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Example of Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2, 3) in gray, (-3, 1) in red, (-1.5, -2.5) in blue, and the original (0, 0) in purple. .

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## What Is 3/0

In geometry, the Cartesian coordinate system (UK: /k ɑːˈt iː zj ə n / , US: /k ɑːr ˈ t i ʒ ə n / ) is a coordinate system on a plane that uniquely identifies each point with a real pair. The numbers called coordinates are the distances to a point drawn from two fixed vertically oriented lines called coordinate lines, coordinate axes, or simply system axes (plural of axes). The point where they meet is called the origin, and its coordinates are (0, 0).

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Thus, the location of any point in three-dimensional space can be indicated by three Cartesian coordinates, which are the distances drawn from the point to three mutually perpendicular planes. More generally, n Cartesian coordinates define a point in n-dimensional Euclidean space for any dimension n. These coordinates are the distances drawn between n points, which are fixed hyperplanes.

Cartesian coordinate system with a circle of radius 2 at the origin marked in red. The equation of the circle is (x − a)2 + (y − b)2 = r2, where a and b are the coordinates of the center (a, b) and r is the radius.

Cartesian coordinates are named after Re Descartes, whose invention revolutionized mathematics in the 17th century, providing the first systematic link between geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by equations involving the coordinates of points on the shape. For example, a circle of radius 2 centered at the origin of the plane can be defined as the set of all points whose x and y coordinates satisfy the equation x2 + y2 = 4.

Cartesian coordinates are the foundation of analytic geometry, and they provide vivid geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, calculus of variables, group theory, and others. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for many applied fields that deal with geometry, including astronomy, physics, engineering, and more. They are the most common coordinate systems used in computer graphics, computer-aided geometric design, and other geometry-related data processing.

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Descartes refers to the French mathematician and philosopher Ray Descartes, who published the idea in 1637 while living in the Netherlands. It was originally discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery.

The French priest Nicolas Oresimi used a construction similar to Cartesian coordinates long before the time of Descartes and Fermat.

Both Descartes and Fermat used a single axis in their treatment and measured variable length relative to that axis. The concept of using pairs of axes was introduced later, after Frans van Schot and his Studt translated Descartes’ La Géometrie into Latin in 1649. These commentators introduced several concepts in an attempt to clarify the ideas contained in Descartes’ work.

The development of the Cartesian coordinate system by Isaac Newton and Gottfried Wilhelm Leibniz would play an important role in the development of calculus.

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Many other coordinate systems have been developed since Descartes, such as plane polar coordinates and spherical and cylindrical coordinates in three-dimensional space.

Choosing a Cartesian coordinate system for a one-dimensional space, that is, a line, involves choosing the point O (origin) of the line, the units of lgth, and the direction of the line. The orientation determines which of the two half-lines defined by O is positive and which is negative; We say that the line “points” (or “points”) from the negative side to the positive side. Every point on the line P can be described by its distance from O, so by the + or − sign, depending on which half of the line P contains.

A line with the chos Cartesian system is called a number line. Each real number has a unique location on the line. Conversely, each point on the line can be interpreted as a number on an ordered sequence, such as a real number.

Cartesian coordinate system in two dimensions (also called rectangular coordinate system or orthogonal coordinate system

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) is defined by an ordered pair of longitudinal lines (axes), a unit of length for both axes, and a direction for each axis. The point where the axes meet is the starting point for both, thus transforming each axis into a number line. For any point P, a line is drawn through P perpendicular to each axis, and the point where it meets the axis is interpreted as a number. These two numbers are in the order of chos the Cartesian coordinates of P. The inverse construction allows to determine the point P with its coordinates.

The first and second coordinates are called abscissa and ordinate respectively. And the point where the axes meet is called the origin of the coordinate system. Coordinates are usually written as two numbers in fractions, separated by commas, such as (3, −10.5). Thus, the coordinates at the origin are (0, 0), and points on the positive semi-axis that are one unit away from the origin have coordinates (1, 0) and (0, 1).

In mathematics, physics, and engineering, the first axis is usually defined or expressed as horizontal and oriented to the right, while the second axis is vertical and directed upward. (However, in some computer graphics contexts, the coordinate axis may be oriented downward.) The origin is often denoted by O, and the two coordinates are often denoted by the letters X and Y or x and y. goes The axes are called X axis and Y axis. The choice of characters is based on the original concept of using the last part of the alphabet to represent unknown values. The first part of the alphabet is used to denote known values.

The Euclidean plane with the chos Cartesian coordinate system is called the Cartesian plane. Geometric figures of linear representation can be defined in the Cartesian plane, such as the unit circle (with radius equal to the length of the triangle and the origin cter), the unit square (whose diagonals are depoints (0, 0) and (1, 1)), unit hyperbola etc.

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Both axes divide the plane into four right angles, called quadrants. Quadrants can be named or numbered in different ways, but the quadrant in which all coordinates are positive is usually called the first quadrant.

If the coordinates of a point are (x, y), then its distances from the X axis and from the Y axis are |y| and |x|, respectively; where | ·| Determines the absolute value of a number.

A three-dimensional Cartesian coordinate system with origin O and axis lines X, Y and Z shown by arrows. Marks on the axis are spaced a long unit apart. The black dot represents the point with coordinates x = 2, y = 3 and z = 4 or (2, 3, 4).

A Cartesian coordinate system for three-dimensional space consists of three aligned lines (axes) that pass through a common point (the origin) and are perpendicular to each other. orientation for each axis; And a long unit for all three axes. As in the two-dimensional case, each axis becomes a number line. For any point P in space, a hyperplane through P is assumed to be tangent to each coordinate axis, and the point where this hyperplane intersects the axis is interpreted as an integer. The Cartesian coordinates of P are these three numbers in chos order. The inverse construction describes the point P by its three coordinates.

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Alternatively, each coordinate of a point P can be considered as the distance from P to the hyperplane defined by the other two axes, with sign determined by the direction of the corresponding axis.

Each pair of axes defines a coordinate hyperplane. These hyperplanes divide space into eight octants. The octants are:

Coordinates are usually written as three numbers (or algebraic formulas) surrounded by parentheses and commas, such as (3, −2.5, 1) or (t, u + v, π/2 ). Thus, the origin has coordinates (0, 0, 0) and the unit points on the three axes are (1, 0, 0), (0, 1, 0) and (0, 0, 1).

There are no standard names for the three-axis coordinates (although the terms abscissa, ordinate, and applicator are sometimes used). Coordinates are often denoted by the letters X, Y and Z or x, y and z. Axes can do th

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