A Cartesian coordinate system (UK: , US: ) is
a coordinate system that specifies each point
uniquely in a plane by a set of numerical
coordinates, which are the signed distances
to the point from two fixed perpendicular
oriented lines, measured in the same unit
of length. Each reference line is called a
coordinate axis or just axis (plural axes)
of the system, and the point where they meet
is its origin, at ordered pair (0, 0). The
coordinates can also be defined as the positions
of the perpendicular projections of the point
onto the two axes, expressed as 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, n Cartesian coordinates
(an element of real n-space) specify the point
in an n-dimensional Euclidean space for any
dimension n. These coordinates are equal,
up to sign, to distances from the point to
n mutually perpendicular hyperplanes.
The invention of Cartesian coordinates in
the 17th century by René Descartes (Latinized
name: Cartesius) 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, centered
at the origin of the plane, may be described
as the set of all points whose coordinates
x and y satisfy the equation x2 + y2 = 4.
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.
== History ==
The adjective Cartesian refers to the French
mathematician and philosopher René Descartes,
who published this idea in 1637. It was independently
discovered by Pierre de Fermat, who also worked
in three dimensions, although Fermat did not
publish the discovery. The French cleric Nicole
Oresme used constructions similar to Cartesian
coordinates well before the time of Descartes
and Fermat.Both Descartes and Fermat used
a single axis in their treatments 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 Géométrie was translated into Latin in
1649 by Frans van Schooten and his students.
These commentators introduced several concepts
while trying to clarify the ideas contained
in Descartes' work.The development of the
Cartesian coordinate system would play a fundamental
role in the development of the calculus by
Isaac Newton and Gottfried Wilhelm Leibniz.
The two-coordinate description of the plane
was later generalized into the concept of
vector spaces.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.
== Description ==
=== One dimension ===
Choosing a Cartesian coordinate system for
a one-dimensional space—that is, for a straight
line—involves choosing a point O of the
line (the origin), a unit of length, and an
orientation for the line. An orientation chooses
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 has a unique
location on the line. Conversely, every point
on the line can be interpreted as a number
in an ordered continuum such as the real numbers.
=== Two dimensions ===
A Cartesian coordinate system in two dimensions
(also called a rectangular coordinate system
or an orthogonal 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. The point
where the axes meet is taken as the origin
for both, thus turning each axis into a number
line. For any point P, a line is drawn through
P perpendicular to each axis, and the position
where it meets the axis is interpreted as
a number. The two numbers, in that chosen
order, are the Cartesian coordinates of P.
The reverse construction allows one to determine
the point P given its coordinates.
The first and second coordinates are called
the abscissa and the ordinate of P, respectively;
and the point where the axes meet is called
the origin of the coordinate system. The coordinates
are usually written as two numbers in parentheses,
in that order, separated by a comma, as in
(3, −10.5). Thus the origin has coordinates
(0, 0), and the points on the positive half-axes,
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 depicted
as horizontal and oriented to the right, and
the second axis is vertical and oriented upwards.
(However, in some computer graphics contexts,
the ordinate axis may be oriented downwards.)
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 may then be
referred to as the X-axis and Y-axis. 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.
A Euclidean plane with a chosen Cartesian
coordinate system is called a Cartesian plane.
In a Cartesian plane one can define canonical
representatives of certain geometric figures,
such as the unit circle (with radius equal
to the length unit, and center at the origin),
the unit square (whose diagonal has endpoints
at (0, 0) and (1, 1)), the unit hyperbola,
and so on.
The two axes divide the plane into four right
angles, called quadrants. The quadrants may
be named or numbered in various ways, but
the quadrant where 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 |...| denotes the absolute value of
a number.
=== Three dimensions ===
A Cartesian coordinate system for a three-dimensional
space consists of an ordered triplet of lines
(the axes) that go through a common point
(the origin), and are pair-wise perpendicular;
an orientation for each axis; and a single
unit of length for all three axes. As in the
two-dimensional case, each axis becomes a
number line. For any point P of space, one
considers a plane through P perpendicular
to each coordinate axis, and interprets the
point where that plane cuts the axis as a
number. The Cartesian coordinates of P are
those three numbers, in the chosen order.
The reverse construction determines the point
P given its three coordinates.
Alternatively, each coordinate of a point
P can be taken as the distance from P to the
plane defined by the other two axes, with
the sign determined by the orientation of
the corresponding axis.
Each pair of axes defines a coordinate plane.
These planes divide space into eight trihedra,
called octants.
The 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, 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 coordinates
in the three axes (however, the terms abscissa,
ordinate and applicate are sometimes used).
The coordinates are often denoted by the letters
X, Y, and Z, or x, y, and z. The axes may
then be referred to as the X-axis, Y-axis,
and Z-axis, respectively. Then the coordinate
planes can be referred to as the XY-plane,
YZ-plane, and XZ-plane.
In mathematics, physics, and engineering contexts,
the first two axes are often defined or depicted
as horizontal, with the third axis pointing
up. In that case the third coordinate may
be called height or altitude. The orientation
is usually chosen so that the 90 degree angle
from the first axis to the second axis looks
counter-clockwise when seen from the point
(0, 0, 1); a convention that is commonly called
the right hand rule.
=== Higher dimensions ===
Since Cartesian coordinates are unique and
non-ambiguous, the points of a Cartesian plane
can be identified with pairs of real numbers;
that is with the Cartesian product
R
2
=
R
×
R
{\displaystyle \mathbb {R} ^{2}=\mathbb {R}
\times \mathbb {R} }
, where
R
{\displaystyle \mathbb {R} }
is the set of all real numbers. In the same
way, the points in any Euclidean space of
dimension n be identified with the tuples
(lists) of n real numbers, that is, with the
Cartesian product
R
n
{\displaystyle \mathbb {R} ^{n}}
.
=== Generalizations ===
The concept of Cartesian coordinates generalizes
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 such an oblique
coordinate system the computations of distances
and angles must be modified from that in standard
Cartesian systems, and many standard formulas
(such as the Pythagorean formula for the distance)
do not hold (see affine plane).
== 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, y) in
the plane, and (x, y, z) in three-dimensional
space. This custom comes from a convention
of algebra, which uses 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,
although other letters may be used. For example,
in a graph showing how a pressure varies with
time, the graph coordinates may be denoted
p and t. Each axis is usually named after
the coordinate which is measured along it;
so one says the x-axis, the y-axis, the t-axis,
etc.
Another common convention for coordinate naming
is to use subscripts, as (x1, x2, ..., xn)
for the n coordinates in an n-dimensional
space, especially when n is greater than 3
or unspecified. Some authors 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, instead of a record, the subscript
can serve to index the coordinates.
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. Young children learning the
Cartesian system, commonly learn the order
to read the values before cementing the x-,
y-, and z-axis concepts, by starting with
2D mnemonics (e.g. 'Walk along the hall then
up the stairs' akin to straight across the
x-axis then up vertically along the y-axis).Computer
graphics and image processing, however, often
use a coordinate system with the y-axis oriented
downwards on the computer display. This convention
developed in the 1960s (or earlier) from the
way that images were originally stored in
display buffers.
For three-dimensional systems, a convention
is to portray the xy-plane horizontally, with
the z-axis added to represent height (positive
up). Furthermore, there is a convention to
orient the x-axis toward the viewer, biased
either to the right or left. If a diagram
(3D projection or 2D perspective drawing)
shows the x- and y-axis horizontally and vertically,
respectively, then the z-axis should be shown
pointing "out of the page" towards the viewer
or camera. In such a 2D diagram of a 3D coordinate
system, the z-axis would appear as a line
or ray pointing down and to the left or down
and to the right, depending on the presumed
viewer or camera perspective. In any diagram
or display, the orientation of the three axes,
as a whole, is arbitrary. However, the orientation
of the axes relative to each other should
always comply with the right-hand rule, unless
specifically stated otherwise. All laws of
physics and math assume this right-handedness,
which ensures consistency.
For 3D diagrams, the names "abscissa" and
"ordinate" are rarely used for x and y, respectively.
When they are, the z-coordinate is sometimes
called the applicate. The words abscissa,
ordinate and applicate are sometimes used
to refer to coordinate axes rather than the
coordinate values.
=== 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 I (+,+), II (−,+),
III (−,−), and IV (+,−). When the axes
are drawn according to the mathematical custom,
the numbering goes counter-clockwise starting
from the upper right ("north-east") 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 convention
used for naming a specific octant is to list
its signs, e.g. (+ + +) or (− + −). The
generalization of the quadrant and octant
to an arbitrary number of dimensions is the
orthant, and a similar naming system applies.
== Cartesian formulae for the plane ==
=== Distance between two points ===
The Euclidean distance between two points
of the plane with Cartesian coordinates
(
x
1
,
y
1
)
{\displaystyle (x_{1},y_{1})}
and
(
x
2
,
y
2
)
{\displaystyle (x_{2},y_{2})}
is
d
=
(
x
2
−
x
1
)
2
+
(
y
2
−
y
1
)
2
.
{\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.}
This is the Cartesian version of Pythagoras's
theorem. In three-dimensional space, the distance
between points
(
x
1
,
y
1
,
z
1
)
{\displaystyle (x_{1},y_{1},z_{1})}
and
(
x
2
,
y
2
,
z
2
)
{\displaystyle (x_{2},y_{2},z_{2})}
is
d
=
(
x
2
−
x
1
)
2
+
(
y
2
−
y
1
)
2
+
(
z
2
−
z
1
)
2
,
{\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}},}
which can be obtained by two consecutive applications
of Pythagoras' theorem.
=== Euclidean transformations ===
The Euclidean transformations or Euclidean
motions are the (bijective) mappings of points
of the Euclidean plane to themselves which
preserve distances between points. There are
four types of these mappings (also called
isometries): translations, rotations, reflections
and glide reflections.
==== 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 (a, b) 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
(
x
′
,
y
′
)
=
(
x
+
a
,
y
+
b
)
.
{\displaystyle (x',y')=(x+a,y+b).}
==== Rotation ====
To rotate a figure counterclockwise around
the origin by some angle
θ
{\displaystyle \theta }
is equivalent to replacing every point with
coordinates (x,y) by the point with coordinates
(x',y'), where
x
′
=
x
cos
⁡
θ
−
y
sin
⁡
θ
{\displaystyle x'=x\cos \theta -y\sin \theta
}
y
′
=
x
sin
⁡
θ
+
y
cos
⁡
θ
.
{\displaystyle y'=x\sin \theta +y\cos \theta
.}
Thus:
(
x
′
,
y
′
)
=
(
(
x
cos
⁡
θ
−
y
sin
⁡
θ
)
,
(
x
sin
⁡
θ
+
y
cos
⁡
θ
)
)
.
{\displaystyle (x',y')=((x\cos \theta -y\sin
\theta \,),(x\sin \theta +y\cos \theta \,)).}
==== 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). In more generality, reflection
across a line through the origin making an
angle
θ
{\displaystyle \theta }
with the x-axis, is equivalent to replacing
every point with coordinates (x, y) by the
point with coordinates (x′,y′), where
x
′
=
x
cos
⁡
2
θ
+
y
sin
⁡
2
θ
{\displaystyle x'=x\cos 2\theta +y\sin 2\theta
}
y
′
=
x
sin
⁡
2
θ
−
y
cos
⁡
2
θ
.
{\displaystyle y'=x\sin 2\theta -y\cos 2\theta
.}
Thus:
(
x
′
,
y
′
)
=
(
(
x
cos
⁡
2
θ
+
y
sin
⁡
2
θ
)
,
(
x
sin
⁡
2
θ
−
y
cos
⁡
2
θ
)
)
.
{\displaystyle (x',y')=((x\cos 2\theta +y\sin
2\theta \,),(x\sin 2\theta -y\cos 2\theta
\,)).}
==== Glide reflection ====
A glide reflection is the composition of a
reflection across a line followed by a translation
in the direction of that line. It can be seen
that the order of these operations does not
matter (the translation can come first, followed
by the reflection).
==== General matrix form of the transformations
====
These Euclidean transformations of the plane
can all be described in a uniform way by using
matrices. The result
(
x
′
,
y
′
)
{\displaystyle (x',y')}
of applying a Euclidean transformation to
a point
(
x
,
y
)
{\displaystyle (x,y)}
is given by the formula
(
x
′
,
y
′
)
=
(
x
,
y
)
A
+
b
{\displaystyle (x',y')=(x,y)A+b}
where A is a 2×2 orthogonal matrix and b
= (b1, b2) is an arbitrary ordered pair of
numbers; that is,
x
′
=
x
A
11
+
y
A
21
+
b
1
{\displaystyle x'=xA_{11}+yA_{21}+b_{1}}
y
′
=
x
A
12
+
y
A
22
+
b
2
,
{\displaystyle y'=xA_{12}+yA_{22}+b_{2},}
where
A
=
(
A
11
A
12
A
21
A
22
)
.
{\displaystyle A={\begin{pmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{pmatrix}}.}
[The row vectors are used for point coordinates,
and the matrix is written on the right.]To
be orthogonal, the matrix A must have orthogonal
rows with same Euclidean length of one, that
is,
A
11
A
21
+
A
12
A
22
=
0
{\displaystyle A_{11}A_{21}+A_{12}A_{22}=0}
and
A
11
2
+
A
12
2
=
A
21
2
+
A
22
2
=
1.
{\displaystyle A_{11}^{2}+A_{12}^{2}=A_{21}^{2}+A_{22}^{2}=1.}
This is equivalent to saying that A times
its transpose must be the identity matrix.
If these conditions do not hold, the formula
describes a more general affine transformation
of the plane provided that the determinant
of A is not zero.
The formula defines 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
A
11
A
22
−
A
21
A
12
=
1.
{\displaystyle A_{11}A_{22}-A_{21}A_{12}=1.}
A reflection or glide reflection is obtained
when,
A
11
A
22
−
A
21
A
12
=
−
1.
{\displaystyle A_{11}A_{22}-A_{21}A_{12}=-1.}
Assuming that translation is not used transformations
can be combined by simply multiplying the
associated transformation matrices.
==== Affine transformation ====
Another way to represent coordinate transformations
in Cartesian coordinates is through affine
transformations. In affine transformations
an extra dimension is added and all points
are given a value of 1 for this extra dimension.
The advantage of doing this is that point
translations can be specified in the final
column of matrix A. In this way, all of the
euclidean transformations become transactable
as matrix point multiplications. The affine
transformation is given by:
(
A
11
A
21
b
1
A
12
A
22
b
2
0
0
1
)
(
x
y
1
)
=
(
x
′
y
′
1
)
.
{\displaystyle {\begin{pmatrix}A_{11}&A_{21}&b_{1}\\A_{12}&A_{22}&b_{2}\\0&0&1\end{pmatrix}}{\begin{pmatrix}x\\y\\1\end{pmatrix}}={\begin{pmatrix}x'\\y'\\1\end{pmatrix}}.}
[Note the matrix A from above was transposed.
The matrix is on the left and column vectors
for point coordinates are used.]Using affine
transformations multiple different euclidean
transformations including translation can
be combined by simply multiplying the corresponding
matrices.
==== Scaling ====
An example of an affine transformation which
is not a Euclidean motion is given by 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
(
x
′
,
y
′
)
=
(
m
x
,
m
y
)
.
{\displaystyle (x',y')=(mx,my).}
If m is greater than 1, the figure becomes
larger; if m is between 0 and 1, it becomes
smaller.
==== Shearing ====
A shearing transformation will push the top
of a square sideways to form a parallelogram.
Horizontal shearing is defined by:
(
x
′
,
y
′
)
=
(
x
+
y
s
,
y
)
{\displaystyle (x',y')=(x+ys,y)}
Shearing can also be applied vertically:
(
x
′
,
y
′
)
=
(
x
,
x
s
+
y
)
{\displaystyle (x',y')=(x,xs+y)}
== Orientation and handedness ==
=== 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 plane, 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 plane 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 towards positive, the curvature
of the fingers indicates a positive rotation
along that axis.
Regardless of the rule used to orient the
plane, rotating the coordinate system will
preserve the orientation. Switching any two
axes will reverse the orientation, but switching
both will leave the orientation unchanged.
=== 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 orientation
for 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 orientation 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 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 space. 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 position
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
r
{\displaystyle \mathbf {r} }
. In two dimensions, the vector from the origin
to the point with Cartesian coordinates (x,
y) can be written as:
r
=
x
i
+
y
j
{\displaystyle \mathbf {r} =x\mathbf {i} +y\mathbf
{j} }
where
i
=
(
1
0
)
{\displaystyle \mathbf {i} ={\begin{pmatrix}1\\0\end{pmatrix}}}
, and
j
=
(
0
1
)
{\displaystyle \mathbf {j} ={\begin{pmatrix}0\\1\end{pmatrix}}}
are unit vectors in the direction of the x-axis
and y-axis respectively, generally referred
to as the standard basis (in some application
areas these may also be referred to as versors).
Similarly, in three dimensions, the vector
from the origin to the point with Cartesian
coordinates
(
x
,
y
,
z
)
{\displaystyle (x,y,z)}
can be written as:
r
=
x
i
+
y
j
+
z
k
{\displaystyle \mathbf {r} =x\mathbf {i} +y\mathbf
{j} +z\mathbf {k} }
where
k
=
(
0
0
1
)
{\displaystyle \mathbf {k} ={\begin{pmatrix}0\\0\\1\end{pmatrix}}}
is the unit vector in the direction of the
z-axis.
There is no natural interpretation of multiplying
vectors to obtain another vector that works
in all dimensions, however there is a way
to use complex numbers to provide such a multiplication.
In a two dimensional cartesian plane, identify
the point with coordinates (x, y) with the
complex number z = x + iy. Here, i is the
imaginary unit and is identified with the
point with coordinates (0, 1), so it is not
the unit vector in the direction of the x-axis.
Since the complex numbers can be multiplied
giving another complex number, this identification
provides a means to "multiply" vectors. In
a three dimensional cartesian space a similar
identification can be made with a subset of
the quaternions.
== Applications ==
Cartesian coordinates are an abstraction that
have a multitude of possible applications
in the real world. However, three constructive
steps are involved in superimposing coordinates
on a problem application. 1) Units of distance
must be decided defining the spatial size
represented by the numbers used as coordinates.
2) An origin must be assigned to a specific
spatial location or landmark, and 3) the orientation
of the axes must be defined using available
directional cues for all but one axis.
Consider as an example superimposing 3D Cartesian
coordinates over all points on the Earth (i.e.
geospatial 3D). What units make sense? Kilometers
are a good choice, since the original definition
of the kilometer was geospatial...10 000 km
equalling the surface distance from the Equator
to the North Pole. Where to place the origin?
Based on symmetry, the gravitational center
of the Earth suggests a natural landmark (which
can be sensed via satellite orbits). Finally,
how to orient X-, Y- and Z-axis? The axis
of Earth's spin provides a natural orientation
strongly associated with "up vs. down", so
positive Z can adopt the direction from geocenter
to North Pole. A location on the Equator is
needed to define the X-axis, and the prime
meridian stands out as a reference orientation,
so the X-axis takes the orientation from geocenter
out to [ 0 degrees longitude, 0 degrees latitude
]. Note that with 3 dimensions, and two perpendicular
axes orientations pinned down for X and Z,
the Y-axis is determined by the first two
choices. In order to obey the right-hand rule,
the Y-axis must point out from the geocenter
to [ 90 degrees longitude, 0 degrees latitude
]. So what are the geocentric coordinates
of the Empire State Building in New York City?
Using [ longitude = −73.985656, latitude
= 40.748433 ], Earth radius = 40,000/2π,
and transforming from spherical --> Cartesian
coordinates, you can estimate the geocentric
coordinates of the Empire State Building,
[ x, y, z ] = [ 1330.53 km, –4635.75 km,
4155.46 km ]. GPS navigation relies on such
geocentric coordinates.
In engineering projects, agreement on the
definition of coordinates is a crucial foundation.
One cannot assume that coordinates come predefined
for a novel application, so knowledge of how
to erect a coordinate system where there is
none is essential to applying René Descartes'
thinking.
While spatial apps employ identical units
along all axes, in business and scientific
apps, each axis may have different units of
measurement associated with it (such as kilograms,
seconds, pounds, etc.). 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.) 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 of many
non-spatial variables.
The graph of a function or relation is the
set of all points satisfying that function
or relation. For a function of one variable,
f, the set of all points (x, y), where y = f(x)
is the graph of the function f. For a function
g of two variables, the set of all points
(x, y, z), where z = g(x, y) is the graph
of the function g. A sketch of the graph of
such a function or relation would consist
of all the salient parts of the function or
relation which would include its relative
extrema, its concavity and points of inflection,
any points of discontinuity and its end behavior.
All of these terms are more fully defined
in calculus. Such graphs are useful in calculus
to understand the nature and behavior of a
function or relation.
== See also ==
Horizontal and vertical
Jones diagram, which plots four variables
rather than two
Orthogonal coordinates
Polar coordinate system
Spherical coordinate system
