Illustration of a Cartesian coordinate
plane. Four points are marked and
labeled with their coordinates: (2,3) in
green, ( 3,1) in red, ( 1.5, 2.5) in blue,
and the origin (0,0) in purple.
Cartesian coordinate system with a circle
of radius 2 centered at the origin marked
in red. The equation of a circle is x
2 + y2
= r
2.
Cartesian coordinate system
From Wikipedia, the free encyclopedia
A
Cartesian coordinate system specifies each point
uniquely in a plane by a pair of numerical coordinates,
which are the signed distances from the point to two fixed
perpendicular directed lines, measured in the same unit of
length.
Each reference line is called a coordinate axis or just axis
of the system, and the point where they meet is its
origin.
The coordinates can also be defined as the positions of the
perpendicular projections of the point onto the two axes,
expressed as a signed distances from the origin.
One can use the same principle to specify the position of
any point in three-dimensional space by three Cartesian
coordinates, its signed distances to three mutually
perpendicular planes (or, equivalently, by its perpendicular
projection onto three mutually perpendicular lines). In
general, one can specify a point in a space of any dimension
n
by use of n Cartesian coordinates, the signed distances
from n mutually perpendicular hyperplanes.
The invention of Cartesian coordinates in the 17th century
by René Descartes revolutionized mathematics by providing
the first systematic link between Euclidean geometry and
algebra. Using the Cartesian coordinate system, geometric
shapes (such as curves) can be described by Cartesian
equations: algebraic equations involving the coordinates of
the points lying on the shape. For example, a circle of radius
2 may be described as the set of all points whose
coordinates x and y satisfy the equation x2 + y2 = 22.
Cartesian coordinates are the foundation 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
familiar example is the concept of the graph of a function.
Cartesian coordinates are also essential tools for most
applied disciplines that deal with geometry, including
astronomy, physics, engineering, and many more. They are
the most common coordinate system used in computer
graphics, computer-aided geometric design, and other
geometry-related data processing.
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Contents
1 History
2 Definitions
2.1 Number line
2.2 Cartesian coordinates in two dimensions
2.3 Cartesian coordinates in three dimensions
2.4 Generalizations
3 Notations and conventions
3.1 Quadrants and octants
4 Cartesian space
5 Cartesian formulas for the plane
5.1 Distance between two points
5.2 Euclidean transformations
5.2.1 Translation
5.2.2 Scaling
5.2.3 Rotation
5.2.4 Reflection
5.2.5 General transformations
6 Orientation and handedness
6.1 In two dimensions
6.2 In three dimensions
7 Representing a vector in the standard basis
8 Applications
9 See also
10 Notes
11 References
12 External links
History
The adjective
Cartesian refers to the French mathematician and philosopher René Descartes (who used
the name Cartesius in Latin).
The idea of this system was developed in 1637 in two writings by Descartes and independently by Pierre
de Fermat, although Fermat used three dimensions and did not publish the discovery.[1] 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.[citation needed] In La Géométrie, he
further explores the above-mentioned concepts.[2]
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.
[citation needed] (See perspective,
projective geometry.) The development of the Cartesian coordinate system enabled the development of
calculus by Isaac Newton and Gottfried Wilhelm Leibniz.[3]
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Nicole Oresme, a French philosopher of the 14th Century, used constructions similar to Cartesian
coordinates well before the time of Descartes.
Many other coordinate systems have been developed since Descartes, such as the polar coordinates for
the plane, and the spherical and cylindrical coordinates for three-dimensional space.
Definitions
Number line
Main article: Number line
Choosing a Cartesian coordinate system for a one-dimensional space that is, for a straight line means
choosing a point
O of the line (the origin), a unit of length, and an orientation for the line. The latter
means choosing which of the two half-lines determined by O is the positive, and which is negative; we
then say that the line is oriented (or points) from the negative half towards the positive half. Then each
point p of the line can be specified by its distance from O, taken with a + or sign depending on which
half-line contains p.
A line with a chosen Cartesian system is called a number line. Every real number, whether integer,
rational, or irrational, has a unique location on the line. Conversely, every point on the line can be
interpreted as a number in an ordered continuum which includes the real numbers.
Cartesian coordinates in two dimensions
The modern Cartesian coordinate system in two dimensions (also called a rectangular coordinate
system) is defined by an ordered pair of perpendicular lines (axes), a single unit of length for both axes,
and an orientation for each axis. (Early systems allowed "oblique" axes, that is, axes that did not meet at
right angles.) The lines are commonly referred to as the
x and y-axes where the x-axis is taken to be
horizontal and the y-axis is taken to be vertical. The point where the axes meet is taken as the origin for
both, thus turning each axis into a number line. For a given point P, a line is drawn through P
perpendicular to the
x-axis to meet it at X and second line is drawn through P perpendicular to the y-axis
to meet it at Y. The coordinates of P are then X and Y interpreted as numbers x and y on the
corresponding number lines. The coordinates are written as an ordered pair (x, y).
The point where the axes meet is the common origin of the two number lines and is simply called the
origin
. It is often labeled O and if so then the axes are called Ox and Oy. A plane with x and y-axes
defined is often referred to as the Cartesian plane or xy plane. The value of x is called the x-coordinate or
abscissa and the value of y is called the y-coordinate or ordinate.
The choices of letters come from the original convention, which is to use the latter part of the alphabet
to indicate unknown values. The first part of the alphabet was used to designate known values.
Cartesian coordinates in three dimensions
Choosing a Cartesian coordinate system for a three-dimensional space means choosing an ordered triplet
of lines (axes), any two of them being perpendicular; a single unit of length for all three axes; and an
orientation for each axis. As in the two-dimensional case, each axis becomes a number line. The
coordinates of a point
p are obtained by drawing a line through p perpendicular to each coordinate axis,
and reading the points where these lines meet the axes as three numbers of these number lines.
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A three dimensional Cartesian
coordinate system, with origin
O and
axis lines X, Y and Z, oriented as shown
by the arrows. The tic marks on the
axes are one length unit apart. The
black dot shows the point with
coordinates X = 2, Y = 3, and Z = 4, or
(2,3,4).
The coordinate surfaces of the
Cartesian coordinates (x, y, z). The zaxis
is vertical and the x-axis is
highlighted in green. Thus, the red
plane shows the points with x=1, the
blue plane shows the points with z=1,
and the yellow plane shows the points
with y=-1. The three surfaces intersect
at the point P (shown as a black sphere)
with the Cartesian coordinates (1, -1,
1).
Alternatively, the coordinates of a point
p can also be taken as
the (signed) distances from p to the three planes defined by
the three axes. If the axes are named x, y, and z, then the x
coordinate is the distance from the plane defined by the
y and
z
axes. The distance is to be taken with the + or sign,
depending on which of the two half-spaces separated by that
plane contains p. The y and z coordinates can be obtained in
the same way from the (x,z) and (x,y) planes, respectively.
Generalizations
One can generalize the concept of Cartesian coordinates to
allow axes that are not perpendicular to each other, and/or
different units along each axis. In that case, each coordinate is
obtained by projecting the point onto one axis along a
direction that is parallel to the other axis (or, in general, to the
hyperplane defined by all the other axes). In those
oblique
coordinate systems the computations of distances and angles
is more complicated than in standard Cartesian systems, and
many standard formulas (such as the Pythagorean formula for
the distance) do not hold.
Notations and conventions
The Cartesian coordinates of a point are usually written in
parentheses and separated by commas, as in (10,5) or (3,5,7).
The origin is often labelled with the capital letter
O. In
analytic geometry, unknown or generic coordinates are often
denoted by the letters x and y on the plane, and x, y, and z in
three-dimensional space. w is often used for four-dimensional
space, but the rarity of such usage precludes concrete
convention here. This custom comes from an old convention
of algebra, to use letters near the end of the alphabet for
unknown values (such as were the coordinates of points in
many geometric problems), and letters near the beginning for
given quantities.
These conventional names are often used in other domains,
such as physics and engineering. However, other letters may
be used too. For example, in a graph showng how a pressure
varies with time, the graph coordinates may be denoted t and
P
. Each axis is usually named after the coordinate which is
measured along it; so one says the x-axis, the y-axis, the taxis,
etc.
Another common convention for coordinate naming is to use
subscripts, as in x1, x2, ... xn for the n coordinates in an ndimensional
space; especially when n is greater than 3, or
variable. Some authors (and many programmers) prefer the numbering x0, x1, ... xn 1. These notations are
especially advantageous in computer programming: by storing the coordinates of a point as an array,
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The four quadrants of a Cartesian
coordinate system.
instead of a record, one can use iterative commands or procedure parameters instead of repeating the
same commands for each coordinate.
In mathematical illustrations of two-dimensional Cartesian systems, the first coordinate (traditionally
called the abscissa) is measured along a horizontal axis, oriented from left to right. The second
coordinate (the ordinate) is then measured along a vertical axis, usually oriented from bottom to top.
However, in computer graphics and image processing one often uses a coordinate system with the
y axis
pointing down (as displayed on the computer's screen). This convention developed in the 1960s (or
earlier) from the way that images were originally stored in display buffers.
For three-dimensional systems, mathematicians usually draw the z axis as vertical and pointing up, so
that the x and y axes lie on an horizontal plane. There is no prevalent convention for the directions of
these two axes, but the orientations are usually chosen according to the right-hand rule. In three
dimensions, the names "abscissa" and "ordinate" are rarely used for x and y, respectively. When they
are, the z-coordinate is sometimes called the applicate.
Quadrants and octants
The axes of a two-dimensional Cartesian system divide the
plane into four infinite regions, called
quadrants, each
bounded by two half-axes. These are often numbered from 1st
to 4th and denoted by Roman numerals: I (where the signs of
the two coordinates are +,+), II ( ,+), III ( , ), and IV (+, ).
When the axes are drawn according to the mathematical
custom, the numbering goes counter-clockwise starting from
the upper right ("northeast") quadrant.
Similarly, a three-dimensional Cartesian system defines a
division of space into eight regions or octants, according to
the signs of the coordinates of the points. The octant where all
three coordinates are positive is sometimes called the first
octant; however, there is no established nomenclature for the
other octants. The n-dimensional generalization of the
quadrant and octant is the orthant.
Cartesian space
A Euclidean plane with a chosen Cartesian system is called a
Cartesian plane. Since Cartesian
coordinates are unique and non-ambiguous, the points of a Cartesian plane can be identified with all
possible pairs of real numbers; that is with the Cartesian product , where is the set of
all reals. In the same way one defines a Cartesian space of any dimension n, whose points can be
identified with the tuples (lists) of n real numbers, that is, with .
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Cartesian formulas for the plane
Distance between two points
The Euclidean distance between two points of the plane with Cartesian coordinates
(x1,y1) and
(x
2,y2) is
This is the Cartesian version of Pythagoras' theorem. In three-dimensional space, the distance between
points (x1,y1,z1) and (x2,y2,z2) is
which can be obtained by two consecutive applications of Pythagoras' theorem.
Euclidean transformations
Translation
Translating a set of points of the plane, preserving the distances and directions between them, is
equivalent to adding a fixed pair of numbers (
X,Y) to the Cartesian coordinates of every point in the set.
That is, if the original coordinates of a point are (x,y), after the translation they will be
Scaling
To make a figure larger or smaller is equivalent to multiplying the Cartesian coordinates of every point
by the same positive number
m. If (x,y) are the coordinates of a point on the original figure, the
corresponding point on the scaled figure has coordinates
If m is greater than 1, the figure becomes larger; if m is between 0 and 1, it becomes smaller.
Rotation
To rotate a figure counterclockwise around the origin by some angle is equivalent to replacing every
point with coordinates (
Reflection
If (
x, y) are the Cartesian coordinates of a point, then ( x, y) are the coordinates of its reflection across
the second coordinate axis (the Y axis), as if that line were a mirror. Likewise, (x, y) are the
coordinates of its reflection across the first coordinate axis (the X axis).
General transformations
The Euclidean transformations of the plane are the translations, rotations, scalings, reflections, and
arbitrary compositions thereof. The result
(x',y') of applying a Euclidean transformation to a point
(x,y)
is given by the formula
where A is a 2×2 matrix and b is a pair of numbers, that depend on the transformation; that is,
The matrix A must have orthogonal rows with same Euclidean length, that is,
and
This is equivalent to saying that A times its transpose must be a diagonal matrix. If these conditions do
not hold, the formula describes a more general affine transformation of the plane.
The formulas define a translation if and only if A is the identity matrix. The transformation is a rotation
around some point if and only if A is a rotation matrix, meaning that
Orientation and handedness
Main article: Orientation (mathematics)
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
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The right hand rule.
Fig. 7 The left-handed orientation is
shown on the left, and the right-handed
on the right. Fig. 8 The right-handed Cartesian
coordinate system indicating the
coordinate planes.
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.
When pointing the thumb away from the origin along an axis, the curvature of the fingers indicates a
positive rotation along that axis.
Regardless of the rule used to orient the axes, rotating the coordinate system will preserve the
orientation. Switching any two axes 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.
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Figure 7 depicts 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 . In three dimensions, the vector from the origin to the point with Cartesian
coordinates
(x,y,z) is sometimes written as[4]:
where , , and are unit vectors and the respective versors of x, y, and z axes. This is the quaternion
representation of the vector, and was introduced by Sir William Rowan Hamilton. The unit vectors , ,
and are called the versors of the coordinate system, and are the vectors of the standard basis in threedimensions.