264:
5369:
5440:
4678:
1397:
33:
1916:
5377:
893:
5260:
4918:
1276:
1593:
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
1931:: I (where the coordinates both have positive signs), II (where the abscissa is negative − and the ordinate is positive +), III (where both the abscissa and the ordinate are −), and IV (abscissa +, ordinate −). When the axes are drawn according to the mathematical custom, the numbering goes
579:
length along the line can be chosen as a unit, with the orientation indicating the correspondence between directions along the line and positive or negative numbers. Each point corresponds to its signed distance from the origin (a number with an absolute value equal to the distance and a
4695:
3113:
994:
4667:
2763:
3869:
1652:) 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 the coordinates of points in many geometric problems), and letters near the beginning for given quantities.
4507:
3261:
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).
5559:, 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
3252:
5494:
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
5530:
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
3521:
2892:
6092:-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
3632:
2440:
5359:
Regardless of the rule used to orient the plane, rotating the coordinate system will preserve the orientation. Switching any one axis will reverse the orientation, but switching both will leave the orientation unchanged.
4977:
An example of an affine transformation which is not
Euclidean 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
5850:
4512:
6034:
5973:
5913:
4113:
2990:
783:-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.
2170:
1516:
5732:
5681:
2652:
5400:-axis should lie, but there are two possible orientations for this line. The two possible coordinate systems, which result are called 'right-handed' and 'left-handed'. The standard orientation, where the
5294:-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
3699:
482:
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'
751:
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
5628:
5539:"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
4913:{\displaystyle {\begin{pmatrix}A_{1,1}&A_{2,1}&b_{1}\\A_{1,2}&A_{2,2}&b_{2}\\0&0&1\end{pmatrix}}{\begin{pmatrix}x\\y\\1\end{pmatrix}}={\begin{pmatrix}x'\\y'\\1\end{pmatrix}}.}
3704:
1374:
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
3709:
3353:
2995:
2657:
1604:
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
999:
949:), 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
4344:
4249:
3997:
1271:{\displaystyle {\begin{aligned}(+x,+y,+z)&&(-x,+y,+z)&&(+x,-y,+z)&&(+x,+y,-z)\\(+x,-y,-z)&&(-x,+y,-z)&&(-x,-y,+z)&&(-x,-y,-z)\end{aligned}}}
2594:
5229:
5156:
5063:
6862:
4405:
3120:
1858:. 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
2294:
2235:
3432:
3395:
2772:
1583:
3526:
4392:
2063:
2017:
645:
5579:
1538:
689:
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
2301:
6270:
or half-lines resulting from splitting the line at the origin. One of the half-lines can be assigned to positive numbers, and the other half-line to negative numbers.
5792:
4953:
2959:
2629:
649:
taking a specific point's coordinate in one system to its coordinate in the other system. Choosing a coordinate system for each of two different lines establishes an
4692:
are transformations that map lines to lines, but may change distances and angles. As said in the preceding section, they can be represented with augmented matrices:
2941:
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
1655:
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
3427:
3304:
563:
in the choice of
Cartesian coordinate system for a line, which can be specified by choosing two distinct points along the line and assigning them to two distinct
7036:
3895:
5797:
6855:
1780:
axis, usually oriented from bottom to top. Young children learning the
Cartesian system, commonly learn the order to read the values before cementing the
5356:
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.
1811:-axis oriented downwards on the computer display. This convention developed in the 1960s (or earlier) from the way that images were originally stored in
4002:
1382:. 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
2070:
697:
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
7031:
6848:
5586:
1942:, according to the signs of the coordinates of the points. The convention used for naming a specific octant is to list its signs; for example,
6759:
6733:
6645:
6614:
6590:
6527:
6504:
6485:
6433:
6208:
6550:
2926:
5978:
5917:
5857:
957:
perpendicular to each coordinate axis, and interprets the point where that plane cuts the axis as a number. The
Cartesian coordinates of
912:, oriented as shown by the arrows. The tick marks on the axes are one length unit apart. The black dot shows the point with coordinates
732:
of the coordinate system. The coordinates are usually written as two numbers in parentheses, in that order, separated by a comma, as in
322:, whose invention of them in the 17th century revolutionized mathematics by allowing the expression of problems of geometry in terms of
4256:
1854:-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
4164:
1476:
1280:
The coordinates are usually written as three numbers (or algebraic formulas) surrounded by parentheses and separated by commas, as in
5686:
5635:
3912:
6941:
6571:
6466:
2495:
a set of points of the plane, preserving the distances and directions between them, is equivalent to adding a fixed pair of numbers
829:. The quadrants may be named or numbered in various ways, but the quadrant where all coordinates are positive is usually called the
1850:-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
3641:
493:
and his students. These commentators introduced several concepts while trying to clarify the ideas contained in
Descartes's work.
6966:
1792:-axis concepts, by starting with 2D mnemonics (for example, 'Walk along the hall then up the stairs' akin to straight across the
7021:
6981:
849:
437:
6956:
6185:
4662:{\displaystyle A'={\begin{pmatrix}A_{1,1}&A_{1,2}&b_{1}\\A_{2,1}&A_{2,2}&b_{2}\\0&0&1\end{pmatrix}}.}
6042:
interpretation of multiplying vectors to obtain another vector that works in all dimensions, however there is a way to use
3108:{\displaystyle {\begin{aligned}x'&=x\cos 2\theta +y\sin 2\theta \\y'&=x\sin 2\theta -y\cos 2\theta .\end{aligned}}}
607:
the line corresponds to multiplication. Any two
Cartesian coordinate systems on the line can be related to each-other by a
6951:
6931:
1466:
Since
Cartesian coordinates are unique and non-ambiguous, the points of a Cartesian plane can be identified with pairs of
524:
167:
3312:
6871:
4681:
Effect of applying various 2D affine transformation matrices on a unit square (reflections are special cases of scaling)
2524:
2468:
to themselves which preserve distances between points. There are four types of these mappings (also called isometries):
267:
Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is
2758:{\displaystyle {\begin{aligned}x'&=x\cos \theta -y\sin \theta \\y'&=x\sin \theta +y\cos \theta .\end{aligned}}}
6936:
6287:
6136:
5165:
5092:
4999:
596:
520:
4994:
are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates
7026:
7016:
6910:
6105:
1764:
In mathematical illustrations of two-dimensional Cartesian systems, the first coordinate (traditionally called the
976:
to the plane defined by the other two axes, with the sign determined by the orientation of the corresponding axis.
5581:. In two dimensions, the vector from the origin to the point with Cartesian coordinates (x, y) can be written as:
653:
from one line to the other taking each point on one line to the point on the other line with the same coordinate.
6905:
1910:
1754:
1600:
556:. Every point on the line has a real-number coordinate, and every real number represents some point on the line.
505:
420:. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including
6946:
6415:
3874:
3864:{\displaystyle {\begin{aligned}x'&=xA_{1,1}+yA_{1,1}+b_{1}\\y'&=xA_{2,1}+yA_{2,2}+b_{2}.\end{aligned}}}
2477:
2453:
873:
592:
4669:
With this trick, the composition of affine transformations is obtained by multiplying the augmented matrices.
263:
6976:
6961:
6890:
6126:
6110:
6046:
to provide such a multiplication. In a two-dimensional cartesian plane, identify the point with coordinates
5368:
2507:
to the Cartesian coordinates of every point in the set. That is, if the original coordinates of a point are
2175:
1906:
1855:
1835:
985:
887:
516:
218:
5338:. Placing a somewhat closed right hand on the plane with the thumb pointing up, the fingers point from the
4398:
by simply multiplying the associated transformation matrices. In the general case, it is useful to use the
798:. In a Cartesian plane, one can define canonical representatives of certain geometric figures, such as the
496:
The development of the Cartesian coordinate system would play a fundamental role in the development of the
6121:
2492:
2473:
2469:
600:
441:
2240:
2181:
941:
A Cartesian coordinate system for a three-dimensional space consists of an ordered triplet of lines (the
36:
Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates:
6900:
6894:
4685:
4132:
3271:
664:
650:
475:, who also worked in three dimensions, although Fermat did not publish the discovery. The French cleric
405:
225:, which are the signed distances from the point to three mutually perpendicular planes. More generally,
6824:
5756:). Similarly, in three dimensions, the vector from the origin to the point with Cartesian coordinates
5439:
1559:
6991:
6971:
6800:
6606:
6542:
4963:
4395:
1746:
568:
396:, and provide enlightening geometric interpretations for many other branches of mathematics, such as
203:
194:
3274:
of the plane can be described in a uniform way by using matrices. For this purpose, the coordinates
6986:
5087:
will push the top of a square sideways to form a parallelogram. Horizontal shearing is defined by:
4677:
4351:
2604:
2022:
1976:
1923:
The axes of a two-dimensional Cartesian system divide the plane into four infinite regions, called
615:
417:
5562:
1938:
Similarly, a three-dimensional Cartesian system defines a division of space into eight regions or
1521:
479:
used constructions similar to Cartesian coordinates well before the time of Descartes and Fermat.
4969:
Some affine transformations that are not Euclidean transformations have received specific names.
3902:
1970:
1777:
604:
490:
6726:
Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions
4502:{\displaystyle {\begin{pmatrix}x'\\y'\\1\end{pmatrix}}=A'{\begin{pmatrix}x\\y\\1\end{pmatrix}},}
1826:-axis added to represent height (positive up). Furthermore, there is a convention to orient the
1396:
485:
6676:
6807:
6782:
6765:
6755:
6729:
6712:
6688:
6680:
6651:
6641:
6631:
6610:
6586:
6567:
6546:
6523:
6500:
6481:
6462:
6429:
6375:
6214:
6204:
4956:
3898:
3247:{\displaystyle (x',y')=((x\cos 2\theta +y\sin 2\theta \,),(x\sin 2\theta -y\cos 2\theta \,)).}
1671:. Each axis is usually named after the coordinate which is measured along it; so one says the
1471:
1401:
752:
576:
560:
464:
433:
393:
343:
319:
151:
32:
5759:
5372:
Fig. 7 – The left-handed orientation is shown on the left, and the right-handed on the right.
4925:
4922:
The Euclidean transformations are the affine transformations such that the 2×2 matrix of the
3358:
2944:
2614:
6728:(corrected 2nd, 3rd print ed.). New York: Springer-Verlag. pp. 9–11 (Table 1.01).
6421:
5556:
5463:
placed at a right angle to both, the three fingers indicate the relative orientation of the
5249:
4399:
2481:
1950:. The generalization of the quadrant and octant to an arbitrary number of dimensions is the
1804:
1769:
961:
are those three numbers, in the chosen order. The reverse construction determines the point
740:, and the points on the positive half-axes, one unit away from the origin, have coordinates
472:
401:
207:
155:
147:
114:
108:
73:
3516:{\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}=A{\begin{pmatrix}x\\y\end{pmatrix}}+b,}
3400:
3277:
6810:
6747:
6743:
6664:
5448:
5393:
5334:
5264:
5245:
4689:
4158:
4148:
4128:
2887:{\displaystyle (x',y')=((x\cos \theta -y\sin \theta \,),(x\sin \theta +y\cos \theta \,)).}
2465:
1915:
1859:
1750:
1541:
1389:
787:
608:
238:
176:
67:
6640:. Translated by Paul J. Oscamp (Revised ed.). Indianapolis, IN: Hackett Publishing.
5491:-axis. Conversely, if the same is done with the left hand, a left-handed system results.
2929:
across the second coordinate axis (the y-axis), as if that line were a mirror. Likewise,
1927:, each bounded by two half-axes. These are often numbered from 1st to 4th and denoted by
1632:. In analytic geometry, unknown or generic coordinates are often denoted by the letters (
5555:
A point in space in a Cartesian coordinate system may also be represented by a position
3627:{\displaystyle A={\begin{pmatrix}A_{1,1}&A_{1,2}\\A_{2,1}&A_{2,2}\end{pmatrix}}}
508:. The two-coordinate description of the plane was later generalized into the concept of
6705:
6700:
6669:
6516:
6267:
6077:
6043:
5748:
4139:
3906:
3880:
815:
686:
567:(most commonly zero and one). Other points can then be uniquely assigned to numbers by
397:
5380:
Fig. 8 – The right-handed Cartesian coordinate system indicating the coordinate planes
7010:
6157:
6115:
5456:
5283:
5239:
5084:
3635:
3307:
1928:
1831:
682:
571:. Equivalently, one point can be assigned to a specific real number, for instance an
509:
476:
457:
171:
5376:
755:
contexts, the ordinate axis may be oriented downwards.) The origin is often labeled
6131:
5735:
5452:
4966:
of two affine transformations is obtained by multiplying their augmented matrices.
1605:
678:
501:
413:
378:
6825:
Coordinate Converter – converts between polar, Cartesian and spherical coordinates
2435:{\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}},}
1897:
are sometimes used to refer to coordinate axes rather than the coordinate values.
17:
6635:
6600:
6561:
6456:
1862:, unless specifically stated otherwise. All laws of physics and math assume this
6840:
5259:
1812:
1617:
1467:
892:
822:
803:
799:
564:
549:
543:
468:
461:
429:
159:
6836:
open source JavaScript class for 2D/3D Cartesian coordinate system manipulation
6835:
515:
Many other coordinate systems have been developed since Descartes, such as the
6829:
6425:
6312:
6093:
2444:
which can be obtained by two consecutive applications of Pythagoras' theorem.
1595:
386:
257:
6655:
6815:
6692:
6218:
4124:
2608:
2461:
1932:
1758:
421:
374:
242:
1830:-axis toward the viewer, biased either to the right or left. If a diagram (
6537:
Hughes-Hallett, Deborah; McCallum, William G.; Gleason, Andrew M. (2013).
6198:
5536:
5416:-axis form a positively oriented two-dimensional coordinate system in the
1686:
Another common convention for coordinate naming is to use subscripts, as (
1359:-axis, respectively. Then the coordinate planes can be referred to as the
342:
of radius 2, centered at the origin of the plane, may be described as the
2961:
with the x-axis, is equivalent to replacing every point with coordinates
1869:
For 3D diagrams, the names "abscissa" and "ordinate" are rarely used for
1773:
1765:
1656:
1428:-axis is highlighted in green. Thus, the red plane shows the points with
720:
714:
497:
409:
382:
335:
327:
57:
5532:
4131:. If these conditions do not hold, the formula describes a more general
5845:{\displaystyle \mathbf {r} =x\mathbf {i} +y\mathbf {j} +z\mathbf {k} ,}
4402:
of the transformation; that is, to rewrite the transformation formula
1952:
872:|, respectively; where | · | denotes the
425:
323:
5332:
A commonly used mnemonic for defining the positive orientation is the
802:(with radius equal to the length unit, and center at the origin), the
5753:
1721:
is greater than 3 or unspecified. Some authors prefer the numbering (
339:
6439:
4151:. The transformation is a rotation around some point if and only if
6029:{\displaystyle \mathbf {k} ={\begin{pmatrix}0\\0\\1\end{pmatrix}}.}
5968:{\displaystyle \mathbf {j} ={\begin{pmatrix}0\\1\\0\end{pmatrix}},}
5908:{\displaystyle \mathbf {i} ={\begin{pmatrix}1\\0\\0\end{pmatrix}},}
1540:
is the set of all real numbers. In the same way, the points in any
330:. Using the Cartesian coordinate system, geometric shapes (such as
5460:
5438:
5375:
5367:
5258:
4676:
4108:{\displaystyle A_{1,1}^{2}+A_{2,1}^{2}=A_{1,2}^{2}+A_{2,2}^{2}=1.}
1549:
1395:
432:
and many more. They are the most common coordinate system used in
331:
262:
31:
5353:, placing the left hand on the plane with the thumb pointing up.
338:
involving the coordinates of points of the shape. For example, a
6786:
6769:
6724:
Moon P, Spencer DE (1988). "Rectangular Coordinates (x, y, z)".
6716:
6684:
2165:{\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.}
1660:
1511:{\displaystyle \mathbb {R} ^{2}=\mathbb {R} \times \mathbb {R} }
728:, respectively; and the point where the axes meet is called the
370:
6844:
6476:
Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1998).
5727:{\displaystyle \mathbf {j} ={\begin{pmatrix}0\\1\end{pmatrix}}}
5676:{\displaystyle \mathbf {i} ={\begin{pmatrix}1\\0\end{pmatrix}}}
669:
A Cartesian coordinate system in two dimensions (also called a
467:, who published this idea in 1637 while he was resident in the
250:. These coordinates are the signed distances from the point to
1818:
For three-dimensional systems, a convention is to portray the
1772:
axis, oriented from left to right. The second coordinate (the
132:
129:
88:
6336:
896:
A three dimensional Cartesian coordinate system, with origin
705:. The reverse construction allows one to determine the point
192:) of the system. The point where the axes meet is called the
5752:(in some application areas these may also be referred to as
3694:{\displaystyle b={\begin{pmatrix}b_{1}\\b_{2}\end{pmatrix}}}
1616:
The Cartesian coordinates of a point are usually written in
120:
79:
1973:
between two points of the plane with Cartesian coordinates
759:, and the two coordinates are often denoted by the letters
135:
94:
91:
5499:
the observer, whereas the "middle"-axis is meant to point
2178:. In three-dimensional space, the distance between points
1846:-axis horizontally and vertically, respectively, then the
5623:{\displaystyle \mathbf {r} =x\mathbf {i} +y\mathbf {j} ,}
2631:
is equivalent to replacing every point with coordinates (
1453:(shown as a black sphere) with the Cartesian coordinates
1323:
Standard names for the coordinates in the three axes are
381:
at any point can be computed from this equation by using
5301:
The usual way of orienting the plane, with the positive
6560:
Kent, Alexander J.; Vujakovic, Peter (4 October 2017).
1935:
starting from the upper right ("north-east") quadrant.
1628:. The origin is often labelled with the capital letter
6637:
Discourse on Method, Optics, Geometry, and Meteorology
6599:
Anton, Howard; Bivens, Irl C.; Davis, Stephen (2021).
5995:
5934:
5874:
5703:
5652:
5349:
The other way of orienting the plane is following the
4869:
4833:
4704:
4532:
4468:
4414:
3656:
3541:
3483:
3441:
3321:
790:
with a chosen Cartesian coordinate system is called a
552:
with a chosen Cartesian coordinate system is called a
6203:(3rd ed.). Boston: Addison-Wesley. p. 484.
5981:
5920:
5860:
5800:
5762:
5689:
5638:
5589:
5565:
5168:
5095:
5002:
4928:
4698:
4515:
4408:
4354:
4259:
4167:
4005:
3915:
3883:
3707:
3644:
3529:
3435:
3403:
3361:
3315:
3280:
3123:
2993:
2947:
2775:
2655:
2617:
2527:
2304:
2243:
2184:
2073:
2025:
1979:
1562:
1524:
1479:
997:
618:
138:
117:
97:
76:
6832:– interactive tool to explore coordinates of a point
126:
85:
6919:
6878:
5346:-axis, in a positively oriented coordinate system.
4253:A reflection or glide reflection is obtained when,
3348:{\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}.}
1919:
The four quadrants of a Cartesian coordinate system
1335:. The coordinates are often denoted by the letters
416:and more. A familiar example is the concept of the
123:
82:
6704:
6671:Mathematical Handbook for Scientists and Engineers
6668:
6515:
6080:and is identified with the point with coordinates
6028:
5967:
5907:
5844:
5786:
5726:
5675:
5622:
5573:
5519:-axis (in both cases). Hence the red arrow passes
5223:
5150:
5057:
4947:
4912:
4661:
4501:
4386:
4348:Assuming that translations are not used (that is,
4338:
4243:
4107:
3991:
3889:
3863:
3693:
3626:
3515:
3421:
3389:
3347:
3298:
3246:
3107:
2953:
2886:
2757:
2623:
2588:
2434:
2288:
2229:
2164:
2057:
2011:
1807:, however, often use a coordinate system with the
1745:). These notations are especially advantageous in
1577:
1556:real numbers; that is, with the Cartesian product
1532:
1510:
1270:
639:
6563:The Routledge Handbook of Mapping and Cartography
6386:
6359:
5746:-axis respectively, generally referred to as the
5071:is greater than 1, the figure becomes larger; if
4339:{\displaystyle A_{1,1}A_{2,2}-A_{2,1}A_{1,2}=-1.}
6675:(1st ed.). New York: McGraw-Hill. pp.
6420:. Undergraduate Texts in Mathematics. Springer.
6371:
4244:{\displaystyle A_{1,1}A_{2,2}-A_{2,1}A_{1,2}=1.}
3397:of applying an affine transformation to a point
712:The first and second coordinates are called the
5547:the observer and thus seeing a concave corner.
3992:{\displaystyle A_{1,1}A_{1,2}+A_{2,1}A_{2,2}=0}
2913:are the Cartesian coordinates of a point, then
1863:
3877:are characterized by the fact that the matrix
1749:: by storing the coordinates of a point as an
231:Cartesian coordinates specify the point in an
210:. The combination of origin and basis forms a
6856:
6181:
1442:, and the yellow plane shows the points with
389:, in a way that can be applied to any curve.
8:
6585:(5th ed.), Pacific Grove: Brooks/Cole,
6118:, which plots four variables rather than two
5459:bent inward at a right angle to it, and the
5317:-axis the "second" axis), is considered the
1449:. The three surfaces intersect at the point
1308:, and the unit points on the three axes are
392:Cartesian coordinates are the foundation of
6337:Hughes-Hallett, McCallum & Gleason 2013
5551:Representing a vector in the standard basis
3306:of a point are commonly represented as the
1598:defined by all the other axes). In such an
6863:
6849:
6841:
6497:The History of Mathematics/An Introduction
5455:of the right hand is pointed forward, the
3266:General matrix form of the transformations
1347:. The axes may then be referred to as the
968:Alternatively, each coordinate of a point
775:. The axes may then be referred to as the
6480:. Cambridge: Cambridge University Press.
6398:
6243:
6200:A history of mathematics: an introduction
5990:
5982:
5980:
5929:
5921:
5919:
5869:
5861:
5859:
5834:
5823:
5812:
5801:
5799:
5761:
5698:
5690:
5688:
5647:
5639:
5637:
5612:
5601:
5590:
5588:
5566:
5564:
5167:
5160:Shearing can also be applied vertically:
5094:
5001:
4962:The augmented matrix that represents the
4933:
4927:
4864:
4828:
4797:
4779:
4761:
4747:
4729:
4711:
4699:
4697:
4625:
4607:
4589:
4575:
4557:
4539:
4527:
4514:
4463:
4409:
4407:
4372:
4359:
4353:
4315:
4299:
4280:
4264:
4258:
4223:
4207:
4188:
4172:
4166:
4093:
4082:
4069:
4058:
4045:
4034:
4021:
4010:
4004:
3971:
3955:
3936:
3920:
3914:
3882:
3848:
3829:
3807:
3775:
3756:
3734:
3708:
3706:
3677:
3663:
3651:
3643:
3604:
3586:
3566:
3548:
3536:
3528:
3478:
3436:
3434:
3402:
3360:
3316:
3314:
3279:
3234:
3191:
3122:
2994:
2992:
2946:
2874:
2837:
2774:
2656:
2654:
2616:
2526:
2421:
2411:
2398:
2382:
2372:
2359:
2343:
2333:
2320:
2311:
2303:
2277:
2264:
2251:
2242:
2218:
2205:
2192:
2183:
2151:
2141:
2128:
2112:
2102:
2089:
2080:
2072:
2046:
2033:
2024:
2000:
1987:
1978:
1569:
1565:
1564:
1561:
1526:
1525:
1523:
1504:
1503:
1496:
1495:
1486:
1482:
1481:
1478:
998:
996:
617:
603:of the line corresponds to addition, and
6779:Mathematische Hilfsmittel des Ingenieurs
6707:The Mathematics of Physics and Chemistry
6288:"Cartesian orthogonal coordinate system"
6088:the unit vector in the direction of the
5392:-axes are specified, they determine the
5290:-axis through the point marked 0 on the
5075:is between 0 and 1, it becomes smaller.
1914:
953:of space, one considers a plane through
891:
821:The two axes divide the plane into four
217:Similarly, the position of any point in
27:Most common coordinate system (geometry)
6499:(7th ed.). New York: McGraw-Hill.
6148:
5511:-plane and indicates rotation from the
1956:, and a similar naming system applies.
1663:, the graph coordinates may be denoted
1435:, the blue plane shows the points with
1386:; a convention that is commonly called
983:. These planes divide space into eight
6752:Methods of Theoretical Physics, Part I
6231:
5305:-axis pointing right and the positive
3873:Among the affine transformations, the
945:) that go through a common point (the
170:distances to the point from two fixed
6347:
6254:
5503:from the observer. The red circle is
2519:, after the translation they will be
489:was translated into Latin in 1649 by
471:. It was independently discovered by
7:
7037:Three-dimensional coordinate systems
6461:. Knopf Doubleday Publishing Group.
6282:
6280:
6278:
6276:
4161:, meaning that it is orthogonal and
4138:The transformation is a translation
1881:-coordinate is sometimes called the
1761:can serve to index the coordinates.
1717:-dimensional space, especially when
595:of the line can be represented by a
575:point corresponding to zero, and an
318:Cartesian coordinates are named for
5278:-axis up to direction. Namely, the
2289:{\displaystyle (x_{2},y_{2},z_{2})}
2230:{\displaystyle (x_{1},y_{1},z_{1})}
1877:, respectively. When they are, the
1796:-axis then up vertically along the
1304:. Thus, the origin has coordinates
6539:Calculus: Single and Multivariable
6156:Bix, Robert A.; D'Souza, Harry J.
5443:3D Cartesian coordinate handedness
4117:This is equivalent to saying that
2589:{\displaystyle (x',y')=(x+a,y+b).}
972:can be taken as the distance from
836:If the coordinates of a point are
736:. Thus the origin has coordinates
298:are the coordinates of the center
25:
2639:) by the point with coordinates (
2174:This is the Cartesian version of
806:(whose diagonal has endpoints at
588:sign chosen based on direction).
5983:
5922:
5862:
5835:
5824:
5813:
5802:
5691:
5640:
5613:
5602:
5591:
5567:
5487:-axis and the middle finger the
5479:system. The thumb indicates the
5313:-axis being the "first" and the
5224:{\displaystyle (x',y')=(x,xs+y)}
5151:{\displaystyle (x',y')=(x+ys,y)}
5058:{\displaystyle (x',y')=(mx,my).}
2611:around the origin by some angle
1960:Cartesian formulae for the plane
1578:{\displaystyle \mathbb {R} ^{n}}
442:geometry-related data processing
346:of all points whose coordinates
113:
72:
6518:Introduction to Electrodynamics
6387:Brannan, Esplen & Gray 1998
6360:Brannan, Esplen & Gray 1998
1620:and separated by commas, as in
438:computer-aided geometric design
6372:Anton, Bivens & Davis 2021
5781:
5763:
5218:
5197:
5191:
5169:
5145:
5124:
5118:
5096:
5049:
5031:
5025:
5003:
3416:
3404:
3384:
3362:
3293:
3281:
3238:
3235:
3198:
3192:
3155:
3152:
3146:
3124:
2973:by the point with coordinates
2878:
2875:
2844:
2838:
2807:
2804:
2798:
2776:
2580:
2556:
2550:
2528:
2418:
2391:
2379:
2352:
2340:
2313:
2283:
2244:
2224:
2185:
2148:
2121:
2109:
2082:
2052:
2026:
2006:
1980:
1822:-plane horizontally, with the
1261:
1234:
1228:
1201:
1195:
1168:
1162:
1135:
1128:
1101:
1095:
1068:
1062:
1035:
1029:
1002:
622:
1:
7032:Orthogonal coordinate systems
6872:Orthogonal coordinate systems
6781:. New York: Springer Verlag.
6711:. New York: D. van Nostrand.
5404:-plane is horizontal and the
5325:orientation, also called the
4387:{\displaystyle b_{1}=b_{2}=0}
3701:is a column matrix. That is,
2058:{\displaystyle (x_{2},y_{2})}
2012:{\displaystyle (x_{1},y_{1})}
1866:, which ensures consistency.
1404:of the Cartesian coordinates
965:given its three coordinates.
671:rectangular coordinate system
640:{\displaystyle x\mapsto ax+b}
527:for three-dimensional space.
256:mutually perpendicular fixed
6514:Griffiths, David J. (1999).
6408:General and cited references
5574:{\displaystyle \mathbf {r} }
5483:-axis, the index finger the
2464:) mappings of points of the
1533:{\displaystyle \mathbb {R} }
979:Each pair of axes defines a
675:orthogonal coordinate system
6801:Cartesian Coordinate System
6292:Encyclopedia of Mathematics
6137:Spherical coordinate system
5309:-axis pointing up (and the
3901:; that is, its columns are
2925:are the coordinates of its
1965:Distance between two points
1776:) is then measured along a
597:function of a real variable
62:Cartesian coordinate system
7053:
6106:Cartesian coordinate robot
5447:The name derives from the
5298:) of the Cartesian plane.
5243:
5237:
5234:Orientation and handedness
1904:
1424:-axis is vertical and the
885:
693:, a line is drawn through
662:
541:
221:can be specified by three
6777:Sauer R, Szabó I (1967).
6754:. New York: McGraw-Hill.
6495:Burton, David M. (2011).
6455:Berlinski, David (2011).
6426:10.1007/978-3-319-11080-6
6417:Linear Algebra Done Right
6389:, Appendix 2, pp. 377–382
6182:Kent & Vujakovic 2017
5408:-axis points up (and the
5282:-axis is necessarily the
4394:) transformations can be
3875:Euclidean transformations
2454:Euclidean transformations
2448:Euclidean transformations
1911:Quadrant (plane geometry)
1612:Notations and conventions
1601:oblique coordinate system
506:Gottfried Wilhelm Leibniz
202:as coordinates. The axes
6581:Smart, James R. (1998),
6197:Katz, Victor J. (2009).
6058:with the complex number
5738:in the direction of the
5420:-plane if observed from
3429:is given by the formula
593:geometric transformation
48:in blue, and the origin
6811:"Cartesian Coordinates"
6602:Calculus: Multivariable
6414:Axler, Sheldon (2015).
6313:"Cartesian coordinates"
6162:Encyclopædia Britannica
6127:Polar coordinate system
6111:Horizontal and vertical
5787:{\displaystyle (x,y,z)}
5270:Fixing or choosing the
5085:shearing transformation
4948:{\displaystyle A_{i,j}}
3390:{\displaystyle (x',y')}
2954:{\displaystyle \theta }
2624:{\displaystyle \theta }
1907:Octant (solid geometry)
1548:be identified with the
888:Three-dimensional space
709:given its coordinates.
685:lines (axes), a single
525:cylindrical coordinates
519:for the plane, and the
219:three-dimensional space
174:oriented lines, called
7022:Elementary mathematics
6830:Coordinates of a point
6458:A Tour of the Calculus
6122:Orthogonal coordinates
6030:
5969:
5909:
5846:
5788:
5728:
5677:
5624:
5575:
5444:
5381:
5373:
5267:
5225:
5152:
5059:
4949:
4914:
4686:Affine transformations
4682:
4663:
4503:
4388:
4340:
4245:
4109:
3993:
3891:
3865:
3695:
3628:
3517:
3423:
3391:
3349:
3300:
3272:affine transformations
3248:
3109:
2955:
2888:
2759:
2625:
2590:
2436:
2290:
2231:
2166:
2059:
2013:
1920:
1836:2D perspective drawing
1803:Computer graphics and
1768:) is measured along a
1579:
1534:
1512:
1458:
1272:
938:
641:
611:(function of the form
334:) can be described by
315:
158:uniquely by a pair of
53:
6607:John Wiley & Sons
6543:John Wiley & Sons
6031:
5970:
5910:
5847:
5789:
5729:
5678:
5625:
5576:
5442:
5379:
5371:
5274:-axis determines the
5262:
5226:
5153:
5060:
4950:
4915:
4680:
4673:Affine transformation
4664:
4504:
4389:
4341:
4246:
4133:affine transformation
4110:
3994:
3909:one, or, explicitly,
3892:
3866:
3696:
3629:
3518:
3424:
3422:{\displaystyle (x,y)}
3392:
3350:
3301:
3299:{\displaystyle (x,y)}
3249:
3110:
2956:
2889:
2760:
2626:
2591:
2437:
2291:
2232:
2167:
2060:
2014:
1918:
1901:Quadrants and octants
1640:) in the plane, and (
1580:
1535:
1513:
1399:
1273:
895:
886:Further information:
699:Cartesian coordinates
665:Two-dimensional space
663:Further information:
642:
456:refers to the French
406:differential geometry
358:satisfy the equation
266:
223:Cartesian coordinates
35:
6967:Elliptic cylindrical
6703:, Murphy GM (1956).
5979:
5918:
5858:
5798:
5760:
5687:
5636:
5587:
5563:
5166:
5093:
5000:
4926:
4696:
4513:
4406:
4352:
4257:
4165:
4003:
3913:
3881:
3705:
3642:
3527:
3433:
3401:
3359:
3313:
3278:
3121:
2991:
2945:
2773:
2653:
2615:
2525:
2302:
2241:
2182:
2176:Pythagoras's theorem
2071:
2023:
1977:
1747:computer programming
1560:
1522:
1477:
1470:; that is, with the
995:
616:
569:linear interpolation
154:that specifies each
6982:Bipolar cylindrical
6158:"Analytic geometry"
5794:can be written as:
5364:In three dimensions
4098:
4074:
4050:
4026:
1402:coordinate surfaces
989:. The octants are:
856:-axis and from the
677:) is defined by an
418:graph of a function
6957:Prolate spheroidal
6808:Weisstein, Eric W.
6026:
6017:
5965:
5956:
5905:
5896:
5842:
5784:
5724:
5718:
5673:
5667:
5620:
5571:
5543:-axis as pointing
5507:to the horizontal
5445:
5428:-plane) is called
5382:
5374:
5268:
5221:
5148:
5055:
4945:
4910:
4901:
4855:
4822:
4683:
4659:
4650:
4499:
4490:
4446:
4384:
4336:
4241:
4105:
4078:
4054:
4030:
4006:
3989:
3903:orthogonal vectors
3887:
3861:
3859:
3691:
3685:
3624:
3618:
3513:
3498:
3466:
3419:
3387:
3345:
3336:
3296:
3244:
3105:
3103:
2951:
2884:
2755:
2753:
2621:
2586:
2432:
2286:
2227:
2162:
2055:
2009:
1971:Euclidean distance
1921:
1713:coordinates in an
1575:
1530:
1508:
1459:
1268:
1266:
939:
637:
561:degrees of freedom
491:Frans van Schooten
316:
54:
18:Cartesian equation
7017:Analytic geometry
7004:
7003:
6952:Oblate spheroidal
6920:Three dimensional
6761:978-0-07-043316-8
6735:978-0-387-18430-2
6647:978-0-87220-567-3
6616:978-1-119-77798-4
6592:978-0-534-35188-5
6583:Modern Geometries
6529:978-0-13-805326-0
6522:. Prentice Hall.
6506:978-0-07-338315-6
6487:978-0-521-59787-6
6435:978-3-319-11079-0
6266:Consider the two
6210:978-0-321-38700-4
5255:In two dimensions
3890:{\displaystyle A}
2482:glide reflections
2458:Euclidean motions
2427:
2157:
1933:counter-clockwise
1472:Cartesian product
1462:Higher dimensions
866:| and |
753:computer graphics
517:polar coordinates
434:computer graphics
394:analytic geometry
152:coordinate system
16:(Redirected from
7044:
6865:
6858:
6851:
6842:
6821:
6820:
6790:
6773:
6739:
6720:
6710:
6696:
6674:
6659:
6620:
6595:
6577:
6556:
6552:978-0470-88861-2
6541:(6th ed.).
6533:
6521:
6510:
6491:
6472:
6451:
6449:
6447:
6438:. Archived from
6401:
6396:
6390:
6384:
6378:
6369:
6363:
6357:
6351:
6345:
6339:
6334:
6328:
6327:
6325:
6323:
6309:
6303:
6302:
6300:
6298:
6284:
6271:
6264:
6258:
6252:
6246:
6241:
6235:
6229:
6223:
6222:
6194:
6188:
6179:
6173:
6172:
6170:
6168:
6153:
6083:
6071:
6057:
6035:
6033:
6032:
6027:
6022:
6021:
5986:
5974:
5972:
5971:
5966:
5961:
5960:
5925:
5914:
5912:
5911:
5906:
5901:
5900:
5865:
5851:
5849:
5848:
5843:
5838:
5827:
5816:
5805:
5793:
5791:
5790:
5785:
5733:
5731:
5730:
5725:
5723:
5722:
5694:
5682:
5680:
5679:
5674:
5672:
5671:
5643:
5629:
5627:
5626:
5621:
5616:
5605:
5594:
5580:
5578:
5577:
5572:
5570:
5396:along which the
5250:Axes conventions
5230:
5228:
5227:
5222:
5190:
5179:
5157:
5155:
5154:
5149:
5117:
5106:
5064:
5062:
5061:
5056:
5024:
5013:
4993:
4954:
4952:
4951:
4946:
4944:
4943:
4919:
4917:
4916:
4911:
4906:
4905:
4891:
4879:
4860:
4859:
4827:
4826:
4802:
4801:
4790:
4789:
4772:
4771:
4752:
4751:
4740:
4739:
4722:
4721:
4668:
4666:
4665:
4660:
4655:
4654:
4630:
4629:
4618:
4617:
4600:
4599:
4580:
4579:
4568:
4567:
4550:
4549:
4523:
4508:
4506:
4505:
4500:
4495:
4494:
4462:
4451:
4450:
4436:
4424:
4400:augmented matrix
4393:
4391:
4390:
4385:
4377:
4376:
4364:
4363:
4345:
4343:
4342:
4337:
4326:
4325:
4310:
4309:
4291:
4290:
4275:
4274:
4250:
4248:
4247:
4242:
4234:
4233:
4218:
4217:
4199:
4198:
4183:
4182:
4156:
4146:
4122:
4114:
4112:
4111:
4106:
4097:
4092:
4073:
4068:
4049:
4044:
4025:
4020:
3998:
3996:
3995:
3990:
3982:
3981:
3966:
3965:
3947:
3946:
3931:
3930:
3896:
3894:
3893:
3888:
3870:
3868:
3867:
3862:
3860:
3853:
3852:
3840:
3839:
3818:
3817:
3792:
3780:
3779:
3767:
3766:
3745:
3744:
3719:
3700:
3698:
3697:
3692:
3690:
3689:
3682:
3681:
3668:
3667:
3633:
3631:
3630:
3625:
3623:
3622:
3615:
3614:
3597:
3596:
3577:
3576:
3559:
3558:
3522:
3520:
3519:
3514:
3503:
3502:
3471:
3470:
3463:
3451:
3428:
3426:
3425:
3420:
3396:
3394:
3393:
3388:
3383:
3372:
3354:
3352:
3351:
3346:
3341:
3340:
3305:
3303:
3302:
3297:
3257:Glide reflection
3253:
3251:
3250:
3245:
3145:
3134:
3114:
3112:
3111:
3106:
3104:
3057:
3005:
2984:
2972:
2960:
2958:
2957:
2952:
2940:
2924:
2912:
2893:
2891:
2890:
2885:
2797:
2786:
2764:
2762:
2761:
2756:
2754:
2713:
2667:
2630:
2628:
2627:
2622:
2609:counterclockwise
2595:
2593:
2592:
2587:
2549:
2538:
2518:
2506:
2441:
2439:
2438:
2433:
2428:
2426:
2425:
2416:
2415:
2403:
2402:
2387:
2386:
2377:
2376:
2364:
2363:
2348:
2347:
2338:
2337:
2325:
2324:
2312:
2295:
2293:
2292:
2287:
2282:
2281:
2269:
2268:
2256:
2255:
2236:
2234:
2233:
2228:
2223:
2222:
2210:
2209:
2197:
2196:
2171:
2169:
2168:
2163:
2158:
2156:
2155:
2146:
2145:
2133:
2132:
2117:
2116:
2107:
2106:
2094:
2093:
2081:
2064:
2062:
2061:
2056:
2051:
2050:
2038:
2037:
2018:
2016:
2015:
2010:
2005:
2004:
1992:
1991:
1949:
1945:
1864:right-handedness
1805:image processing
1627:
1623:
1584:
1582:
1581:
1576:
1574:
1573:
1568:
1539:
1537:
1536:
1531:
1529:
1517:
1515:
1514:
1509:
1507:
1499:
1491:
1490:
1485:
1456:
1448:
1441:
1434:
1419:
1385:
1319:
1315:
1311:
1307:
1303:
1283:
1277:
1275:
1274:
1269:
1267:
1232:
1199:
1166:
1099:
1066:
1033:
981:coordinate plane
936:
932:
925:
918:
882:Three dimensions
871:
865:
860:-axis are |
847:
813:
809:
796:
795:
747:
743:
739:
735:
648:
646:
644:
643:
638:
587:
583:
473:Pierre de Fermat
402:complex analysis
368:
357:
351:
309:
289:
255:
249:
236:
230:
208:orthogonal basis
201:
177:coordinate lines
166:, which are the
145:
144:
141:
140:
137:
134:
131:
128:
125:
122:
119:
112:
104:
103:
100:
99:
96:
93:
90:
87:
84:
81:
78:
71:
51:
47:
43:
39:
21:
7052:
7051:
7047:
7046:
7045:
7043:
7042:
7041:
7007:
7006:
7005:
7000:
6915:
6879:Two dimensional
6874:
6869:
6806:
6805:
6797:
6776:
6762:
6742:
6736:
6723:
6699:
6662:
6648:
6632:Descartes, René
6630:
6627:
6625:Further reading
6617:
6609:. p. 657.
6598:
6593:
6580:
6574:
6559:
6553:
6536:
6530:
6513:
6507:
6494:
6488:
6475:
6469:
6454:
6445:
6443:
6436:
6413:
6410:
6405:
6404:
6397:
6393:
6385:
6381:
6370:
6366:
6358:
6354:
6346:
6342:
6335:
6331:
6321:
6319:
6311:
6310:
6306:
6296:
6294:
6286:
6285:
6274:
6265:
6261:
6253:
6249:
6242:
6238:
6230:
6226:
6211:
6196:
6195:
6191:
6180:
6176:
6166:
6164:
6155:
6154:
6150:
6145:
6102:
6081:
6059:
6047:
6044:complex numbers
6016:
6015:
6009:
6008:
6002:
6001:
5991:
5977:
5976:
5955:
5954:
5948:
5947:
5941:
5940:
5930:
5916:
5915:
5895:
5894:
5888:
5887:
5881:
5880:
5870:
5856:
5855:
5796:
5795:
5758:
5757:
5717:
5716:
5710:
5709:
5699:
5685:
5684:
5666:
5665:
5659:
5658:
5648:
5634:
5633:
5585:
5584:
5561:
5560:
5553:
5449:right-hand rule
5366:
5335:right-hand rule
5265:right-hand rule
5257:
5252:
5246:Right-hand rule
5242:
5236:
5183:
5172:
5164:
5163:
5110:
5099:
5091:
5090:
5081:
5017:
5006:
4998:
4997:
4983:
4975:
4929:
4924:
4923:
4900:
4899:
4893:
4892:
4884:
4881:
4880:
4872:
4865:
4854:
4853:
4847:
4846:
4840:
4839:
4829:
4821:
4820:
4815:
4810:
4804:
4803:
4793:
4791:
4775:
4773:
4757:
4754:
4753:
4743:
4741:
4725:
4723:
4707:
4700:
4694:
4693:
4690:Euclidean plane
4675:
4649:
4648:
4643:
4638:
4632:
4631:
4621:
4619:
4603:
4601:
4585:
4582:
4581:
4571:
4569:
4553:
4551:
4535:
4528:
4516:
4511:
4510:
4489:
4488:
4482:
4481:
4475:
4474:
4464:
4455:
4445:
4444:
4438:
4437:
4429:
4426:
4425:
4417:
4410:
4404:
4403:
4368:
4355:
4350:
4349:
4311:
4295:
4276:
4260:
4255:
4254:
4219:
4203:
4184:
4168:
4163:
4162:
4159:rotation matrix
4152:
4149:identity matrix
4142:
4129:identity matrix
4118:
4001:
4000:
3967:
3951:
3932:
3916:
3911:
3910:
3879:
3878:
3858:
3857:
3844:
3825:
3803:
3793:
3785:
3782:
3781:
3771:
3752:
3730:
3720:
3712:
3703:
3702:
3684:
3683:
3673:
3670:
3669:
3659:
3652:
3640:
3639:
3617:
3616:
3600:
3598:
3582:
3579:
3578:
3562:
3560:
3544:
3537:
3525:
3524:
3497:
3496:
3490:
3489:
3479:
3465:
3464:
3456:
3453:
3452:
3444:
3437:
3431:
3430:
3399:
3398:
3376:
3365:
3357:
3356:
3335:
3334:
3328:
3327:
3317:
3311:
3310:
3276:
3275:
3268:
3259:
3138:
3127:
3119:
3118:
3102:
3101:
3058:
3050:
3047:
3046:
3006:
2998:
2989:
2988:
2974:
2962:
2943:
2942:
2930:
2914:
2902:
2899:
2790:
2779:
2771:
2770:
2752:
2751:
2714:
2706:
2703:
2702:
2668:
2660:
2651:
2650:
2613:
2612:
2601:
2542:
2531:
2523:
2522:
2508:
2496:
2490:
2466:Euclidean plane
2450:
2417:
2407:
2394:
2378:
2368:
2355:
2339:
2329:
2316:
2300:
2299:
2273:
2260:
2247:
2239:
2238:
2214:
2201:
2188:
2180:
2179:
2147:
2137:
2124:
2108:
2098:
2085:
2069:
2068:
2042:
2029:
2021:
2020:
1996:
1983:
1975:
1974:
1967:
1962:
1947:
1943:
1913:
1905:Main articles:
1903:
1860:right-hand rule
1813:display buffers
1753:, instead of a
1744:
1734:
1727:
1708:
1699:
1692:
1625:
1621:
1614:
1591:
1589:Generalizations
1563:
1558:
1557:
1542:Euclidean space
1520:
1519:
1480:
1475:
1474:
1464:
1454:
1443:
1436:
1429:
1405:
1390:right-hand rule
1383:
1317:
1313:
1309:
1305:
1285:
1281:
1265:
1264:
1231:
1198:
1165:
1132:
1131:
1098:
1065:
1032:
993:
992:
934:
927:
920:
913:
900:and axis lines
890:
884:
867:
861:
837:
811:
807:
794:Cartesian plane
793:
792:
788:Euclidean plane
745:
741:
737:
733:
667:
661:
614:
613:
612:
609:linear function
585:
581:
546:
540:
533:
450:
408:, multivariate
359:
353:
347:
299:
268:
251:
245:
239:Euclidean space
232:
226:
212:Cartesian frame
199:
182:coordinate axes
116:
107:
106:
75:
66:
65:
49:
45:
41:
37:
28:
23:
22:
15:
12:
11:
5:
7050:
7048:
7040:
7039:
7034:
7029:
7027:René Descartes
7024:
7019:
7009:
7008:
7002:
7001:
6999:
6998:
6996:
6994:
6989:
6984:
6979:
6974:
6969:
6964:
6959:
6954:
6949:
6944:
6939:
6934:
6929:
6923:
6921:
6917:
6916:
6914:
6913:
6908:
6903:
6898:
6888:
6882:
6880:
6876:
6875:
6870:
6868:
6867:
6860:
6853:
6845:
6839:
6838:
6833:
6827:
6822:
6803:
6796:
6795:External links
6793:
6792:
6791:
6774:
6760:
6740:
6734:
6721:
6697:
6660:
6646:
6626:
6623:
6622:
6621:
6615:
6596:
6591:
6578:
6572:
6557:
6551:
6534:
6528:
6511:
6505:
6492:
6486:
6473:
6467:
6452:
6442:on 27 May 2022
6434:
6409:
6406:
6403:
6402:
6399:Griffiths 1999
6391:
6379:
6364:
6352:
6340:
6329:
6317:planetmath.org
6304:
6272:
6259:
6247:
6244:Berlinski 2011
6236:
6224:
6209:
6189:
6174:
6147:
6146:
6144:
6141:
6140:
6139:
6134:
6129:
6124:
6119:
6113:
6108:
6101:
6098:
6078:imaginary unit
6025:
6020:
6014:
6011:
6010:
6007:
6004:
6003:
6000:
5997:
5996:
5994:
5989:
5985:
5964:
5959:
5953:
5950:
5949:
5946:
5943:
5942:
5939:
5936:
5935:
5933:
5928:
5924:
5904:
5899:
5893:
5890:
5889:
5886:
5883:
5882:
5879:
5876:
5875:
5873:
5868:
5864:
5841:
5837:
5833:
5830:
5826:
5822:
5819:
5815:
5811:
5808:
5804:
5783:
5780:
5777:
5774:
5771:
5768:
5765:
5749:standard basis
5721:
5715:
5712:
5711:
5708:
5705:
5704:
5702:
5697:
5693:
5670:
5664:
5661:
5660:
5657:
5654:
5653:
5651:
5646:
5642:
5619:
5615:
5611:
5608:
5604:
5600:
5597:
5593:
5569:
5552:
5549:
5365:
5362:
5351:left-hand rule
5256:
5253:
5238:Main article:
5235:
5232:
5220:
5217:
5214:
5211:
5208:
5205:
5202:
5199:
5196:
5193:
5189:
5186:
5182:
5178:
5175:
5171:
5147:
5144:
5141:
5138:
5135:
5132:
5129:
5126:
5123:
5120:
5116:
5113:
5109:
5105:
5102:
5098:
5080:
5077:
5054:
5051:
5048:
5045:
5042:
5039:
5036:
5033:
5030:
5027:
5023:
5020:
5016:
5012:
5009:
5005:
4974:
4971:
4942:
4939:
4936:
4932:
4909:
4904:
4898:
4895:
4894:
4890:
4887:
4883:
4882:
4878:
4875:
4871:
4870:
4868:
4863:
4858:
4852:
4849:
4848:
4845:
4842:
4841:
4838:
4835:
4834:
4832:
4825:
4819:
4816:
4814:
4811:
4809:
4806:
4805:
4800:
4796:
4792:
4788:
4785:
4782:
4778:
4774:
4770:
4767:
4764:
4760:
4756:
4755:
4750:
4746:
4742:
4738:
4735:
4732:
4728:
4724:
4720:
4717:
4714:
4710:
4706:
4705:
4703:
4674:
4671:
4658:
4653:
4647:
4644:
4642:
4639:
4637:
4634:
4633:
4628:
4624:
4620:
4616:
4613:
4610:
4606:
4602:
4598:
4595:
4592:
4588:
4584:
4583:
4578:
4574:
4570:
4566:
4563:
4560:
4556:
4552:
4548:
4545:
4542:
4538:
4534:
4533:
4531:
4526:
4522:
4519:
4498:
4493:
4487:
4484:
4483:
4480:
4477:
4476:
4473:
4470:
4469:
4467:
4461:
4458:
4454:
4449:
4443:
4440:
4439:
4435:
4432:
4428:
4427:
4423:
4420:
4416:
4415:
4413:
4383:
4380:
4375:
4371:
4367:
4362:
4358:
4335:
4332:
4329:
4324:
4321:
4318:
4314:
4308:
4305:
4302:
4298:
4294:
4289:
4286:
4283:
4279:
4273:
4270:
4267:
4263:
4240:
4237:
4232:
4229:
4226:
4222:
4216:
4213:
4210:
4206:
4202:
4197:
4194:
4191:
4187:
4181:
4178:
4175:
4171:
4140:if and only if
4104:
4101:
4096:
4091:
4088:
4085:
4081:
4077:
4072:
4067:
4064:
4061:
4057:
4053:
4048:
4043:
4040:
4037:
4033:
4029:
4024:
4019:
4016:
4013:
4009:
3988:
3985:
3980:
3977:
3974:
3970:
3964:
3961:
3958:
3954:
3950:
3945:
3942:
3939:
3935:
3929:
3926:
3923:
3919:
3907:Euclidean norm
3886:
3856:
3851:
3847:
3843:
3838:
3835:
3832:
3828:
3824:
3821:
3816:
3813:
3810:
3806:
3802:
3799:
3796:
3794:
3791:
3788:
3784:
3783:
3778:
3774:
3770:
3765:
3762:
3759:
3755:
3751:
3748:
3743:
3740:
3737:
3733:
3729:
3726:
3723:
3721:
3718:
3715:
3711:
3710:
3688:
3680:
3676:
3672:
3671:
3666:
3662:
3658:
3657:
3655:
3650:
3647:
3621:
3613:
3610:
3607:
3603:
3599:
3595:
3592:
3589:
3585:
3581:
3580:
3575:
3572:
3569:
3565:
3561:
3557:
3554:
3551:
3547:
3543:
3542:
3540:
3535:
3532:
3512:
3509:
3506:
3501:
3495:
3492:
3491:
3488:
3485:
3484:
3482:
3477:
3474:
3469:
3462:
3459:
3455:
3454:
3450:
3447:
3443:
3442:
3440:
3418:
3415:
3412:
3409:
3406:
3386:
3382:
3379:
3375:
3371:
3368:
3364:
3344:
3339:
3333:
3330:
3329:
3326:
3323:
3322:
3320:
3295:
3292:
3289:
3286:
3283:
3267:
3264:
3258:
3255:
3243:
3240:
3237:
3233:
3230:
3227:
3224:
3221:
3218:
3215:
3212:
3209:
3206:
3203:
3200:
3197:
3194:
3190:
3187:
3184:
3181:
3178:
3175:
3172:
3169:
3166:
3163:
3160:
3157:
3154:
3151:
3148:
3144:
3141:
3137:
3133:
3130:
3126:
3100:
3097:
3094:
3091:
3088:
3085:
3082:
3079:
3076:
3073:
3070:
3067:
3064:
3061:
3059:
3056:
3053:
3049:
3048:
3045:
3042:
3039:
3036:
3033:
3030:
3027:
3024:
3021:
3018:
3015:
3012:
3009:
3007:
3004:
3001:
2997:
2996:
2950:
2898:
2895:
2883:
2880:
2877:
2873:
2870:
2867:
2864:
2861:
2858:
2855:
2852:
2849:
2846:
2843:
2840:
2836:
2833:
2830:
2827:
2824:
2821:
2818:
2815:
2812:
2809:
2806:
2803:
2800:
2796:
2793:
2789:
2785:
2782:
2778:
2750:
2747:
2744:
2741:
2738:
2735:
2732:
2729:
2726:
2723:
2720:
2717:
2715:
2712:
2709:
2705:
2704:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2671:
2669:
2666:
2663:
2659:
2658:
2620:
2600:
2597:
2585:
2582:
2579:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2555:
2552:
2548:
2545:
2541:
2537:
2534:
2530:
2489:
2486:
2449:
2446:
2431:
2424:
2420:
2414:
2410:
2406:
2401:
2397:
2393:
2390:
2385:
2381:
2375:
2371:
2367:
2362:
2358:
2354:
2351:
2346:
2342:
2336:
2332:
2328:
2323:
2319:
2315:
2310:
2307:
2285:
2280:
2276:
2272:
2267:
2263:
2259:
2254:
2250:
2246:
2226:
2221:
2217:
2213:
2208:
2204:
2200:
2195:
2191:
2187:
2161:
2154:
2150:
2144:
2140:
2136:
2131:
2127:
2123:
2120:
2115:
2111:
2105:
2101:
2097:
2092:
2088:
2084:
2079:
2076:
2054:
2049:
2045:
2041:
2036:
2032:
2028:
2008:
2003:
1999:
1995:
1990:
1986:
1982:
1966:
1963:
1961:
1958:
1929:Roman numerals
1902:
1899:
1739:
1732:
1725:
1704:
1697:
1690:
1613:
1610:
1590:
1587:
1572:
1567:
1528:
1506:
1502:
1498:
1494:
1489:
1484:
1463:
1460:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1179:
1176:
1173:
1170:
1167:
1164:
1161:
1158:
1155:
1152:
1149:
1146:
1143:
1140:
1137:
1134:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
1000:
883:
880:
874:absolute value
831:first quadrant
816:unit hyperbola
687:unit of length
660:
659:Two dimensions
657:
636:
633:
630:
627:
624:
621:
599:, for example
559:There are two
542:Main article:
539:
536:
532:
529:
465:René Descartes
452:The adjective
449:
446:
398:linear algebra
320:René Descartes
314:is the radius.
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7049:
7038:
7035:
7033:
7030:
7028:
7025:
7023:
7020:
7018:
7015:
7014:
7012:
6997:
6995:
6993:
6990:
6988:
6985:
6983:
6980:
6978:
6975:
6973:
6970:
6968:
6965:
6963:
6960:
6958:
6955:
6953:
6950:
6948:
6945:
6943:
6940:
6938:
6935:
6933:
6930:
6928:
6925:
6924:
6922:
6918:
6912:
6909:
6907:
6904:
6902:
6899:
6896:
6892:
6889:
6887:
6884:
6883:
6881:
6877:
6873:
6866:
6861:
6859:
6854:
6852:
6847:
6846:
6843:
6837:
6834:
6831:
6828:
6826:
6823:
6818:
6817:
6812:
6809:
6804:
6802:
6799:
6798:
6794:
6788:
6784:
6780:
6775:
6771:
6767:
6763:
6757:
6753:
6749:
6745:
6741:
6737:
6731:
6727:
6722:
6718:
6714:
6709:
6708:
6702:
6698:
6694:
6690:
6686:
6682:
6678:
6673:
6672:
6666:
6661:
6657:
6653:
6649:
6643:
6639:
6638:
6633:
6629:
6628:
6624:
6618:
6612:
6608:
6604:
6603:
6597:
6594:
6588:
6584:
6579:
6575:
6573:9781317568216
6569:
6566:. Routledge.
6565:
6564:
6558:
6554:
6548:
6544:
6540:
6535:
6531:
6525:
6520:
6519:
6512:
6508:
6502:
6498:
6493:
6489:
6483:
6479:
6474:
6470:
6468:9780307789730
6464:
6460:
6459:
6453:
6441:
6437:
6431:
6427:
6423:
6419:
6418:
6412:
6411:
6407:
6400:
6395:
6392:
6388:
6383:
6380:
6377:
6373:
6368:
6365:
6361:
6356:
6353:
6349:
6344:
6341:
6338:
6333:
6330:
6318:
6314:
6308:
6305:
6293:
6289:
6283:
6281:
6279:
6277:
6273:
6269:
6263:
6260:
6256:
6251:
6248:
6245:
6240:
6237:
6233:
6228:
6225:
6220:
6216:
6212:
6206:
6202:
6201:
6193:
6190:
6187:
6183:
6178:
6175:
6163:
6159:
6152:
6149:
6142:
6138:
6135:
6133:
6130:
6128:
6125:
6123:
6120:
6117:
6116:Jones diagram
6114:
6112:
6109:
6107:
6104:
6103:
6099:
6097:
6095:
6091:
6087:
6079:
6075:
6070:
6066:
6062:
6055:
6051:
6045:
6041:
6036:
6023:
6018:
6012:
6005:
5998:
5992:
5987:
5962:
5957:
5951:
5944:
5937:
5931:
5926:
5902:
5897:
5891:
5884:
5877:
5871:
5866:
5852:
5839:
5831:
5828:
5820:
5817:
5809:
5806:
5778:
5775:
5772:
5769:
5766:
5755:
5751:
5750:
5745:
5741:
5737:
5719:
5713:
5706:
5700:
5695:
5668:
5662:
5655:
5649:
5644:
5630:
5617:
5609:
5606:
5598:
5595:
5582:
5558:
5550:
5548:
5546:
5542:
5538:
5534:
5528:
5526:
5522:
5518:
5515:-axis to the
5514:
5510:
5506:
5502:
5498:
5492:
5490:
5486:
5482:
5478:
5474:
5470:
5466:
5462:
5458:
5457:middle finger
5454:
5450:
5441:
5437:
5435:
5431:
5427:
5423:
5419:
5415:
5411:
5407:
5403:
5399:
5395:
5391:
5387:
5378:
5370:
5363:
5361:
5357:
5354:
5352:
5347:
5345:
5342:-axis to the
5341:
5337:
5336:
5330:
5329:orientation.
5328:
5324:
5320:
5316:
5312:
5308:
5304:
5299:
5297:
5293:
5289:
5285:
5284:perpendicular
5281:
5277:
5273:
5266:
5261:
5254:
5251:
5247:
5241:
5240:Orientability
5233:
5231:
5215:
5212:
5209:
5206:
5203:
5200:
5194:
5187:
5184:
5180:
5176:
5173:
5161:
5158:
5142:
5139:
5136:
5133:
5130:
5127:
5121:
5114:
5111:
5107:
5103:
5100:
5088:
5086:
5078:
5076:
5074:
5070:
5065:
5052:
5046:
5043:
5040:
5037:
5034:
5028:
5021:
5018:
5014:
5010:
5007:
4995:
4991:
4987:
4981:
4972:
4970:
4967:
4965:
4960:
4958:
4940:
4937:
4934:
4930:
4920:
4907:
4902:
4896:
4888:
4885:
4876:
4873:
4866:
4861:
4856:
4850:
4843:
4836:
4830:
4823:
4817:
4812:
4807:
4798:
4794:
4786:
4783:
4780:
4776:
4768:
4765:
4762:
4758:
4748:
4744:
4736:
4733:
4730:
4726:
4718:
4715:
4712:
4708:
4701:
4691:
4687:
4679:
4672:
4670:
4656:
4651:
4645:
4640:
4635:
4626:
4622:
4614:
4611:
4608:
4604:
4596:
4593:
4590:
4586:
4576:
4572:
4564:
4561:
4558:
4554:
4546:
4543:
4540:
4536:
4529:
4524:
4520:
4517:
4496:
4491:
4485:
4478:
4471:
4465:
4459:
4456:
4452:
4447:
4441:
4433:
4430:
4421:
4418:
4411:
4401:
4397:
4381:
4378:
4373:
4369:
4365:
4360:
4356:
4346:
4333:
4330:
4327:
4322:
4319:
4316:
4312:
4306:
4303:
4300:
4296:
4292:
4287:
4284:
4281:
4277:
4271:
4268:
4265:
4261:
4251:
4238:
4235:
4230:
4227:
4224:
4220:
4214:
4211:
4208:
4204:
4200:
4195:
4192:
4189:
4185:
4179:
4176:
4173:
4169:
4160:
4155:
4150:
4145:
4141:
4136:
4134:
4130:
4126:
4121:
4115:
4102:
4099:
4094:
4089:
4086:
4083:
4079:
4075:
4070:
4065:
4062:
4059:
4055:
4051:
4046:
4041:
4038:
4035:
4031:
4027:
4022:
4017:
4014:
4011:
4007:
3986:
3983:
3978:
3975:
3972:
3968:
3962:
3959:
3956:
3952:
3948:
3943:
3940:
3937:
3933:
3927:
3924:
3921:
3917:
3908:
3904:
3900:
3884:
3876:
3871:
3854:
3849:
3845:
3841:
3836:
3833:
3830:
3826:
3822:
3819:
3814:
3811:
3808:
3804:
3800:
3797:
3795:
3789:
3786:
3776:
3772:
3768:
3763:
3760:
3757:
3753:
3749:
3746:
3741:
3738:
3735:
3731:
3727:
3724:
3722:
3716:
3713:
3686:
3678:
3674:
3664:
3660:
3653:
3648:
3645:
3637:
3619:
3611:
3608:
3605:
3601:
3593:
3590:
3587:
3583:
3573:
3570:
3567:
3563:
3555:
3552:
3549:
3545:
3538:
3533:
3530:
3510:
3507:
3504:
3499:
3493:
3486:
3480:
3475:
3472:
3467:
3460:
3457:
3448:
3445:
3438:
3413:
3410:
3407:
3380:
3377:
3373:
3369:
3366:
3342:
3337:
3331:
3324:
3318:
3309:
3308:column matrix
3290:
3287:
3284:
3273:
3265:
3263:
3256:
3254:
3241:
3231:
3228:
3225:
3222:
3219:
3216:
3213:
3210:
3207:
3204:
3201:
3195:
3188:
3185:
3182:
3179:
3176:
3173:
3170:
3167:
3164:
3161:
3158:
3149:
3142:
3139:
3135:
3131:
3128:
3115:
3098:
3095:
3092:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3060:
3054:
3051:
3043:
3040:
3037:
3034:
3031:
3028:
3025:
3022:
3019:
3016:
3013:
3010:
3008:
3002:
2999:
2986:
2982:
2978:
2970:
2966:
2948:
2938:
2934:
2928:
2922:
2918:
2910:
2906:
2896:
2894:
2881:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2850:
2847:
2841:
2834:
2831:
2828:
2825:
2822:
2819:
2816:
2813:
2810:
2801:
2794:
2791:
2787:
2783:
2780:
2768:
2765:
2748:
2745:
2742:
2739:
2736:
2733:
2730:
2727:
2724:
2721:
2718:
2716:
2710:
2707:
2699:
2696:
2693:
2690:
2687:
2684:
2681:
2678:
2675:
2672:
2670:
2664:
2661:
2648:
2646:
2642:
2638:
2634:
2618:
2610:
2606:
2598:
2596:
2583:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2553:
2546:
2543:
2539:
2535:
2532:
2520:
2516:
2512:
2504:
2500:
2494:
2487:
2485:
2483:
2479:
2475:
2471:
2467:
2463:
2459:
2455:
2447:
2445:
2442:
2429:
2422:
2412:
2408:
2404:
2399:
2395:
2388:
2383:
2373:
2369:
2365:
2360:
2356:
2349:
2344:
2334:
2330:
2326:
2321:
2317:
2308:
2305:
2297:
2278:
2274:
2270:
2265:
2261:
2257:
2252:
2248:
2219:
2215:
2211:
2206:
2202:
2198:
2193:
2189:
2177:
2172:
2159:
2152:
2142:
2138:
2134:
2129:
2125:
2118:
2113:
2103:
2099:
2095:
2090:
2086:
2077:
2074:
2066:
2047:
2043:
2039:
2034:
2030:
2001:
1997:
1993:
1988:
1984:
1972:
1964:
1959:
1957:
1955:
1954:
1941:
1936:
1934:
1930:
1926:
1917:
1912:
1908:
1900:
1898:
1896:
1892:
1888:
1884:
1880:
1876:
1872:
1867:
1865:
1861:
1857:
1853:
1849:
1845:
1841:
1837:
1833:
1832:3D projection
1829:
1825:
1821:
1816:
1814:
1810:
1806:
1801:
1799:
1795:
1791:
1787:
1783:
1779:
1775:
1771:
1767:
1762:
1760:
1756:
1752:
1748:
1742:
1738:
1731:
1724:
1720:
1716:
1712:
1707:
1703:
1696:
1689:
1684:
1682:
1678:
1674:
1670:
1666:
1662:
1658:
1653:
1651:
1647:
1643:
1639:
1635:
1631:
1619:
1611:
1609:
1607:
1603:
1602:
1597:
1588:
1586:
1570:
1555:
1551:
1547:
1544:of dimension
1543:
1500:
1492:
1487:
1473:
1469:
1461:
1452:
1446:
1439:
1432:
1427:
1423:
1417:
1413:
1409:
1403:
1398:
1394:
1392:
1391:
1381:
1377:
1372:
1370:
1366:
1362:
1358:
1354:
1350:
1346:
1342:
1338:
1334:
1330:
1326:
1321:
1301:
1297:
1293:
1289:
1278:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1125:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1092:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
990:
988:
987:
982:
977:
975:
971:
966:
964:
960:
956:
952:
948:
944:
930:
923:
916:
911:
907:
903:
899:
894:
889:
881:
879:
877:
876:of a number.
875:
870:
864:
859:
855:
851:
845:
841:
834:
832:
828:
824:
819:
818:, and so on.
817:
805:
801:
797:
789:
784:
782:
778:
774:
770:
766:
762:
758:
754:
749:
731:
727:
723:
722:
717:
716:
710:
708:
704:
700:
696:
692:
688:
684:
683:perpendicular
680:
676:
672:
666:
658:
656:
654:
652:
634:
631:
628:
625:
619:
610:
606:
602:
598:
594:
589:
578:
574:
570:
566:
562:
557:
555:
551:
545:
538:One dimension
537:
535:
530:
528:
526:
522:
518:
513:
511:
510:vector spaces
507:
503:
499:
494:
492:
488:
487:
480:
478:
477:Nicole Oresme
474:
470:
466:
463:
459:
458:mathematician
455:
447:
445:
443:
439:
435:
431:
427:
423:
419:
415:
411:
407:
403:
399:
395:
390:
388:
384:
380:
376:
372:
366:
362:
356:
350:
345:
341:
337:
333:
329:
325:
321:
313:
307:
303:
297:
293:
288:
284:
280:
276:
272:
265:
261:
259:
254:
248:
244:
240:
237:-dimensional
235:
229:
224:
220:
215:
213:
209:
206:represent an
205:
197:
196:
191:
187:
183:
179:
178:
173:
172:perpendicular
169:
165:
161:
157:
153:
149:
143:
110:
102:
69:
63:
59:
34:
30:
19:
6947:Paraboloidal
6926:
6885:
6814:
6778:
6751:
6725:
6706:
6670:
6636:
6601:
6582:
6562:
6538:
6517:
6496:
6477:
6457:
6444:. Retrieved
6440:the original
6416:
6394:
6382:
6367:
6355:
6343:
6332:
6320:. Retrieved
6316:
6307:
6295:. Retrieved
6291:
6262:
6250:
6239:
6227:
6199:
6192:
6177:
6165:. Retrieved
6161:
6151:
6132:Regular grid
6089:
6085:
6073:
6068:
6064:
6060:
6053:
6049:
6039:
6038:There is no
6037:
5853:
5747:
5743:
5739:
5736:unit vectors
5631:
5583:
5554:
5544:
5540:
5529:
5524:
5520:
5516:
5512:
5508:
5504:
5500:
5496:
5493:
5488:
5484:
5480:
5477:right-handed
5476:
5472:
5468:
5464:
5453:index finger
5446:
5433:
5430:right-handed
5429:
5425:
5421:
5417:
5413:
5409:
5405:
5401:
5397:
5389:
5385:
5383:
5358:
5355:
5350:
5348:
5343:
5339:
5333:
5331:
5327:right-handed
5326:
5322:
5318:
5314:
5310:
5306:
5302:
5300:
5295:
5291:
5287:
5279:
5275:
5271:
5269:
5162:
5159:
5089:
5082:
5072:
5068:
5066:
4996:
4989:
4985:
4979:
4976:
4968:
4961:
4921:
4684:
4347:
4252:
4153:
4143:
4137:
4119:
4116:
3872:
3269:
3260:
3116:
2987:
2980:
2976:
2968:
2964:
2936:
2932:
2920:
2916:
2908:
2904:
2900:
2769:
2766:
2649:
2644:
2640:
2636:
2632:
2602:
2521:
2514:
2510:
2502:
2498:
2491:
2470:translations
2457:
2451:
2443:
2298:
2173:
2067:
1968:
1951:
1939:
1937:
1924:
1922:
1894:
1890:
1886:
1885:. The words
1882:
1878:
1874:
1870:
1868:
1851:
1847:
1843:
1839:
1838:) shows the
1827:
1823:
1819:
1817:
1808:
1802:
1797:
1793:
1789:
1785:
1781:
1763:
1740:
1736:
1729:
1722:
1718:
1714:
1710:
1705:
1701:
1694:
1687:
1685:
1680:
1676:
1672:
1668:
1664:
1659:varies with
1654:
1649:
1645:
1641:
1637:
1633:
1629:
1615:
1606:affine plane
1599:
1592:
1553:
1545:
1468:real numbers
1465:
1450:
1444:
1437:
1430:
1425:
1421:
1415:
1411:
1407:
1387:
1379:
1375:
1373:
1368:
1367:-plane, and
1364:
1360:
1356:
1352:
1348:
1344:
1340:
1336:
1332:
1328:
1324:
1322:
1299:
1295:
1291:
1287:
1282:(3, −2.5, 1)
1279:
991:
984:
980:
978:
973:
969:
967:
962:
958:
954:
950:
946:
942:
940:
928:
921:
914:
909:
905:
901:
897:
878:
868:
862:
857:
853:
843:
839:
835:
830:
826:
823:right angles
820:
791:
785:
780:
776:
772:
768:
764:
760:
756:
750:
729:
725:
719:
713:
711:
706:
702:
698:
694:
690:
679:ordered pair
674:
670:
668:
655:
590:
572:
565:real numbers
558:
553:
547:
534:
514:
502:Isaac Newton
495:
486:La Géométrie
484:
481:
453:
451:
414:group theory
391:
379:tangent line
364:
360:
354:
348:
317:
311:
305:
301:
295:
291:
286:
282:
278:
274:
270:
252:
246:
233:
227:
222:
216:
211:
193:
189:
185:
181:
175:
163:
160:real numbers
61:
55:
46:(−1.5, −2.5)
29:
6977:Bispherical
6962:Ellipsoidal
6932:Cylindrical
6257:, p. 1
6232:Burton 2011
6094:quaternions
6084:, so it is
5535:cube and a
5521:in front of
5475:-axes in a
4964:composition
3355:The result
2493:Translating
2488:Translation
2478:reflections
1856:perspective
1618:parentheses
1552:(lists) of
1355:-axis, and
848:, then its
804:unit square
800:unit circle
601:translation
554:number line
550:affine line
544:Number line
531:Description
469:Netherlands
462:philosopher
430:engineering
387:derivatives
258:hyperplanes
188:(plural of
164:coordinates
7011:Categories
6748:Feshbach H
6701:Margenau H
6374:, p.
6348:Smart 1998
6255:Axler 2015
5742:-axis and
5412:- and the
5296:handedness
5244:See also:
4957:orthogonal
4123:times its
3899:orthogonal
2927:reflection
2897:Reflection
1770:horizontal
1709:) for the
1596:hyperplane
779:-axis and
734:(3, −10.5)
651:affine map
440:and other
204:directions
52:in purple.
40:in green,
6942:Parabolic
6937:Spherical
6927:Cartesian
6901:Parabolic
6895:Log-polar
6886:Cartesian
6816:MathWorld
6663:Korn GA,
6656:488633510
6350:, Chap. 2
6322:25 August
6234:, p. 374.
6143:Citations
5451:. If the
5384:Once the
4331:−
4293:−
4201:−
4125:transpose
3634:is a 2×2
3232:θ
3226:
3217:−
3214:θ
3208:
3189:θ
3183:
3171:θ
3165:
3096:θ
3090:
3081:−
3078:θ
3072:
3044:θ
3038:
3026:θ
3020:
2949:θ
2872:θ
2869:
2857:θ
2854:
2835:θ
2832:
2823:−
2820:θ
2817:
2746:θ
2743:
2731:θ
2728:
2700:θ
2697:
2688:−
2685:θ
2682:
2647:), where
2619:θ
2607:a figure
2474:rotations
2462:bijective
2460:are the (
2405:−
2366:−
2327:−
2135:−
2096:−
1925:quadrants
1895:applicate
1883:applicate
1759:subscript
1626:(3, 5, 7)
1501:×
1455:(1, −1, 1
1384:(0, 0, 1)
1333:applicate
1318:(0, 0, 1)
1314:(0, 1, 0)
1310:(1, 0, 0)
1306:(0, 0, 0)
1256:−
1247:−
1238:−
1214:−
1205:−
1190:−
1172:−
1157:−
1148:−
1123:−
1081:−
1039:−
935:(2, 3, 4)
852:from the
850:distances
827:quadrants
825:, called
623:↦
521:spherical
454:Cartesian
422:astronomy
383:integrals
375:perimeter
336:equations
243:dimension
6992:6-sphere
6972:Toroidal
6911:Elliptic
6787:67-25285
6770:52-11515
6750:(1953).
6744:Morse PM
6717:55-10911
6693:19959906
6685:59-14456
6667:(1961).
6634:(2001).
6478:Geometry
6446:17 April
6362:, pg. 49
6297:6 August
6219:71006826
6167:6 August
6100:See also
6072:. Here,
5505:parallel
5434:positive
5323:standard
5319:positive
5188:′
5177:′
5115:′
5104:′
5079:Shearing
5022:′
5011:′
4889:′
4877:′
4521:′
4460:′
4434:′
4422:′
4396:composed
3790:′
3717:′
3461:′
3449:′
3381:′
3370:′
3143:′
3132:′
3055:′
3003:′
2985:, where
2795:′
2784:′
2711:′
2665:′
2599:Rotation
2547:′
2536:′
1891:ordinate
1887:abscissa
1800:-axis).
1778:vertical
1774:ordinate
1766:abscissa
1657:pressure
1518:, where
1380:altitude
1371:-plane.
1363:-plane,
1329:ordinate
1325:abscissa
721:ordinate
718:and the
715:abscissa
577:oriented
498:calculus
410:calculus
377:and the
328:calculus
241:for any
198:and has
184:or just
58:geometry
44:in red,
6987:Conical
6906:Bipolar
6665:Korn TM
6076:is the
6040:natural
5754:versors
5545:towards
5537:concave
5527:-axis.
5497:towards
5471:-, and
5286:to the
4973:Scaling
4688:of the
4509:where
4147:is the
4127:is the
1953:orthant
1948:(− + −)
1944:(+ + +)
1940:octants
1788:-, and
1735:, ...,
1700:, ...,
1683:, etc.
1622:(10, 5)
1351:-axis,
986:octants
814:), the
605:scaling
448:History
426:physics
324:algebra
162:called
146:) in a
42:(−3, 1)
6785:
6768:
6758:
6732:
6715:
6691:
6683:
6654:
6644:
6613:
6589:
6570:
6549:
6526:
6503:
6484:
6465:
6432:
6217:
6207:
6184:, See
6082:(0, 1)
5854:where
5632:where
5557:vector
5533:convex
5388:- and
3636:matrix
3523:where
3117:Thus:
2767:Thus:
2605:rotate
1842:- and
1757:, the
1755:record
1681:t-axis
1679:, the
1677:y-axis
1675:, the
1673:x-axis
1550:tuples
1420:. The
1376:height
1343:, and
1316:, and
947:origin
926:, and
812:(1, 1)
808:(0, 0)
746:(0, 1)
742:(1, 0)
738:(0, 0)
730:origin
673:or an
573:origin
373:, the
369:; the
340:circle
332:curves
290:where
200:(0, 0)
195:origin
168:signed
50:(0, 0)
38:(2, 3)
6891:Polar
6677:55–79
5461:thumb
5422:above
4982:. If
4157:is a
1751:array
933:, or
767:, or
277:) + (
156:point
150:is a
148:plane
6783:LCCN
6766:LCCN
6756:ISBN
6730:ISBN
6713:LCCN
6689:OCLC
6681:LCCN
6652:OCLC
6642:ISBN
6611:ISBN
6587:ISBN
6568:ISBN
6547:ISBN
6524:ISBN
6501:ISBN
6482:ISBN
6463:ISBN
6448:2022
6430:ISBN
6324:2024
6299:2017
6268:rays
6215:OCLC
6205:ISBN
6186:here
6169:2017
5975:and
5734:are
5683:and
5523:the
5501:away
5424:the
5394:line
5263:The
5248:and
3999:and
3638:and
3270:All
2480:and
2452:The
2237:and
2019:and
1969:The
1909:and
1893:and
1873:and
1667:and
1661:time
1447:= −1
1400:The
1388:the
1331:and
943:axes
908:and
810:and
771:and
763:and
744:and
523:and
504:and
460:and
385:and
371:area
352:and
326:and
310:and
294:and
285:) =
190:axis
186:axes
60:, a
6422:doi
6376:657
6086:not
5467:-,
5432:or
5321:or
5067:If
4955:is
3905:of
3897:is
3223:cos
3205:sin
3180:sin
3162:cos
3087:cos
3069:sin
3035:sin
3017:cos
2935:, −
2901:If
2866:cos
2851:sin
2829:sin
2814:cos
2740:cos
2725:sin
2694:sin
2679:cos
2603:To
2456:or
2296:is
2065:is
1946:or
1834:or
1784:-,
1624:or
1608:).
1440:= 1
1433:= 1
1378:or
1302:/2)
1284:or
931:= 4
924:= 3
917:= 2
724:of
701:of
681:of
584:or
548:An
500:by
367:= 4
344:set
121:ɑːr
80:ɑːr
56:In
7013::
6813:.
6764:.
6746:,
6687:.
6679:.
6650:.
6605:.
6545:.
6428:.
6315:.
6290:.
6275:^
6213:.
6160:.
6096:.
6069:iy
6067:+
6063:=
6052:,
5509:xy
5436:.
5426:xy
5418:xy
5402:xy
5083:A
4988:,
4959:.
4334:1.
4239:1.
4135:.
4103:1.
2983:′)
2979:′,
2967:,
2919:,
2915:(−
2907:,
2645:y'
2641:x'
2513:,
2501:,
2484:.
2476:,
2472:,
1889:,
1820:xy
1815:.
1743:−1
1728:,
1693:,
1648:,
1644:,
1636:,
1585:.
1457:).
1414:,
1410:,
1393:.
1369:xz
1365:yz
1361:xy
1339:,
1327:,
1320:.
1312:,
1298:,
1294:+
1290:,
919:,
904:,
842:,
833:.
786:A
748:.
591:A
512:.
444:.
436:,
428:,
424:,
412:,
404:,
400:,
363:+
304:,
281:−
273:−
260:.
214:.
180:,
130:iː
111::
109:US
105:,
92:zj
89:iː
70::
68:UK
6897:)
6893:(
6864:e
6857:t
6850:v
6819:.
6789:.
6772:.
6738:.
6719:.
6695:.
6658:.
6619:.
6576:.
6555:.
6532:.
6509:.
6490:.
6471:.
6450:.
6424::
6326:.
6301:.
6221:.
6171:.
6090:x
6074:i
6065:x
6061:z
6056:)
6054:y
6050:x
6048:(
6024:.
6019:)
6013:1
6006:0
5999:0
5993:(
5988:=
5984:k
5963:,
5958:)
5952:0
5945:1
5938:0
5932:(
5927:=
5923:j
5903:,
5898:)
5892:0
5885:0
5878:1
5872:(
5867:=
5863:i
5840:,
5836:k
5832:z
5829:+
5825:j
5821:y
5818:+
5814:i
5810:x
5807:=
5803:r
5782:)
5779:z
5776:,
5773:y
5770:,
5767:x
5764:(
5744:y
5740:x
5720:)
5714:1
5707:0
5701:(
5696:=
5692:j
5669:)
5663:0
5656:1
5650:(
5645:=
5641:i
5618:,
5614:j
5610:y
5607:+
5603:i
5599:x
5596:=
5592:r
5568:r
5541:x
5525:z
5517:y
5513:x
5489:z
5485:y
5481:x
5473:z
5469:y
5465:x
5414:y
5410:x
5406:z
5398:z
5390:y
5386:x
5344:y
5340:x
5315:y
5311:x
5307:y
5303:x
5292:x
5288:x
5280:y
5276:y
5272:x
5219:)
5216:y
5213:+
5210:s
5207:x
5204:,
5201:x
5198:(
5195:=
5192:)
5185:y
5181:,
5174:x
5170:(
5146:)
5143:y
5140:,
5137:s
5134:y
5131:+
5128:x
5125:(
5122:=
5119:)
5112:y
5108:,
5101:x
5097:(
5073:m
5069:m
5053:.
5050:)
5047:y
5044:m
5041:,
5038:x
5035:m
5032:(
5029:=
5026:)
5019:y
5015:,
5008:x
5004:(
4992:)
4990:y
4986:x
4984:(
4980:m
4941:j
4938:,
4935:i
4931:A
4908:.
4903:)
4897:1
4886:y
4874:x
4867:(
4862:=
4857:)
4851:1
4844:y
4837:x
4831:(
4824:)
4818:1
4813:0
4808:0
4799:2
4795:b
4787:2
4784:,
4781:2
4777:A
4769:2
4766:,
4763:1
4759:A
4749:1
4745:b
4737:1
4734:,
4731:2
4727:A
4719:1
4716:,
4713:1
4709:A
4702:(
4657:.
4652:)
4646:1
4641:0
4636:0
4627:2
4623:b
4615:2
4612:,
4609:2
4605:A
4597:1
4594:,
4591:2
4587:A
4577:1
4573:b
4565:2
4562:,
4559:1
4555:A
4547:1
4544:,
4541:1
4537:A
4530:(
4525:=
4518:A
4497:,
4492:)
4486:1
4479:y
4472:x
4466:(
4457:A
4453:=
4448:)
4442:1
4431:y
4419:x
4412:(
4382:0
4379:=
4374:2
4370:b
4366:=
4361:1
4357:b
4328:=
4323:2
4320:,
4317:1
4313:A
4307:1
4304:,
4301:2
4297:A
4288:2
4285:,
4282:2
4278:A
4272:1
4269:,
4266:1
4262:A
4236:=
4231:2
4228:,
4225:1
4221:A
4215:1
4212:,
4209:2
4205:A
4196:2
4193:,
4190:2
4186:A
4180:1
4177:,
4174:1
4170:A
4154:A
4144:A
4120:A
4100:=
4095:2
4090:2
4087:,
4084:2
4080:A
4076:+
4071:2
4066:2
4063:,
4060:1
4056:A
4052:=
4047:2
4042:1
4039:,
4036:2
4032:A
4028:+
4023:2
4018:1
4015:,
4012:1
4008:A
3987:0
3984:=
3979:2
3976:,
3973:2
3969:A
3963:1
3960:,
3957:2
3953:A
3949:+
3944:2
3941:,
3938:1
3934:A
3928:1
3925:,
3922:1
3918:A
3885:A
3855:.
3850:2
3846:b
3842:+
3837:2
3834:,
3831:2
3827:A
3823:y
3820:+
3815:1
3812:,
3809:2
3805:A
3801:x
3798:=
3787:y
3777:1
3773:b
3769:+
3764:1
3761:,
3758:1
3754:A
3750:y
3747:+
3742:1
3739:,
3736:1
3732:A
3728:x
3725:=
3714:x
3687:)
3679:2
3675:b
3665:1
3661:b
3654:(
3649:=
3646:b
3620:)
3612:2
3609:,
3606:2
3602:A
3594:1
3591:,
3588:2
3584:A
3574:2
3571:,
3568:1
3564:A
3556:1
3553:,
3550:1
3546:A
3539:(
3534:=
3531:A
3511:,
3508:b
3505:+
3500:)
3494:y
3487:x
3481:(
3476:A
3473:=
3468:)
3458:y
3446:x
3439:(
3417:)
3414:y
3411:,
3408:x
3405:(
3385:)
3378:y
3374:,
3367:x
3363:(
3343:.
3338:)
3332:y
3325:x
3319:(
3294:)
3291:y
3288:,
3285:x
3282:(
3242:.
3239:)
3236:)
3229:2
3220:y
3211:2
3202:x
3199:(
3196:,
3193:)
3186:2
3177:y
3174:+
3168:2
3159:x
3156:(
3153:(
3150:=
3147:)
3140:y
3136:,
3129:x
3125:(
3099:.
3093:2
3084:y
3075:2
3066:x
3063:=
3052:y
3041:2
3032:y
3029:+
3023:2
3014:x
3011:=
3000:x
2981:y
2977:x
2975:(
2971:)
2969:y
2965:x
2963:(
2939:)
2937:y
2933:x
2931:(
2923:)
2921:y
2917:x
2911:)
2909:y
2905:x
2903:(
2882:.
2879:)
2876:)
2863:y
2860:+
2848:x
2845:(
2842:,
2839:)
2826:y
2811:x
2808:(
2805:(
2802:=
2799:)
2792:y
2788:,
2781:x
2777:(
2749:.
2737:y
2734:+
2722:x
2719:=
2708:y
2691:y
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