The Cartesian coordinate system (Cartesian coordinates, French: les coordonn carté siennes) isCartesian coordinate systemand oblique coordinate system collectively.

Two lines that intersect at the originNumber axis, constitute the plane radiation coordinate system. If the units of measurement on the two axes are equal, the radial coordinate system is called the Cartesian coordinate system. A Cartesian coordinate system in which two axes are perpendicular to each other, called CartesianRight Anglecoordinate system, otherwise known as a Cartesian oblique coordinate system.

Chinese name: Cartesian coordinate system

外文名：Cartesian coordinates

Field: Mathematics

Interpretation: Cartesian coordinate system and oblique coordinate system collectively

attribute: a kind of radial coordinate system

Generalization: Spatial Cartesian Coordinate System

 

Introduction to Coordinate System

The Cartesian coordinate system isCartesian coordinate systemand oblique coordinate system collectively. Two number axes intersecting at the origin, forming a plane radiationcoordinate system. If the units of measurement on the two axes are equal, the radial coordinate system is called the Cartesian coordinate system. The Cartesian coordinate system with two axes perpendicular to each other is called Cartesian Cartesian coordinate system, otherwise it is called Cartesian oblique coordinate system. It should be pointed out that please compare the Cartesian coordinate system in mathematics with the movie 《different dimensional kill arrayThe definition in the movie is different from that in mathematics, so please do not confuse it.

The two-dimensional Cartesian coordinate system is composed of two axes perpendicular to each other and coinciding with 0 points. In the plane, the coordinates of any point are set based on the coordinates of the corresponding point on the number axis. In the plane, the correspondence between any point and coordinates is similar to the correspondence between points and coordinates on the number axis. Using Cartesian coordinates, geometric shapes Kowey used.AlgebraThe formula is clearly expressed. The Cartesian coordinates of each point of the geometric shape must obey this algebraic formula.

Cartesian coordinate system Two-dimensional coordinate system

A two-dimensional Cartesian coordinate system is usually set by two mutually perpendicular coordinate axes, often referred to as the x-axis.

and y-axis; the intersection of the two coordinate axes, calledOrigin, Usually marked as O, which means both "zero" and the first letter of the English "Origin. Each axis points in a particular direction. The coordinate axes of these two different lines determine a plane, called the xy-plane, also known as the Cartesian plane. Usually, as long as the two axes are perpendicular to each other, where they point has no effect on the analysis problem, but habitually (see the right figure), the x-axis is placed horizontally, called the horizontal axis, usually pointing to the right; The y-axis is placed vertically and is called the vertical axis, usually pointing upward. Such a positional relationship between the two axes is called two-dimensionalright-handed coordinate systemOr right hand. If this right hand is drawn on a piece of transparent paper, it will be in the plane anyway.RotateIt, the obtained are called right-handed system; but if the paper is turned over, the coordinate system seen on the back is called "left-handed system". This is related to the nature of the left and right when looking in the mirror.

To know the distance from the origin at any point on the coordinate axis. Suppose, we can depict the values on the axes. Then, starting from the origin, in the direction of the axis, every other unit length, the value is depicted on the axis. This value is the number of times depicted and is also a positive value from the origin.IntegerDistance; Similarly, carrying the direction pointed by the coordinate axis, we can also depict the negative integer distance from the origin. The value depicted on the x-axis is called the x-coordinate, also known as the abscissa, and the value depicted on the y-axis is called the y-Coordinates, also calledordinate. Although, here, both coordinates are integers, corresponding to specific points of the coordinate axis. Proportionally, we can generalizereal numbercoordinates and each point of the coordinate axis to which it corresponds. These two coordinates are the Cartesian coordinates, labeled (x,y).

The position of any point P in the plane Kowey uniquely expressed by rectangular coordinates. Just from point p

Draw a line perpendicular to the x-axis. From the intersection of this line with the x-axis, the x-coordinate of the point P Kowey found. Likewise, the y-coordinate of the point P Kowey found. In this way, we can get the Cartesian coordinates of the point P.

Cartesian coordinate system can also be extendedthree-dimensional space（3 dimension）与high dimensional space(higher dimension) 。

The two axes of the Cartesian coordinate system divide the plane into four parts, calledQuadrant, respectively.roman numeralsI, II, III, IV. By convention, both coordinates of quadrant I are positive; the x-coordinate of quadrant II is negative and the y-coordinate is positive; both coordinates of quadrant III are negative; and the x-coordinate of quadrant IV is positive and the y-coordinate is negative. Therefore, the quadrants are numbered counterclockwise, from quadrant I to quadrant IV.

Cartesian coordinate system three-dimensional coordinate system

The extension of the plane of the radial coordinate system and the Cartesian coordinate system to space: three non-coplanar number axes intersecting at the origin constitute the radial coordinate system of space. A radial coordinate system with equal units of measurement on three axes is called a spatial Cartesian coordinate system. A Cartesian coordinate system in which three axes are perpendicular to each other is called a spatial Cartesian rectangular coordinate system, otherwise it is called a spatial Cartesian oblique coordinate system.

space rectangular coordinate system

In order to communicateSpaceThe study of graphics and numbers, we need to establish the space of points and order

The connection between arrays is achieved by introducing a spatial rectangular coordinate system. Over a fixed point O, three mutually perpendicular number axes, all of which have O as the origin and generally have the sameLength UnitThe three axes are called the x-axis (Horizontal axis), y-axis (vertical axis), z-axis (vertical axis); collectivelycoordinate axis. The x-axis and y-axis are usually arranged on a horizontal plane, while the z-axis is a plumb line; theirPositive directionTo conform to the right-hand rule, that is, to hold the z-axis with the right hand, when the four fingers of the right hand turn from the positive x-axis to the positive y-axis at an angle of π/2, the thumb is pointing to the positive direction of the z-axis, so that the three axes form aspace rectangular coordinate systemThe point O is called the coordinate origin. This constitutes a Cartesian coordinate.

In the three-dimensional Cartesian coordinate system, three planes, xy-plane, yz-plane, xz-plane, the three-dimensional space is divided into eight parts, calledGua Limit(Octant) empty. The three coordinates of each point of the first divination limit are positive values.

Cartesian coordinate system generation process

It is said that one day, a French philosopher and mathematicianDescartesHe was ill in bed and seriously ill, but he still pondered a question over and over again:Geometryis intuitive, and algebraic equations are more abstract, can the geometry andalgebraic equationIn order to achieve this goal, the key is how to hook the points that make up the geometry and each group of "numbers" that satisfy the equation. He pondered hard and tried hard to figure out how to connect the "points" and "numbers. Suddenly, he saw a spider on the corner of the roof, pulling the silk down, and after a while, the spider climbed up the silk again, drawing the silk left and right on the top. The spider's "performance" made Descartes's train of thought suddenly clear. He thought, you can think of the spider as a point, it can move up, down, left and right in the room, can you use one for each position of the spider?Number of groupsWhat about being sure? He thought again, the two adjacent walls of the house and the ground handed over three lines, and if you take the corner of the wall on the ground as a starting point, take the three lines handed over as three.Number axis, then the position of any point in space Kowey used to find the sequential three numbers on these three number axes. On the other hand, any given group of three sequential numbers can also find a point P in space corresponding to it. Similarly, a group of numbers (X, Y) can represent a point on a plane, and a point on a plane can also be represented by a group of two sequential numbers. This is the embryonic form of the coordinate system.

Cartesian coordinate systemThe creation of, in algebra and geometry to build a bridge, it makes the geometric concept to use the number to express, the geometric figure can also use algebraic form to express. Thus Descartes in the establishment of the Cartesian coordinate system on the basis of the creation of the useAlgebraTo study the mathematical branch of geometry-analytic geometry, he boldly imagined that if the geometry is regarded as the motion trajectory of moving points, the geometry Kowey regarded as consisting of points with some common characteristics. For example, we can think of a circle as having an equal distance from a moving point to a fixed point.point trajectoryIf we again think of points as the basic elements that make up geometry and numbers as the componentssolution of equationSo algebra and geometry are one family.

Remarks

In the paper Geometry in the appendix to the book "Methods", Descartes explained the basic principles of analytic geometry and created a Cartesian coordinate system.

Before Descartes, geometry and algebra were two sciences. Geometry studied graphics and algebra studied numbers. Descartes was not satisfied with the abstractness of the isolated studies of these two sciences and attempted to link the two and make them concrete. Through the coordinate system labeling method he designed and his in-depth study of variables, he proved that geometric problems Kowey reduced to algebraic problems, and all algebraic methods Kowey used in solving them. Since then, variables have been introduced into mathematics, becoming a turning point in the development of mathematics and creating conditions for the emergence of calculus. Cartesian coordinate system is widely used in the field of engineering technology and physics.

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