to choose the direction they needed.

Eastern longitude and northern latitude are indicated by numbers with a plus sign, and western longitude and southern latitude are indicated by numbers with a minus sign. Thus, a pair of signed numbers uniquely identifies a point on the globe.

Geographic latitude? - the angle between the plumb line at a given point and the plane of the equator, measured from 0 to 90 on both sides of the equator. Geographic longitude? - the angle between the plane of the meridian passing through a given point and the plane of the origin of the meridian (see Greenwich meridian). Longitudes from 0 to 180 east of the beginning of the meridian are called eastern, and to the west - western.

To find a certain object in a city, in most cases it is enough to know its address. Difficulties arise if you need to explain where, for example, country cottage area, place in the forest. A universal remedy Geographic coordinates serve as location indications.

When hitting emergency situation, a person must first of all be able to navigate the terrain. Sometimes it is necessary to determine the geographic coordinates of your location, for example, to transmit to the rescue service or for other purposes.

Modern navigation uses the WGS-84 worldwide coordinate system as standard. All GPS navigators and major cartographic projects on the Internet operate in this coordinate system. Coordinates in the WGS-84 system are as commonly used and understood by everyone as universal time. Generally available accuracy when working with geographical coordinates is 5 - 10 meters on the ground.

Geographic coordinates are signed numbers (latitude from -90° to +90°, longitude from -180° to +180°) and can be written in various forms: in degrees (ddd.ddddd°); degrees and minutes (ddd° mm.mmm"); degrees, minutes and seconds (ddd° mm" ss.s"). The recording forms can be easily converted into one another (1 degree = 60 minutes, 1 minute = 60 seconds) To indicate the sign of coordinates, letters are often used, based on the names of the cardinal directions: N and E - northern latitude and eastern longitude - positive numbers, S and W - southern latitude and western longitude - negative numbers.

The form of recording coordinates in DEGREES is most convenient for manual entry and coincides with mathematical notation numbers. The form of recording coordinates in DEGREES AND MINUTES is preferred in many cases; this format is set by default in most GPS navigators and is standardly used in aviation and at sea. Classic shape recording coordinates in DEGREES, MINUTES AND SECONDS does not really find much practical use.

§2. Coordinate system in astronomy. Myths about constellations

As mentioned above, the idea of specifying the position of a point on a plane using numbers originated in ancient times among astronomers when drawing up star maps. People needed to count time, predict seasonal phenomena (high tides, seasonal rains, flooding), and needed to navigate the terrain while traveling.

Astronomy is the science of stars, planets, celestial bodies, their structure and development.

Thousands of years have passed, science has stepped far forward, but people still cannot take their eyes off the beauty of the night sky.

Constellations - areas starry sky, characteristic figures formed by bright stars. The entire sky is divided into 88 constellations, which make it easier to navigate among the stars. Most of the names of the constellations come from antiquity.

The most famous constellation is Ursa Major. IN Ancient Egypt it was called “Hippopotamus”, and the Kazakhs called it “Horse on a leash”, although outwardly the constellation does not resemble either one or the other animal. What is it like?

The ancient Greeks had a legend about the constellations Ursa Major and Ursa Minor. The almighty god Zeus decided to marry the beautiful nymph Calisto, one of the servants of the goddess Aphrodite, against the latter’s wishes. To save Kalisto from the persecution of the goddess, Zeus turned Kalisto into Ursa Major, her beloved dog into Ursa Minor and took them to heaven. Transfer the constellations Ursa Major and Ursa Minor from the starry sky to the coordinate plane. . Each of the stars in the Big Dipper has its own name.

URSA GREAT

I recognize it by the BUCKET!

Seven stars sparkle here

Here's what their names are:

DUBHE illuminates the darkness,

MERAK is burning next to him,

On the side is FEKDA with MEGRETZ,

A daring fellow.

From MEGRETZ for departure

ALIOT is located

And behind him - MITZAR with ALCOR

(These two shine in unison.)

Our ladle closes

Incomparable BENETNASH.

He points to the eye

The path to the constellation BOOTES,

Where the beautiful ARCTURUS shines,

Everyone will notice him now!

No less beautiful legend about the constellations Cepheus, Cassiopeia and Andromeda.

Ethiopia was once ruled by King Cepheus. One day his wife, Queen Cassiopeia, had the imprudence to show off her beauty to the inhabitants of the sea - the Nereids. The latter, offended, complained to the god of the sea Poseidon, and the ruler of the seas, enraged by Cassiopeia's insolence, released a sea monster - Whale - onto the shores of Ethiopia. To save his kingdom from destruction, Cepheus, on the advice of the oracle, decided to sacrifice to the monster and give him his beloved daughter Andromeda to be devoured. He chained Andromeda to a coastal rock and left her awaiting the decision of her fate.

And at this time, on the other side of the world, the mythical hero Perseus accomplished a brave feat. He entered a secluded island where gorgons lived - amazing monsters in the form of women whose heads were swarming with snakes instead of hair. The gaze of the gorgons was so terrible that everyone they looked at instantly turned into stone.

Taking advantage of the sleep of these monsters, Perseus cut off the head of one of them, the Gorgon Medusa. At that moment, the horse Pegasus flew out of the severed body of Medusa. Perseus grabbed the head of the jellyfish, jumped on Pegasus and rushed through the air to his homeland. When he flew over Ethiopia, he saw Andromeda chained to a rock. At this moment, the whale had already emerged from the depths of the sea, preparing to swallow its victim. But Perseus, rushing into a mortal battle with Keith, defeated the monster. He showed Keith the head of the jellyfish, which had not yet lost its strength, and the monster petrified, turning into an island. As for Perseus, having unchained Andromeda, he returned her to her father, and Cepheus, moved with happiness, gave Andromeda as a wife to Perseus. This is how this story ended happily, the main characters of which were placed in heaven by the ancient Greeks.

On the star map you can find not only Andromeda with her father, mother and husband, but also the magical horse Pegasus and the culprit of all troubles - the monster Keith.

The constellation Cetus is located below Pegasus and Andromeda. Unfortunately, it is not marked by any characteristic bright stars and therefore belongs to the number of minor constellations.

§3. Using the idea of rectangular coordinates in painting.

Traces of the application of the idea of rectangular coordinates in the form of a square grid (palette) are depicted on the wall of one of the burial chambers of Ancient Egypt. In the burial chamber of the pyramid of Father Ramesses, there is a network of squares on the wall. With their help, the image is transferred in an enlarged form. Renaissance artists also used a rectangular grid.

The word "perspective" is Latin for "seeing clearly." In fine art, linear perspective is the depiction of objects on a plane in accordance with apparent changes in their size. The basis modern theory perspectives were laid by the great artists of the Renaissance - Leonardo da Vinci, Albrecht Durer and others. One of Durer's engravings (Fig. 3) depicts a method of drawing from life through glass with a square grid applied to it. This process can be described as follows: if you stand in front of a window and, without changing your point of view, circle on the glass everything that is visible behind it, then the resulting drawing will be a perspective image of space.

Egyptian design methods that appear to have been based on square grid patterns. There are numerous examples in Egyptian art showing that artists and sculptors first drew a grid on the wall, which had to be painted or carved in order to maintain the established proportions. The simple numerical relationships of these grids are at the core of all great works of art Egyptians

The same method was used by many Renaissance artists, including Leonardo da Vinci. In Ancient Egypt, this was embodied in the Great Pyramid, which is reinforced by its close connection with the pattern on Marlborough Down.

When starting work, the Egyptian artist lined the wall with a grid of straight lines and then carefully transferred the figures onto it. But geometric orderliness did not prevent him from recreating nature with detailed accuracy. The appearance of every fish and every bird is conveyed with such truthfulness that modern zoologists can easily determine their species. Figure 4 shows a detail of the composition from the illustration - a tree with birds captured in Khnumhotep’s net. The movement of the artist's hand was guided not only by the reserves of his skills, but also by the eye, sensitive to the outlines of nature.

Fig.4 Birds on acacia

Chapter II. Coordinate method in mathematics

§1. Application of coordinates in mathematics. Merits

French mathematician René Descartes

For a long time, only geography “land description” used this wonderful invention, and only in the 14th century the French mathematician Nicolas Oresme (1323-1382) tried to apply it to “land measurement” - geometry. He proposed to cover the plane with a rectangular grid and call latitude and longitude what we now call abscissa and ordinate.

Based on this successful innovation, the coordinate method arose, linking geometry with algebra. The main credit for the creation of this method belongs to the great French mathematician Rene Descartes (1596 - 1650). In his honor, such a coordinate system is called Cartesian, indicating the location of any point on the plane by the distances from this point to the “zero latitude” - the abscissa axis and the “zero meridian” - the ordinate axis.

However, this brilliant French scientist and thinker of the 17th century (1596 - 1650) did not immediately find his place in life. Born into a noble family, Descartes received a good education. In 1606, his father sent him to the Jesuit college of La Flèche. Considering not very good health Descartes, he was given some relaxations in the strict regime of this educational institution, for example, they were allowed to get up later than others. Having acquired a lot of knowledge at the college, Descartes at the same time became imbued with antipathy towards scholastic philosophy, which he retained throughout his life.

After graduating from college, Descartes continued his education. In 1616, at the University of Poitiers, he received a bachelor's degree in law. In 1617, Descartes enlisted in the army and traveled extensively throughout Europe.

The year 1619 turned out to be a key year for Descartes scientifically.

It was at this time, as he himself wrote in his diary, that the foundations of a new “most amazing science” were revealed to him. Most likely, Descartes had in mind the discovery of the universal scientific method, which he subsequently fruitfully applied in a variety of disciplines.

In the 1620s, Descartes met the mathematician M. Mersenne, through whom he “kept in touch” with the entire European scientific community for many years.

In 1628, Descartes settled in the Netherlands for more than 15 years, but did not settle in any one place, but changed his place of residence about two dozen times.

In 1633, having learned about the condemnation of Galileo by the church, Descartes refused to publish his natural philosophical work “The World,” which outlined the ideas of the natural origin of the universe according to the mechanical laws of matter.

In 1637 on French Descartes' work “Discourse on Method” is published, with which, as many believe, modern European philosophy began.

Descartes's last philosophical work, The Passions of the Soul, published in 1649, also had a great influence on European thought. In the same year, at the invitation of the Swedish Queen Christina, Descartes went to Sweden. The harsh climate and unusual regime (the queen forced Descartes to get up at 5 a.m. to give her lessons and carry out other assignments) undermined Descartes’ health, and, having caught a cold, he

died of pneumonia.

According to the tradition introduced by Descartes, the “latitude” of a point is denoted by the letter x, “longitude” by the letter y

Many ways of indicating a place are based on this system.

For example, on a cinema ticket there are two numbers: row and seat - they can be considered as the coordinates of the seat in the theater.

Similar coordinates are accepted in chess. Instead of one of the numbers, a letter is taken: the vertical rows of cells are designated by letters of the Latin alphabet, and the horizontal rows by numbers. Thus, each square of the chessboard is assigned a pair of letters and numbers, and chess players are able to record their games. Konstantin Simonov writes about the use of coordinates in his poem “The Artilleryman’s Son.”

All night, walking like a pendulum,

The major did not close his eyes,

Bye on the radio in the morning

The first signal came:

"It's okay, I got there,

The Germans are to the left of me,

Coordinates (3;10),

Let's fire soon!

The guns are loaded

The major calculated everything himself.

And with a roar the first volleys

They hit the mountains.

And again the signal on the radio:

"The Germans are more right than me,

Coordinates (5; 10),

More fire soon!

Earth and rocks flew,

Smoke rose in a column.

It seemed that now from there

No one will leave alive.

Third radio signal:

"The Germans are around me,

Coordinates (4; 10),

Don't spare the fire.

The major turned pale when he heard:

(4;10) - just

The place where his Lyonka

Must sit now.

Konstantin Simonov "Son of an Artilleryman"

§2. Legends about the invention of the coordinate system

There are several legends about the invention of the coordinate system, which bears the name of Descartes.

Legend 1

This story has reached our times.

Visiting Parisian theaters, Descartes never tired of being surprised by the confusion, squabbles, and sometimes even challenges to a duel caused by the lack of an elementary order of distribution of the audience in the auditorium. The numbering system he proposed, in which each seat received a row number and serial number from the edge, immediately removed all reasons for contention and created a real sensation in Parisian high society.

Legend2. One day, Rene Descartes lay in bed all day, thinking about something, and a fly buzzed around and did not allow him to concentrate. He began to think about how to describe the position of a fly at any given time mathematically in order to be able to swat it without missing. And...I came up with Cartesian coordinates, one of greatest inventions in the history of mankind.

Markovtsev Yu.

Once upon a time in an unfamiliar city

Young Descartes arrived.

He was terribly tormented by hunger.

It was a chilly month of March.

I decided to ask a passerby

Descartes, trying to calm the trembling:

Where is the hotel, tell me?

And the lady began to explain:

- Go to the dairy shop

Then to the bakery, behind it

Gypsy woman sells pins

And poison for rats and mice,

You will surely find them

Cheeses, biscuits, fruits

And colorful silks...

I listened to all these explanations

Descartes, shivering from the cold.

He really wanted to eat

- Behind the shops is a pharmacy

(the pharmacist there is a mustachioed Swede),

And the church where at the beginning of the century

It seems my grandfather got married...

When the lady fell silent for a moment,

Suddenly her servant said:

- Walk straight three blocks

And two to the right. Entrance from the corner.

This is the third tale about the incident that gave Descartes the idea of coordinates.

Conclusion

While creating our project, we learned about the use of the coordinate plane in various areas science and Everyday life, some information from the history of the origin of the coordinate plane and mathematicians who made a great contribution to this invention. The material that we collected during the writing of the work can be used in school club classes as additional material to lessons. All this can interest schoolchildren and brighten up the learning process.

And we would like to end with these words:

“Imagine your life as a coordinate plane. The y-axis is your position in society. The x axis is moving forward, towards the goal, towards your dream. And as we know, it is endless... we can fall down, going further and further into minus, we can remain at zero and do nothing, absolutely nothing. We can rise up, we can fall, we can go forward or go back, and all because our whole life is a coordinate plane and the most important thing here is what your coordinate is...”

Bibliography

Glazer G.I. History of mathematics in school: - M.: Prosveshchenie, 1981. - 239 pp., ill.

Lyatker Ya. A. Descartes. M.: Mysl, 1975. - (Thinkers of the past)

Matvievskaya G. P. Rene Descartes, 1596-1650. M.: Nauka, 1976.

A. Savin. Coordinates Quantum. 1977. No. 9

Mathematics - supplement to the newspaper “First of September”, No. 7, No. 20, No. 17, 2003, No. 11, 2000.

Siegel F.Yu. Star alphabet: A manual for students. - M.: Education, 1981. - 191 pp., illus.

Steve Parker, Nicholas Harris. Illustrated encyclopedia for children. Secrets of the universe. Kharkov Belgorod. 2008

Materials from the site http://istina.rin.ru/

Cartesian coordinates of points on the plane. Equation of a circle. Coordinate plane

Point coordinates

Constructing a point at specified coordinates

Point coordinates

Constructing a point at specified coordinates

Rectangular coordinate system on a plane

Orty

Story

see also

Links

See what “Coordinate plane” is in other dictionaries: