Cartesian coordinate system

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A Cartesian coordinate system is a coordinate system that assigns each point uniquely in the plane by a pair of coordinates of the number, which is the signed distance to the point of two perpendicular directed lines, measured in units the same length. Each reference line is called coordinate coordinates or only the axis (the plural axis of ) of the system, and the point where they meet is > origin , in the ordered pairs (0, 0) . Coordinates can also be defined as projection positions perpendicular from the second point of the axis, expressed as the signed distance from the origin.

One can use the same principle to determine the position of each point in three-dimensional space by three Cartesian coordinates, the distance it signs to three perpendicular plane (or, equivalently, by projection perpendicular to three perpendicular lines). In general, n Cartesian coordinates (elements of real n -space) define a point in n -the two Euclidean spaces of any dimension n . These coordinates are the same, up to the mark, to keep from vertical to hyper point of intersection.

The discovery of Cartesian coordinates in the 17th century by Renà © ¨ Descartes (Latinized name: Cartesius ) revolutionized mathematics by providing the first systematic relationship between Euclidean and algebraic geometry. Using a Cartesian coordinate system, geometric shapes (such as curves) can be explained by a Cartesian equation : an algebraic equation involving the coordinates of points located in the form. For example, a circle with radius 2, centered on the origin of the plane, can be described as the set of all points whose coordinates x and y satisfy the equation x ² y ² = 4 .

Cartesian co-ordinates are the foundations of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory, and more. A well-known example is the graph concept of a function. Cartesian coordinates are also an important tool for most of the applied disciplines related to geometry, including astronomy, physics, engineering, and more. They are the most commonly used coordinate systems in computer graphics, computer-aided geometric designs and data processing related to other geometries.

Video Cartesian coordinate system

Histori

The adjective Cartesian refers to the French mathematician and philosopher Renà © ¨ Descartes who published this idea in 1637. He was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish his discovery.. French cleric Nicole Oresme, used a construction similar to Cartesian coordinates long before the days of Descartes and Fermat.

Both Descartes and Fermat use a single axis in their care and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes' La GÃÆ' © omà © Ã… © trie was translated into Latin in 1649 by Frans van Schooten and his disciples. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work.

The development of Cartesian coordinate system will play a fundamental role in the development of calculus by Isaac Newton and Gottfried Wilhelm Leibniz. The description of the two coordinates of the plane is then generalized into the concept of vector space.

Many other coordinate systems have been developed since Descartes, such as pole coordinates for planes, and spherical and cylindrical coordinates for three-dimensional space.

Maps Cartesian coordinate system

Description

One dimension

Selecting a Cartesian coordinate system for a one-dimensional space - that is, for a straight line - involves selecting the point of (i) the line (origin), the unit length, and the orientation for the line. The orientation of choosing which of the two and a half lines defined by O is positive, and the negative; we then say that the line is "oriented" (or "point") of the negative half towards the positive half. Then any point P of the line can be determined by the distance from O , taken with a sign or - depending on half of the line containing P

Lines with selected Cartesian system are called line number . Each real number has a unique location in the phone. In contrast, every point on a line can be interpreted as a number in a regular sequence like a real number.

Two dimensions

The two-dimensional Cartesian coordinate system (also called the rectangular coordinate system or orthogonal coordinate system ) is determined by a perpendicular (axis) line pair, a unit of length for both axes, and orientation for each axis. The point at which the axes meet is taken as the origin for both, thus converting each axis into a number line. For each point P , a line is drawn through P that is perpendicular to each axis, and the position where it encounters the axis is interpreted as a number. Two numbers, in the order selected, are Cartesian coordinates of P . Conversely construction allows one to specify the point P given its coordinates.

The first and second coordinates are called abscissa and ordinate of P , respectively; and the point at which the axis meets is called origin of the coordinate system. Coordinates are usually written as two numbers in parentheses, in that order, separated by commas, as in (3, -10.5) . Thus its origin has the coordinates (0,0) , and the points on the positive half axis, one unit of origin, have coordinates (1.0) and (0.1) .

In mathematics, physics, and engineering, the first axis is usually defined or described as horizontal and oriented to the right, and the axis is both vertical and oriented upward. (However, in some computer graphics context, the ordinate axis may be downward oriented.) The origin is often labeled O , and the two coordinates are often denoted by the letters X and Y , or x and y . The axes can then be referred to as X -axis and Y -axis. The choice of letters comes from the original convention, which uses the end of the alphabet to indicate an unknown value. The first part of the alphabet is used to denote a known value.

Euclidean aircraft with selected Cartesian coordinate system Cartesian plane . In a Cartesian plane one can define a canonical representation of a particular geometric figure, such as a unit circle (with radius equal to unit length, and center at the origin), square units (which have a diagonal endpoint at (0,0) span> and (1,1) ), hyperbola units, and so on.

The two axes divide the plane into four right angles, called the quadrant . Quadrants can be named or numbered in various ways, but quadrants where all positive coordinates are usually called first quadrant .

If the coordinates of a point are ( x , y ) , then the distance from X -axis and from Y -axis is | y | and | x |. respectively; where |... | indicates the absolute value of a number.

Three dimensions

The Cartesian coordinate system for three-dimensional space consists of triplets ordered by lines (axis ) that passes through common points ( origin ), and pair-wise perpendicular; orientation for each axis; and one unit of length for all three axes. As in the case of two dimensions, each axis becomes a number line. For each point of P space, one assumes the field through P is perpendicular to each coordinate axis, and interprets the point at which the plane intersects the axis as a number. Cartesian coordinates of P are all three numbers, in the selected order. Conversely construction determines the point P given three coordinates.

Alternatively, any point coordinate P can be taken as distance from P to a field determined by the other two axes, with a sign determined by the corresponding orientation of the axis.

Each pair of axes defines the coordinates of the plane . These planes divide the space into eight trihedra, called octane .

Coordinates are usually written as three numbers (or algebraic formulas) surrounded by parentheses and separated by commas, as in (3, -2,5,1) or ( t , u v ,?/2) . Thus, the origin has the coordinates (0,0,0) , and the unit points on the three axes are (1,0,0) , (0, 1 , 0) , and (0,0,1) .

There is no standard name for coordinates in three axes. The coordinates are often denoted by the letters X , Y , and Z (or x , y >, and z ), in which case the line is called X -, Y respectively. Then the coordinate plane can be called XY -, YZ -, and XZ-aircraft .

In mathematics, physics, and engineering contexts, the first two axes are often defined or described as horizontal, with the third axis pointing upwards. In this case the third coordinate can be called high or altitude . Orientation is usually chosen so that the angle of 90 degrees from the first axis to the second axis looks counterclockwise when seen from the point (0,0,1) ; a convention commonly called the right hand rule .

Higher dimensions

Pesawat Euclidean dengan sistem Cartesian yang dipilih disebut Cartesian plane . Karena koordinat Cartesian unik dan tidak ambigu, titik-titik dari bidang Cartesian dapat diidentifikasi dengan pasangan bilangan real; yaitu dengan produk Cartesian ${\ displaystyle \ mathbb {R} ^ {2} = \ mathbb {R} \ times \ mathbb {R}}  ÂÂ$ , di mana ${\ displaystyle \ mathbb {R}}  ÂÂ$ adalah himpunan semua real. Dengan cara yang sama, titik-titik di setiap ruang dimensi Euclidean n diidentifikasi dengan tupel (daftar) dari n bilangan real, yaitu, dengan produk Cartesian ${\ displaystyle \ mathbb {R} ^ {n}}  ÂÂ$ .

Generalisasi

Cartesian coordinate generalization concepts to allow the axes that are not perpendicular to each other, and/or different units along each axis. In this case, each coordinate is obtained by projecting the point to one axis along the direction parallel to the other axis (or, in general, to the hyperplane determined by all other axes). In such oblique coordinate systems the distance and angle calculations should be changed from those in standard Cartesian systems, and many standard formulas (such as Pythagoras for distance formulas) do not apply (see afine field).

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Notations and conventions

Cartesian coordinates of a point are usually written in parentheses and separated by commas, as in (10, 5) or (3, 5, 7) . Origins are often labeled with capital letters O . In analytic geometry, unknown or common coordinates are often denoted by letters ( x , y ) on the plane, and ( x , y , z ) in a three-dimensional space. This habit comes from the algebraic convention, which uses letters near the end of the alphabet for unknown values ​​(such as point coordinates in many geometric problems), and the initial near letter for a given number.

These conventional names are often used in other domains, such as physics and engineering, although other letters can be used. For example, in a graph showing how pressure varies with time, the graphic coordinates can be denoted p and t . Each axis is usually named as a measured coordinate along it; so one word x-axis , y-axis , t-axis , etc.

Another common convention for naming coordinates is to use subscripts, such as ( x ₁ , x ₂ ,..., x _n ) for coordinates n in n -space dimensions, especially when < i> n is greater than 3 or not specific. Some authors prefer numbering ( x ₀ , x ₁ ,..., x _{n -1} ). This notation is very advantageous in computer programming: by storing point coordinates as arrays instead of records, subscripts can serve to index coordinates.

In a mathematical illustration of a two-dimensional Cartesian system, the first coordinate (traditionally called abscissa) is measured along the horizontal axis, oriented from left to right. The second coordinate (ordinate) is then measured along the vertical axis, usually oriented from the bottom up. Young children learn Cartesian systems, usually learn commands to read values ​​before cementing x, y, z axis concepts, by starting with 2D mnemonics (eg 'Walking along the hall and going up stairs' is similar to straight across the x-axis then up vertically along the y-axis).

Computer graphics and image processing, however, often use coordinate systems with y -cleanly downward on a computer screen. The Convention was developed in the 1960s (or earlier) of the way the images were originally stored in display buffers.

For a three-dimensional system, a convention is to describe xy -Step horizontally, with z -axis added to represent high (positive up). Next, there is a convention to direct x -axis to the viewer, biased to the right or left. If the diagram (3D projection or 2D perspective image) shows x - and y -axis horizontally and vertically, respectively, then z - The axis should be pointed to pointing "out of the page" toward the viewer or camera. In the 2D diagram â € <â € z -axis will appear as a line or ray that leads down and to the left or down and to the right, depending on the viewing angle considered or camera perspective. In any diagram or display, the orientation of the three axes, in whole, is arbitrary. However, the orientation of the axes relative to each other must always obey the rules of the right hand, unless specifically stated otherwise. All laws of physics and mathematics assume this right hand, which ensures consistency.

For 3D diagrams the names "absis" and "ordinates" are rarely used for x and y , respectively. When they are, z -ordinates are sometimes called applicate . abscissa , ordinat and applicate are sometimes used to refer to coordinate axes rather than coordinate values.

Quadrants and octane

The axis of the two-dimensional Cartesian system divides the plane into four infinite regions, called quadrants , each bordered by two and a half axes. It is often numbered from 1 to 4 and is denoted by Roman numerals: I (where the signs of the two coordinates are I (,), II (-,), III (-, -), and IV (, -) axis is drawn in accordance with math custom, numbering goes counterclockwise from the upper right quadrant ("northeast").

Similarly, the three-dimensional Cartesian system defines the division of space into eight regions or octane , corresponding to the coordinate signs of the dots. The convention used to name a particular octant is to include its signs, e.g. () or (- -) . The generalization of quadrants and octants to a number of arbitrary dimensions is orthant , and the same naming system applies.

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Cartesian formula for the

field

The distance between two points

Jarak Euclidean antara dua titik pesawat dengan koordinat Cartesian ${\ displaystyle (x_ {1}, y_ {1})}  ÂÂ$ dan ${\ displaystyle (x_ {2}, y_ {2})}  ÂÂ$ adalah

${\ displaystyle d = {\ sqrt {(x_ {2} -x_ {1}) ^ {2} (y_ {2} -y_ {1}) ^ { 2}}}.}  ÂÂ$

Ini adalah teorema Pythagoras versi Cartesian. Dalam ruang tiga dimensi, jarak antara titik ${\ displaystyle (x_ {1}, y_ {1}, z_ {1})}  ÂÂ$ dan ${\ displaystyle (x_ {2}, y_ {2}, z_ {2})}  ÂÂ$ adalah

${\ displaystyle d = {\ sqrt {(x_ {2} -x_ {1}) ^ {2} (y_ {2} -y_ {1}) ^ { 2} (z_ {2} -z_ {1}) ^ {2}}},}  ÂÂ$

which can be obtained with two Pythagoras theorem apps in sequence.

Euclidean Transformation

Euclidean transformations or Euclidean motions are the (biased) mapping of Euclidean plane points for themselves that keep the distance between points. There are four types of this mapping (also called isometry): translation, rotation, reflection and glide reflection.

Translation

Translate a set of plane dots, keeping distance and directions between them, equivalent to adding a fixed number of ( a , b ) to the Cartesian coordinates of each point in set. That is, if the original coordinates of a point are ( x , y ) , after their translation will , ${\ displaystyle (x ', y') = (x a, y b).}$

Rotation

Untuk memutar angka berlawanan arah jarum jam di sekitar titik asal dengan beberapa sudut ${\ displaystyle \ theta}  ÂÂ$ setara dengan mengganti setiap titik dengan koordinat ( x , y ) pada titik dengan koordinat (< i> x ', y' ), di mana

${\ displaystyle x '= x \ cos \ theta -y \ sin \ theta}  ÂÂ$

${\ displaystyle y '= x \ sin \ theta y \ cos \ theta.}  ÂÂ$

Thereby:

${\ displaystyle (x ', y') = ((x \ cos \ theta -y \ tanpa \ theta \,), (x \ tanpa \ theta y \ cos \ theta \,)).}  ÂÂ$

Refleksi

If ( x , y ) is the Cartesian coordinates of a point, i> y ) is the reflection coordinate in the second coordinate axis (y axis), as if it were a mirror. Similarly, ( x , - y ) is its reflection coordinates in the first coordinate axis (x axis). In general, the reflection across the line through the starting point makes the angle ${\ displaystyle \ theta}$ with x axis, equivalent to replacing each dot with coordinates ( x ) at a point with the coordinates ( x ?, y ?) , where

$Â\; ÂÂ\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; Âx^{?}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂxÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\cos Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â2Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â?Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Ây}$ mi                 2         ?           {\ displaystyle x '= x \ cos 2 \ theta y \ sin 2 \ theta}  Â

$Â\; ÂÂ\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; Ây^{?}Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; Â\; Âx}$ mi                 2         ?         -         y         cos                 2         ?         .           {\ displaystyle y '= x \ sin 2 \ theta -y \ cos 2 \ theta.}  Â

Dengan demikian: ${\ displaystyle (x ', y') = ((x \ cos 2 \ theta y \ sin 2 \ theta \,), (x \ sin 2 \ theta - y \ cos 2 \ theta \,)).}  ÂÂ$

Glide reflection

Glide reflection is a reflection arrangement on a line followed by a translation toward that line. It can be seen that this sequence of operations is not important (translation can take precedence, followed by reflection).

General matrix form of transformation

Transformasi Euclidean dari pesawat ini semuanya dapat dijelaskan dengan cara yang seragam dengan menggunakan matriks. Hasilnya ${\ displaystyle (x ', y')}  ÂÂ$ untuk menerapkan transformasi Euclidean ke titik ${\ displaystyle (x, y)}  ÂÂ$ diberikan oleh rumus

${\ displaystyle (x ', y') = (x, y) A b}  ÂÂ$

di mana A adalah matriks ortogonal 2ÃÆ' — 2 dan b = ( b ₁ , b ₂ ) adalah pasangan angka acak yang diurutkan; itu adalah,

${\ displaystyle x '= xA_ {11} yA_ {21} b_ {1}}  ÂÂ$

${\ displaystyle y '= xA_ {12} yA_ {22} b_ {2},}  ÂÂ$

dimana

${\ displaystyle A = {\ begin {pmatrix} A_ {11} & amp; A_ {12} \\ A_ {21} & amp; A_ {22} \ end {pmatrix} }.}  ÂÂ$ [Perhatikan penggunaan vektor baris untuk koordinat titik dan bahwa matriks ditulis di sebelah kanan.]

Untuk menjadi orthogonal , matriks A harus memiliki baris ortogonal dengan panjang Euclidean yang sama, yaitu,

${\ displaystyle A_ {11} A_ {21} A_ {12} A_ {22} = 0}  ÂÂ$

dan

${\ displaystyle A_ {11} ^ {2} A_ {12} ^ {2} = A_ {21} ^ {2} A_ {22} ^ {2} = 1.}  ÂÂ$

This is equivalent to saying that A times its transpose must be t

Source of the article : Wikipedia