# Dictionary Definition

ordinate n : the value of a coordinate on the
vertical axis

### Verb

2 bring (components or parts) into proper or
desirable coordination correlation; "align the wheels of my car";
"ordinate similar parts" [syn: align, coordinate]

# User Contributed Dictionary

## English

### Noun

- the value of a coordinate on the vertical (Y) axis

#### Translations

- ttbc Indonesian: ordinat
- ttbc Spanish: ordenar

## Italian

### Noun

ordinate- Plural of ordinata

### Adjective

ordinate p### Verb

ordinate- second-person plural present tense of ordinare
- second-person plural present subjunctive of ordinare
- second-person plural imperative of ordinare
- feminine plural past participle of ordinare

# Extensive Definition

In mathematics, the Cartesian
coordinate system (also called rectangular coordinate system) is
used to determine each point
uniquely in a plane
through two numbers,
usually called the x-coordinate or abscissa and the y-coordinate or
ordinate of the point. To define the coordinates, two perpendicular directed lines (the x-axis, and
the y-axis), are specified, as well as the unit length,
which is marked off on the two axes (see Figure 1). Cartesian
coordinate systems are also used in space
(where three coordinates are used) and in higher
dimensions.

Using the Cartesian coordinate system, geometric
shapes (such as curves)
can be described by algebraic equations, namely equations
satisfied by the coordinates of the points lying on the shape. For
example, the circle of radius 2 may be described by the equation x2
+ y2 = 4 (see Figure 2).

## History

Cartesian means relating to the French mathematician and philosopher René
Descartes (Latin: Cartesius), who, among other things, worked
to merge algebra and
Euclidean
geometry. This work was influential in the development of
analytic
geometry, calculus,
and cartography.

The idea of this system was developed in 1637 in two writings
by Descartes and independently by Pierre de
Fermat, although Fermat did not publish the discovery. In part
two of his Discourse
on Method, Descartes introduces the new idea of specifying the
position of a point or
object on a surface, using two intersecting axes as measuring
guides. In La
Géométrie, he further explores the above-mentioned
concepts.

## Two-dimensional coordinate system

A Cartesian coordinate
system in two dimensions is commonly defined by two axes, at
right
angles to each other, forming a plane (an xy-plane). The
horizontal
axis is normally labeled x, and the vertical
axis is normally labeled y. In a three dimensional coordinate
system, another axis, normally labeled z, is added, providing a
third dimension of space measurement. The axes are commonly defined
as mutually orthogonal to each other
(each at a right angle to the other). (Early systems allowed
"oblique" axes, that is, axes that did not meet at right angles,
and such systems are occasionally used today, although mostly as
theoretical exercises.) All the points in a Cartesian coordinate
system taken together form a so-called Cartesian plane. Equations
that use the Cartesian coordinate system are called Cartesian
equations.

The point of intersection, where the axes meet,
is called the origin normally labeled O. The x and y axes define a
plane that is referred to as the xy plane. Given each axis, choose
a unit length, and mark off each unit along the axis, forming a
grid. To specify a particular point on a two dimensional coordinate
system, indicate the x unit first (abscissa), followed by the y
unit (ordinate) in the form (x,y), an ordered pair.

The choice of letters comes from a convention, to
use the latter part of the alphabet to indicate unknown values. In
contrast, the first part of the alphabet was used to designate
known values.

An example of a point P
on the system is indicated in Figure 3, using the coordinate
(3,5).

The intersection of the two axes creates four
regions, called quadrants, indicated by the Roman numerals I (+,+),
II (−,+), III (−,−), and IV (+,−). Conventionally, the quadrants
are labeled counter-clockwise starting from the upper right
("northeast") quadrant. In the first quadrant, both coordinates are
positive, in the second quadrant x-coordinates are negative and
y-coordinates positive, in the third quadrant both coordinates are
negative and in the fourth quadrant, x-coordinates are positive and
y-coordinates negative (see table below.)

## Three-dimensional coordinate system

The three dimensional Cartesian coordinate system
provides the three physical dimensions of space — length, width,
and height. Figures 4 and 5 show two common ways of representing
it.

The three Cartesian axes defining the system are
perpendicular to each other. The relevant coordinates are of the
form (x,y,z). As an example, figure 4 shows two points
plotted in a three-dimensional Cartesian coordinate system:
P(3,0,5) and Q(−5,−5,7). The axes are depicted in a
"world-coordinates" orientation with the z-axis pointing up.

The x-, y-, and z-coordinates of a point can also
be taken as the distances from the yz-plane, xz-plane, and xy-plane
respectively. Figure 5 shows the distances of point P from the
planes.

The xy-, yz-, and xz-planes divide the
three-dimensional space into eight subdivisions known as octants, similar to the quadrants
of 2D space. While conventions have been established for the
labelling of the four quadrants of the x-y plane, only the first
octant of three dimensional space is labelled. It contains all of
the points whose x, y, and z coordinates are positive.

The z-coordinate is also called applicate.

## Orientation and handedness

- see also: right-hand rule

### In two dimensions

Fixing or choosing the x-axis determines the y-axis up to direction. Namely, the y-axis is necessarily the perpendicular to the x-axis through the point marked 0 on the x-axis. But there is a choice of which of the two half lines on the perpendicular to designate as positive and which as negative. Each of these two choices determines a different orientation (also called handedness) of the Cartesian plane.The usual way of orienting the axes, with the
positive x-axis pointing right and the positive y-axis pointing up
(and the x-axis being the "first" and the y-axis the "second" axis)
is considered the positive or standard orientation, also called the
right-handed orientation.

A commonly used mnemonic for defining the
positive orientation is the right hand
rule. Placing a somewhat closed right hand on the plane with
the thumb pointing up, the fingers point from the x-axis to the
y-axis, in a positively oriented coordinate system.

The other way of orienting the axes is following
the left hand rule, placing the left hand on the plane with the
thumb pointing up.

Regardless of the rule used to orient the axes,
rotating the coordinate system will preserve the orientation.
Switching the role of x and y will reverse the orientation.

### In three dimensions

Once the x- and y-axes are specified, they
determine the line
along which the z-axis should lie, but there are two possible
directions on this line. The two possible coordinate systems which
result are called 'right-handed' and 'left-handed'. The standard
orientation, where the xy-plane is horizontal and the z-axis points
up (and the x- and the y-axis form a positively oriented
two-dimensional coordinate system in the xy-plane if observed from
above the xy''-plane) is called right-handed or positive.

The name derives from the right-hand
rule. If the index finger
of the right hand is pointed forward, the middle
finger bent inward at a right angle to it, and the thumb placed at a right angle to
both, the three fingers indicate the relative directions of the x-,
y-, and z-axes in a right-handed system. The thumb indicates the
x-axis, the index finger the y-axis and the middle finger the
z-axis. Conversely, if the same is done with the left hand, a
left-handed system results.

Figure 7 is an attempt at depicting a left- and a
right-handed coordinate system. Because a three-dimensional object
is represented on the two-dimensional screen, distortion and
ambiguity result. The axis pointing downward (and to the right) is
also meant to point towards the observer, whereas the "middle" axis
is meant to point away from the observer. The red circle is
parallel to the horizontal xy-plane and indicates rotation from the
x-axis to the y-axis (in both cases). Hence the red arrow passes in
front of the z-axis.

Figure 8 is another attempt at depicting a
right-handed coordinate system. Again, there is an ambiguity caused
by projecting the three-dimensional coordinate system into the
plane. Many observers see Figure 8 as "flipping in and out" between
a convex cube and
a concave
"corner". This corresponds to the two possible orientations of the
coordinate system. Seeing the figure as convex gives a left-handed
coordinate system. Thus the "correct" way to view Figure 8 is to
imagine the x-axis as pointing towards the observer and thus seeing
a concave corner.

## Representing a vector in the standard basis

A point in space in a Cartesian coordinate system may also be represented by a vector, which can be thought of as an arrow pointing from the origin of the coordinate system to the point. If the coordinates represent spatial positions (displacements) it is common to represent the vector from the origin to the point of interest as \mathbf. In three dimensions, the vector from the origin to the point with Cartesian coordinates (x,y,z) is sometimes written as:\mathbf = x \mathbf + y \mathbf + z \mathbf

where \mathbf, \mathbf, and \mathbf are unit vectors
that point the same direction as the x, y, and z axes,
respectively. This is the quaternion representation of
the vector, and was introduced by
Sir William Rowan Hamilton. The unit vectors \mathbf, \mathbf,
and \mathbf are called the versors of the coordinate system, and
are the vectors of the standard
basis in three-dimensions.

## Applications

Cartesian coordinates are often used to represent
two or three dimensions of space, but they can also be used to
represent many other quantities (such as mass, time, force, etc.).
In such cases the coordinate axes will typically be labelled with
other letters (such as m, t, F, etc.) in place of x, y, and z. Each
axis may also have different units
of measurement associated with it (such as kilograms, seconds,
pounds, etc.). It is also possible to define coordinate systems
with more than three dimensions to represent relationships between
more than three quantities. Although four- and higher-dimensional
spaces are difficult to visualize, the algebra of Cartesian
coordinates can be extended relatively easily to four or more
variables, so that certain calculations involving many variables
can be done. (This sort of algebraic extension is what is used to
define the geometry of higher-dimensional spaces, which can become
rather complicated.) Conversely, it is often helpful to use the
geometry of Cartesian coordinates in two or three dimensions to
visualize algebraic relationships between two or three (perhaps two
or three of many) non-spatial variables.

## Further notes

In computational
geometry the Cartesian coordinate system is the foundation for
the algebraic manipulation of geometrical shapes. Many other
coordinate systems have been developed since Descartes. One common
set of systems use polar
coordinates; astronomers and physicists often use spherical
coordinates, a type of three-dimensional polar coordinate
system.

It may be interesting to note that some have
indicated that the master artists of the Renaissance
used a grid, in the form of a wire mesh, as a tool for breaking up
the component parts of their subjects they painted. That this may
have influenced Descartes is merely speculative. (See perspective,
projective
geometry.)

## See also

- List of canonical coordinate transformations
- Graph of a function
- Point plotting
- Orientation (mathematics)
- Right-hand rule
- Regular grid
- Taxicab geometry
- Euclidean space
- Curvilinear coordinates
- Stereographic projection
- Point (geometry)
- Line (mathematics)
- Plane (mathematics)
- Integer point
- Complex plane
- Coordinates (mathematics)
- Coordinate systems
- Geocentric coordinates
- Parallel coordinates
- René Descartes
- Discourse on Method
- La Géométrie
- Ordered pair
- Analytic geometry
- Abstraction (mathematics)
- Notation system
- ISO 31-1
- Graph paper

## References

Descartes, René. Oscamp, Paul J. (trans).
Discourse on Method, Optics, Geometry, and Meteorology. 2001.

## Bibliography

- Methods of Theoretical Physics, Part I | pages = p. 656}}

- The Mathematics of Physics and Chemistry | pages = p. 177 }}

- Mathematical Handbook for Scientists and Engineers , ASIN B0000CKZX7 | pages = pp. 55–79}}

- Mathematische Hilfsmittel des Ingenieurs | pages = p. 94}}

- Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions

## External links

ordinate in Afrikaans: Cartesiese
koördinatestelsel

ordinate in Bosnian: Dekartov koordinatni
sistem

ordinate in Bulgarian: Декартова координатна
система

ordinate in Catalan: Sistema de coordenades
cartesianes

ordinate in Czech: Kartézská soustava
souřadnic

ordinate in Danish: Kartesisk
koordinatsystem

ordinate in German: Kartesisches
Koordinatensystem

ordinate in Modern Greek (1453-): Καρτεσιανό
σύστημα συντεταγμένων

ordinate in Spanish: Coordenadas
cartesianas

ordinate in Esperanto: Kartezia koordinato

ordinate in Persian: دستگاه مختصات دکارتی

ordinate in French: Coordonnées
cartésiennes

ordinate in Korean: 직교 좌표계

ordinate in Hindi: कार्तीय निर्देशांक
पद्धति

ordinate in Indonesian: Sistem koordinat
Kartesius

ordinate in Icelandic:
Kartesíusarhnitakerfið

ordinate in Italian: Piano cartesiano

ordinate in Hebrew: מערכת צירים קרטזית

ordinate in Marathi: कार्टेशियन गुणक
पद्धती

ordinate in Malay (macrolanguage): Sistem
koordinat Cartes

ordinate in Dutch: Cartesisch
coördinatenstelsel

ordinate in Japanese: 直交座標系

ordinate in Norwegian: Kartesisk
koordinatsystem

ordinate in Low German: Karteesch
Koordinatensystem

ordinate in Polish: Układ współrzędnych
kartezjańskich

ordinate in Portuguese: Sistema de coordenadas
cartesiano

ordinate in Romanian: Coordonate
carteziene

ordinate in Russian: Прямоугольная система
координат

ordinate in Albanian: Sistemi koordinativ
kartezian

ordinate in Simple English: Cartesian coordinate
system

ordinate in Slovak: Karteziánska sústava
súradníc

ordinate in Slovenian: Kartezični koordinatni
sistem

ordinate in Serbian: Декартов координатни
систем

ordinate in Finnish:
Koordinaatisto#Suorakulmainen_koordinaatisto

ordinate in Swedish: Kartesiskt
koordinatsystem

ordinate in Tamil: காட்டீசியன் ஆள்கூற்று
முறைமை

ordinate in Thai: ระบบพิกัดคาร์ทีเซียน

ordinate in Vietnamese: Hệ tọa độ
Descartes

ordinate in Turkish: Kartezyen koordinat
sistemi

ordinate in Ukrainian: Декартова система
координат

ordinate in Chinese: 笛卡儿坐标系