253:
5358:
5429:
4667:
1386:
22:
1905:
5366:
882:
5249:
4907:
1265:
1582:
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
1920:: 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
568:
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
4684:
3102:
983:
4656:
2752:
3858:
1641:) 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.
4496:
3250:
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).
5548:, 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
3241:
5483:
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
5519:
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
3510:
2881:
6081:-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
3621:
2429:
5348:
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.
4966:
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
5839:
4501:
6023:
5962:
5902:
4102:
2979:
772:-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.
2159:
1505:
5721:
5670:
2641:
5389:-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
5283:-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
3688:
471:
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'
740:
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
5617:
5528:"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
4902:{\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}}.}
3693:
1363:
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
3698:
3342:
2984:
2646:
1593:
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
988:
938:), 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
4333:
4238:
3986:
1260:{\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}}}
2583:
5218:
5145:
5052:
4394:
3109:
1847:. 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
2283:
2224:
3421:
3384:
2761:
1572:
3515:
4381:
2052:
2006:
634:
5568:
1527:
678:
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
2290:
6259:
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.
5781:
4942:
2948:
2618:
638:
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
4681:
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:
2930:
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
1644:
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
3416:
3293:
552:
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
6851:
3884:
5786:
1769:
axis, usually oriented from bottom to top. Young children learning the
Cartesian system, commonly learn the order to read the values before cementing the
5345:
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.
1800:-axis oriented downwards on the computer display. This convention developed in the 1960s (or earlier) from the way that images were originally stored in
3991:
1371:. 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
2059:
686:
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
5575:
1931:, 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,
7025:
6844:
5967:
5906:
5846:
946:
perpendicular to each coordinate axis, and interprets the point where that plane cuts the axis as a number. The
Cartesian coordinates of
901:, 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
721:
of the coordinate system. The coordinates are usually written as two numbers in parentheses, in that order, separated by a comma, as in
311:, whose invention of them in the 17th century revolutionized mathematics by allowing the expression of problems of geometry in terms of
4245:
1843:-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
4153:
1465:
1269:
The coordinates are usually written as three numbers (or algebraic formulas) surrounded by parentheses and separated by commas, as in
5675:
5624:
3901:
2484:
a set of points of the plane, preserving the distances and directions between them, is equivalent to adding a fixed pair of numbers
818:. The quadrants may be named or numbered in various ways, but the quadrant where all coordinates are positive is usually called the
1839:-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
3630:
482:
and his students. These commentators introduced several concepts while trying to clarify the ideas contained in
Descartes's work.
1781:-axis concepts, by starting with 2D mnemonics (for example, 'Walk along the hall then up the stairs' akin to straight across the
7020:
6837:
6748:
6722:
6634:
6603:
6579:
6516:
6493:
6474:
6422:
6197:
6174:
4651:{\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}}.}
6539:
6031:
interpretation of multiplying vectors to obtain another vector that works in all dimensions, however there is a way to use
3097:{\displaystyle {\begin{aligned}x'&=x\cos 2\theta +y\sin 2\theta \\y'&=x\sin 2\theta -y\cos 2\theta .\end{aligned}}}
2915:
596:
the line corresponds to multiplication. Any two
Cartesian coordinate systems on the line can be related to each-other by a
1455:
Since
Cartesian coordinates are unique and non-ambiguous, the points of a Cartesian plane can be identified with pairs of
3301:
4670:
Effect of applying various 2D affine transformation matrices on a unit square (reflections are special cases of scaling)
2513:
2457:
to themselves which preserve distances between points. There are four types of these mappings (also called isometries):
256:
Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is
2747:{\displaystyle {\begin{aligned}x'&=x\cos \theta -y\sin \theta \\y'&=x\sin \theta +y\cos \theta .\end{aligned}}}
6276:
5154:
5081:
4988:
6930:
6560:
6455:
4983:
are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates
6955:
1753:
In mathematical illustrations of two-dimensional Cartesian systems, the first coordinate (traditionally called the
965:
to the plane defined by the other two axes, with the sign determined by the orientation of the corresponding axis.
5570:. In two dimensions, the vector from the origin to the point with Cartesian coordinates (x, y) can be written as:
642:
from one line to the other taking each point on one line to the point on the other line with the same coordinate.
7010:
6970:
838:
545:. Every point on the line has a real-number coordinate, and every real number represents some point on the line.
426:
409:. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including
6945:
6404:
3853:{\displaystyle {\begin{aligned}x'&=xA_{1,1}+yA_{1,1}+b_{1}\\y'&=xA_{2,1}+yA_{2,2}+b_{2}.\end{aligned}}}
4658:
With this trick, the composition of affine transformations is obtained by multiplying the augmented matrices.
6940:
6920:
513:
156:
6035:
to provide such a multiplication. In a two-dimensional cartesian plane, identify the point with coordinates
2496:
to the Cartesian coordinates of every point in the set. That is, if the original coordinates of a point are
6860:
5327:. Placing a somewhat closed right hand on the plane with the thumb pointing up, the fingers point from the
4387:
by simply multiplying the associated transformation matrices. In the general case, it is useful to use the
787:. In a Cartesian plane, one can define canonical representatives of certain geometric figures, such as the
485:
The development of the Cartesian coordinate system would play a fundamental role in the development of the
6925:
6125:
2229:
2170:
930:
A Cartesian coordinate system for a three-dimensional space consists of an ordered triplet of lines (the
585:
509:
25:
Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates:
1390:
464:, who also worked in three dimensions, although Fermat did not publish the discovery. The French cleric
214:, which are the signed distances from the point to three mutually perpendicular planes. More generally,
7005:
6899:
6094:
7015:
6813:
5745:). Similarly, in three dimensions, the vector from the origin to the point with Cartesian coordinates
1548:
6894:
6789:
1899:
1743:
1589:
494:
385:, and provide enlightening geometric interpretations for many other branches of mathematics, such as
3263:
of the plane can be described in a uniform way by using matrices. For this purpose, the coordinates
6935:
5076:
will push the top of a square sideways to form a parallelogram. Horizontal shearing is defined by:
4340:
3863:
2466:
2442:
2011:
1965:
1912:
The axes of a two-dimensional Cartesian system divide the plane into four infinite regions, called
862:
604:
581:
5551:
1927:
Similarly, a three-dimensional Cartesian system defines a division of space into eight regions or
1510:
468:
used constructions similar to Cartesian coordinates well before the time of Descartes and Fermat.
6965:
6950:
6879:
6115:
6099:
4958:
Some affine transformations that are not Euclidean transformations have received specific names.
2164:
1895:
1844:
1824:
974:
876:
505:
207:
6715:
Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions
4491:{\displaystyle {\begin{pmatrix}x'\\y'\\1\end{pmatrix}}=A'{\begin{pmatrix}x\\y\\1\end{pmatrix}},}
1815:-axis added to represent height (positive up). Furthermore, there is a convention to orient the
6665:
6110:
2481:
2462:
2458:
589:
430:
6364:
3236:{\displaystyle (x',y')=((x\cos 2\theta +y\sin 2\theta \,),(x\sin 2\theta -y\cos 2\theta \,)).}
1660:. Each axis is usually named after the coordinate which is measured along it; so one says the
6889:
6883:
5748:
5361:
Fig. 7 – The left-handed orientation is shown on the left, and the right-handed on the right.
4914:
4911:
The Euclidean transformations are the affine transformations such that the 2×2 matrix of the
4674:
4121:
3347:
3260:
2933:
2603:
653:
639:
394:
6717:(corrected 2nd, 3rd print ed.). New York: Springer-Verlag. pp. 9–11 (Table 1.01).
5452:
placed at a right angle to both, the three fingers indicate the relative orientation of the
1939:. The generalization of the quadrant and octant to an arbitrary number of dimensions is the
950:
are those three numbers, in the chosen order. The reverse construction determines the point
729:, and the points on the positive half-axes, one unit away from the origin, have coordinates
6980:
6960:
6595:
6531:
4952:
4384:
3505:{\displaystyle {\begin{pmatrix}x'\\y'\end{pmatrix}}=A{\begin{pmatrix}x\\y\end{pmatrix}}+b,}
3389:
3266:
1735:
557:
192:
183:
6799:
2876:{\displaystyle (x',y')=((x\cos \theta -y\sin \theta \,),(x\sin \theta +y\cos \theta \,)).}
252:
8:
6975:
6629:. Translated by Paul J. Oscamp (Revised ed.). Indianapolis, IN: Hackett Publishing.
5480:-axis. Conversely, if the same is done with the left hand, a left-handed system results.
5357:
2918:
across the second coordinate axis (the y-axis), as if that line were a mirror. Likewise,
2593:
1916:, each bounded by two half-axes. These are often numbered from 1st to 4th and denoted by
1621:. In analytic geometry, unknown or generic coordinates are often denoted by the letters (
465:
406:
5544:
A point in space in a Cartesian coordinate system may also be represented by a position
3616:{\displaystyle A={\begin{pmatrix}A_{1,1}&A_{1,2}\\A_{2,1}&A_{2,2}\end{pmatrix}}}
497:. The two-coordinate description of the plane was later generalized into the concept of
6694:
6658:
6505:
3891:
3869:
1959:
1766:
593:
556:(most commonly zero and one). Other points can then be uniquely assigned to numbers by
479:
5369:
Fig. 8 – The right-handed Cartesian coordinate system indicating the coordinate planes
6796:
6771:
6754:
6744:
6718:
6701:
6677:
6669:
6640:
6630:
6599:
6575:
6556:
6535:
6512:
6489:
6470:
6451:
6418:
6203:
6193:
6146:
4945:
3887:
1460:
741:
565:
560:. Equivalently, one point can be assigned to a specific real number, for instance an
549:
474:
422:
382:
332:
140:
6620:
5428:
744:
contexts, the ordinate axis may be oriented downwards.) The origin is often labeled
453:
308:
6410:
5545:
5238:
4955:
of two affine transformations is obtained by multiplying their augmented matrices.
4388:
2470:
1793:
1758:
461:
390:
196:
144:
136:
103:
97:
62:
6814:
Coordinate Converter – converts between polar, Cartesian and spherical coordinates
4666:
2424:{\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}},}
1886:
are sometimes used to refer to coordinate axes rather than the coordinate values.
6736:
6732:
6653:
6624:
6589:
6550:
6445:
5437:
5382:
5323:
5253:
5234:
4678:
4147:
4137:
4117:
2454:
1851:, unless specifically stated otherwise. All laws of physics and math assume this
1848:
1739:
1530:
1378:
776:
597:
227:
165:
56:
6829:
1385:
6825:
open source JavaScript class for 2D/3D Cartesian coordinate system manipulation
6824:
6689:
6256:
6066:
6032:
5737:
4128:
3895:
804:
675:
504:
Many other coordinate systems have been developed since Descartes, such as the
386:
6818:
6414:
6301:
2433:
which can be obtained by two consecutive applications of Pythagoras' theorem.
21:
6999:
6644:
6104:
5445:
5272:
5228:
5073:
3624:
3296:
1917:
1820:
671:
498:
446:
160:
6681:
6207:
6120:
5724:
5441:
1904:
1819:-axis toward the viewer, biased either to the right or left. If a diagram (
1594:
667:
490:
402:
367:
6526:
Hughes-Hallett, Deborah; McCallum, William G.; Gleason, Andrew M. (2013).
6187:
5525:
5405:-axis form a positively oriented two-dimensional coordinate system in the
1675:
Another common convention for coordinate naming is to use subscripts, as (
1348:-axis, respectively. Then the coordinate planes can be referred to as the
331:
of radius 2, centered at the origin of the plane, may be described as the
2950:
with the x-axis, is equivalent to replacing every point with coordinates
1858:
For 3D diagrams, the names "abscissa" and "ordinate" are rarely used for
1801:
1606:
1456:
1417:-axis is highlighted in green. Thus, the red plane shows the points with
811:
792:
788:
553:
538:
532:
457:
450:
418:
148:
5521:
4120:. If these conditions do not hold, the formula describes a more general
6082:
5834:{\displaystyle \mathbf {r} =x\mathbf {i} +y\mathbf {j} +z\mathbf {k} ,}
5365:
4391:
of the transformation; that is, to rewrite the transformation formula
1584:
861:|, respectively; where | · | denotes the
375:
246:
5321:
A commonly used mnemonic for defining the positive orientation is the
791:(with radius equal to the length unit, and center at the origin), the
6804:
4113:
2597:
2450:
1921:
1747:
1710:
is greater than 3 or unspecified. Some authors prefer the numbering (
410:
363:
231:
6428:
5248:
4140:. The transformation is a rotation around some point if and only if
881:
6018:{\displaystyle \mathbf {k} ={\begin{pmatrix}0\\0\\1\end{pmatrix}}.}
5957:{\displaystyle \mathbf {j} ={\begin{pmatrix}0\\1\\0\end{pmatrix}},}
5897:{\displaystyle \mathbf {i} ={\begin{pmatrix}1\\0\\0\end{pmatrix}},}
1762:
1754:
1645:
1529:
is the set of all real numbers. In the same way, the points in any
709:
703:
486:
398:
371:
324:
319:. Using the Cartesian coordinate system, geometric shapes (such as
316:
46:
4097:{\displaystyle A_{1,1}^{2}+A_{2,1}^{2}=A_{1,2}^{2}+A_{2,2}^{2}=1.}
421:
and many more. They are the most common coordinate system used in
5342:, placing the left hand on the plane with the thumb pointing up.
1941:
414:
327:
involving the coordinates of points of the shape. For example, a
312:
6775:
6758:
6713:
Moon P, Spencer DE (1988). "Rectangular Coordinates (x, y, z)".
6705:
6673:
2154:{\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.}
1500:{\displaystyle \mathbb {R} ^{2}=\mathbb {R} \times \mathbb {R} }
717:, respectively; and the point where the axes meet is called the
6465:
Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1998).
5742:
5716:{\displaystyle \mathbf {j} ={\begin{pmatrix}0\\1\end{pmatrix}}}
5665:{\displaystyle \mathbf {i} ={\begin{pmatrix}1\\0\end{pmatrix}}}
658:
A Cartesian coordinate system in two dimensions (also called a
456:, who published this idea in 1637 while he was resident in the
328:
239:. These coordinates are the signed distances from the point to
6794:
5539:
1807:
For three-dimensional systems, a convention is to portray the
1761:
axis, oriented from left to right. The second coordinate (the
121:
118:
77:
6525:
6325:
5449:
3254:
1538:
885:
A three dimensional Cartesian coordinate system, with origin
694:. The reverse construction allows one to determine the point
320:
181:) of the system. The point where the axes meet is called the
5741:(in some application areas these may also be referred to as
3683:{\displaystyle b={\begin{pmatrix}b_{1}\\b_{2}\end{pmatrix}}}
1605:
The Cartesian coordinates of a point are usually written in
109:
68:
1962:
between two points of the plane with Cartesian coordinates
1649:
748:, and the two coordinates are often denoted by the letters
359:
124:
83:
80:
5488:
the observer, whereas the "middle"-axis is meant to point
2167:. In three-dimensional space, the distance between points
1835:-axis horizontally and vertically, respectively, then the
5612:{\displaystyle \mathbf {r} =x\mathbf {i} +y\mathbf {j} ,}
2620:
is equivalent to replacing every point with coordinates (
1442:(shown as a black sphere) with the Cartesian coordinates
1312:
Standard names for the coordinates in the three axes are
370:
at any point can be computed from this equation by using
5290:
The usual way of orienting the plane, with the positive
6549:
Kent, Alexander J.; Vujakovic, Peter (4 October 2017).
1924:
starting from the upper right ("north-east") quadrant.
1617:. The origin is often labelled with the capital letter
6626:
Discourse on Method, Optics, Geometry, and Meteorology
6588:
Anton, Howard; Bivens, Irl C.; Davis, Stephen (2021).
5984:
5923:
5863:
5692:
5641:
5338:
The other way of orienting the plane is following the
4858:
4822:
4693:
4521:
4457:
4403:
3645:
3530:
3472:
3430:
3310:
779:
with a chosen Cartesian coordinate system is called a
541:
with a chosen Cartesian coordinate system is called a
6192:(3rd ed.). Boston: Addison-Wesley. p. 484.
5970:
5909:
5849:
5789:
5751:
5678:
5627:
5578:
5554:
5157:
5084:
4991:
4917:
4687:
4504:
4397:
4343:
4248:
4156:
3994:
3904:
3872:
3696:
3633:
3518:
3424:
3392:
3350:
3304:
3269:
3112:
2982:
2936:
2764:
2644:
2606:
2516:
2293:
2232:
2173:
2062:
2014:
1968:
1551:
1513:
1468:
986:
607:
127:
106:
86:
65:
6821:– interactive tool to explore coordinates of a point
115:
74:
5335:-axis, in a positively oriented coordinate system.
4242:A reflection or glide reflection is obtained when,
3337:{\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}.}
1948:
1908:
The four quadrants of a Cartesian coordinate system
1324:. The coordinates are often denoted by the letters
405:and more. A familiar example is the concept of the
112:
71:
6693:
6660:Mathematical Handbook for Scientists and Engineers
6657:
6504:
6069:and is identified with the point with coordinates
6017:
5956:
5896:
5833:
5775:
5715:
5664:
5611:
5562:
5508:-axis (in both cases). Hence the red arrow passes
5212:
5139:
5046:
4936:
4901:
4650:
4490:
4375:
4337:Assuming that translations are not used (that is,
4327:
4232:
4096:
3980:
3878:
3852:
3682:
3615:
3504:
3410:
3378:
3336:
3287:
3235:
3096:
2942:
2875:
2746:
2612:
2577:
2423:
2277:
2218:
2153:
2046:
2000:
1796:, however, often use a coordinate system with the
1734:). These notations are especially advantageous in
1566:
1545:real numbers; that is, with the Cartesian product
1521:
1499:
1259:
628:
6859:
6552:The Routledge Handbook of Mapping and Cartography
6464:
6375:
6348:
5735:-axis respectively, generally referred to as the
5060:is greater than 1, the figure becomes larger; if
4328:{\displaystyle A_{1,1}A_{2,2}-A_{2,1}A_{1,2}=-1.}
6997:
6664:(1st ed.). New York: McGraw-Hill. pp.
6587:
6409:. Undergraduate Texts in Mathematics. Springer.
6360:
4233:{\displaystyle A_{1,1}A_{2,2}-A_{2,1}A_{1,2}=1.}
3386:of applying an affine transformation to a point
701:The first and second coordinates are called the
6619:
6396:
5536:the observer and thus seeing a concave corner.
3981:{\displaystyle A_{1,1}A_{1,2}+A_{2,1}A_{2,2}=0}
2902:are the Cartesian coordinates of a point, then
1852:
6688:
3866:are characterized by the fact that the matrix
1953:
1738:: by storing the coordinates of a point as an
220:Cartesian coordinates specify the point in an
199:. The combination of origin and basis forms a
6845:
6731:
6548:
6170:
5222:
1431:, and the yellow plane shows the points with
378:, in a way that can be applied to any curve.
6574:(5th ed.), Pacific Grove: Brooks/Cole,
6107:, which plots four variables rather than two
5448:bent inward at a right angle to it, and the
5306:-axis the "second" axis), is considered the
2436:
1600:
1438:. The three surfaces intersect at the point
1297:, and the unit points on the three axes are
381:Cartesian coordinates are the foundation of
6712:
6326:Hughes-Hallett, McCallum & Gleason 2013
5540:Representing a vector in the standard basis
3295:of a point are commonly represented as the
1587:defined by all the other axes). In such an
6852:
6838:
6765:
6486:The History of Mathematics/An Introduction
6144:
5444:of the right hand is pointed forward, the
3255:General matrix form of the transformations
1336:. The axes may then be referred to as the
957:Alternatively, each coordinate of a point
764:. The axes may then be referred to as the
6502:
6469:. Cambridge: Cambridge University Press.
6443:
6387:
6232:
6189:A history of mathematics: an introduction
5149:Shearing can also be applied vertically:
4951:The augmented matrix that represents the
3223:
3180:
2863:
2826:
1554:
1515:
1493:
1485:
1471:
592:of the line corresponds to addition, and
6768:Mathematische Hilfsmittel des Ingenieurs
6696:The Mathematics of Physics and Chemistry
6651:
6277:"Cartesian orthogonal coordinate system"
6077:the unit vector in the direction of the
5427:
5381:-axes are specified, they determine the
5364:
5356:
5279:-axis through the point marked 0 on the
5247:
5064:is between 0 and 1, it becomes smaller.
4665:
4661:
1903:
1889:
1384:
942:of space, one considers a plane through
880:
810:The two axes divide the plane into four
251:
206:Similarly, the position of any point in
20:
16:Most common coordinate system (geometry)
6488:(7th ed.). New York: McGraw-Hill.
5500:-plane and indicates rotation from the
1945:, and a similar naming system applies.
1652:, the graph coordinates may be denoted
1424:, the blue plane shows the points with
1375:; a convention that is commonly called
972:. These planes divide space into eight
6998:
6741:Methods of Theoretical Physics, Part I
6483:
6220:
5352:
5294:-axis pointing right and the positive
3862:Among the affine transformations, the
934:) that go through a common point (the
159:distances to the point from two fixed
6833:
6795:
6569:
6402:
6336:
6243:
5492:from the observer. The red circle is
2508:, after the translation they will be
478:was translated into Latin in 1649 by
460:. It was independently discovered by
7026:Three-dimensional coordinate systems
6450:. Knopf Doubleday Publishing Group.
6271:
6269:
6267:
6265:
6185:
5243:
4150:, meaning that it is orthogonal and
4127:The transformation is a translation
1870:-coordinate is sometimes called the
1750:can serve to index the coordinates.
1706:-dimensional space, especially when
1450:
584:of the line can be represented by a
564:point corresponding to zero, and an
307:Cartesian coordinates are named for
5267:-axis up to direction. Namely, the
3245:
2278:{\displaystyle (x_{2},y_{2},z_{2})}
2219:{\displaystyle (x_{1},y_{1},z_{1})}
1866:, respectively. When they are, the
1785:-axis then up vertically along the
1293:. Thus, the origin has coordinates
870:
13:
6613:
6528:Calculus: Single and Multivariable
6145:Bix, Robert A.; D'Souza, Harry J.
5432:3D Cartesian coordinate handedness
4106:This is equivalent to saying that
2578:{\displaystyle (x',y')=(x+a,y+b).}
1577:
961:can be taken as the distance from
825:If the coordinates of a point are
725:. Thus the origin has coordinates
287:are the coordinates of the center
14:
7037:
6783:
6262:
2628:) by the point with coordinates (
2163:This is the Cartesian version of
795:(whose diagonal has endpoints at
647:
577:sign chosen based on direction).
5972:
5911:
5851:
5824:
5813:
5802:
5791:
5680:
5629:
5602:
5591:
5580:
5556:
5476:-axis and the middle finger the
5468:system. The thumb indicates the
5302:-axis being the "first" and the
5213:{\displaystyle (x',y')=(x,xs+y)}
5140:{\displaystyle (x',y')=(x+ys,y)}
5047:{\displaystyle (x',y')=(mx,my).}
2600:around the origin by some angle
1949:Cartesian formulae for the plane
1567:{\displaystyle \mathbb {R} ^{n}}
526:
431:geometry-related data processing
335:of all points whose coordinates
102:
61:
6507:Introduction to Electrodynamics
6381:
6376:Brannan, Esplen & Gray 1998
6369:
6354:
6349:Brannan, Esplen & Gray 1998
6342:
6330:
6319:
1609:and separated by commas, as in
427:computer-aided geometric design
6361:Anton, Bivens & Davis 2021
6294:
6249:
6237:
6226:
6214:
6179:
6164:
6138:
5770:
5752:
5207:
5186:
5180:
5158:
5134:
5113:
5107:
5085:
5038:
5020:
5014:
4992:
3405:
3393:
3373:
3351:
3282:
3270:
3227:
3224:
3187:
3181:
3144:
3141:
3135:
3113:
2962:by the point with coordinates
2867:
2864:
2833:
2827:
2796:
2793:
2787:
2765:
2569:
2545:
2539:
2517:
2476:
2407:
2380:
2368:
2341:
2329:
2302:
2272:
2233:
2213:
2174:
2137:
2110:
2098:
2071:
2041:
2015:
1995:
1969:
1811:-plane horizontally, with the
1250:
1223:
1217:
1190:
1184:
1157:
1151:
1124:
1117:
1090:
1084:
1057:
1051:
1024:
1018:
991:
611:
519:
1:
7021:Orthogonal coordinate systems
6861:Orthogonal coordinate systems
6770:. New York: Springer Verlag.
6700:. New York: D. van Nostrand.
5393:-plane is horizontal and the
5314:orientation, also called the
4376:{\displaystyle b_{1}=b_{2}=0}
3690:is a column matrix. That is,
2885:
2047:{\displaystyle (x_{2},y_{2})}
2001:{\displaystyle (x_{1},y_{1})}
1855:, which ensures consistency.
1393:of the Cartesian coordinates
954:given its three coordinates.
660:rectangular coordinate system
629:{\displaystyle x\mapsto ax+b}
516:for three-dimensional space.
245:mutually perpendicular fixed
6503:Griffiths, David J. (1999).
6397:General and cited references
6131:
5563:{\displaystyle \mathbf {r} }
5472:-axis, the index finger the
2453:) mappings of points of the
1522:{\displaystyle \mathbb {R} }
968:Each pair of axes defines a
664:orthogonal coordinate system
7:
6790:Cartesian Coordinate System
6281:Encyclopedia of Mathematics
6126:Spherical coordinate system
6088:
5298:-axis pointing up (and the
5067:
3890:; that is, its columns are
2914:are the coordinates of its
2587:
1954:Distance between two points
1765:) is then measured along a
586:function of a real variable
51:Cartesian coordinate system
10:
7042:
6095:Cartesian coordinate robot
5436:The name derives from the
5287:) of the Cartesian plane.
5232:
5226:
5223:Orientation and handedness
4961:
1893:
1413:-axis is vertical and the
874:
682:, a line is drawn through
651:
530:
436:
210:can be specified by three
6908:
6867:
6766:Sauer R, Szabó I (1967).
6743:. New York: McGraw-Hill.
6484:Burton, David M. (2011).
6444:Berlinski, David (2011).
6415:10.1007/978-3-319-11080-6
6406:Linear Algebra Done Right
6378:, Appendix 2, pp. 377–382
6171:Kent & Vujakovic 2017
5397:-axis points up (and the
5271:-axis is necessarily the
4383:) transformations can be
3864:Euclidean transformations
2443:Euclidean transformations
2437:Euclidean transformations
1900:Quadrant (plane geometry)
1601:Notations and conventions
1590:oblique coordinate system
495:Gottfried Wilhelm Leibniz
191:as coordinates. The axes
6570:Smart, James R. (1998),
6186:Katz, Victor J. (2009).
6047:with the complex number
5727:in the direction of the
5409:-plane if observed from
3418:is given by the formula
582:geometric transformation
37:in blue, and the origin
6800:"Cartesian Coordinates"
6591:Calculus: Multivariable
6403:Axler, Sheldon (2015).
6302:"Cartesian coordinates"
6151:Encyclopædia Britannica
6116:Polar coordinate system
6100:Horizontal and vertical
5776:{\displaystyle (x,y,z)}
5259:Fixing or choosing the
5074:shearing transformation
4937:{\displaystyle A_{i,j}}
3379:{\displaystyle (x',y')}
2943:{\displaystyle \theta }
2613:{\displaystyle \theta }
1896:Octant (solid geometry)
1537:be identified with the
877:Three-dimensional space
698:given its coordinates.
674:lines (axes), a single
514:cylindrical coordinates
508:for the plane, and the
208:three-dimensional space
163:oriented lines, called
7011:Elementary mathematics
6819:Coordinates of a point
6447:A Tour of the Calculus
6111:Orthogonal coordinates
6019:
5958:
5898:
5835:
5777:
5717:
5666:
5613:
5564:
5433:
5370:
5362:
5256:
5214:
5141:
5048:
4938:
4903:
4675:Affine transformations
4671:
4652:
4492:
4377:
4329:
4234:
4098:
3982:
3880:
3854:
3684:
3617:
3506:
3412:
3380:
3338:
3289:
3261:affine transformations
3237:
3098:
2944:
2877:
2748:
2614:
2579:
2425:
2279:
2220:
2155:
2048:
2002:
1909:
1825:2D perspective drawing
1792:Computer graphics and
1757:) is measured along a
1568:
1523:
1501:
1447:
1261:
927:
630:
600:(function of the form
323:) can be described by
304:
147:uniquely by a pair of
42:
6596:John Wiley & Sons
6532:John Wiley & Sons
6020:
5959:
5899:
5836:
5778:
5718:
5667:
5614:
5565:
5431:
5368:
5360:
5263:-axis determines the
5251:
5215:
5142:
5049:
4939:
4904:
4669:
4662:Affine transformation
4653:
4493:
4378:
4330:
4235:
4122:affine transformation
4099:
3983:
3898:one, or, explicitly,
3881:
3855:
3685:
3618:
3507:
3413:
3411:{\displaystyle (x,y)}
3381:
3339:
3290:
3288:{\displaystyle (x,y)}
3238:
3099:
2945:
2878:
2749:
2615:
2580:
2426:
2280:
2221:
2156:
2049:
2003:
1907:
1890:Quadrants and octants
1629:) in the plane, and (
1569:
1524:
1502:
1388:
1262:
884:
875:Further information:
688:Cartesian coordinates
654:Two-dimensional space
652:Further information:
631:
445:refers to the French
395:differential geometry
347:satisfy the equation
255:
212:Cartesian coordinates
24:
6956:Elliptic cylindrical
6692:, Murphy GM (1956).
5968:
5907:
5847:
5787:
5749:
5676:
5625:
5576:
5552:
5155:
5082:
4989:
4915:
4685:
4502:
4395:
4341:
4246:
4154:
3992:
3902:
3870:
3694:
3631:
3516:
3422:
3390:
3348:
3302:
3267:
3110:
2980:
2934:
2762:
2642:
2604:
2514:
2291:
2230:
2171:
2165:Pythagoras's theorem
2060:
2012:
1966:
1736:computer programming
1549:
1511:
1466:
1459:; that is, with the
984:
605:
558:linear interpolation
143:that specifies each
6971:Bipolar cylindrical
6147:"Analytic geometry"
5783:can be written as:
5353:In three dimensions
4087:
4063:
4039:
4015:
1391:coordinate surfaces
978:. The octants are:
845:-axis and from the
666:) is defined by an
407:graph of a function
6946:Prolate spheroidal
6797:Weisstein, Eric W.
6015:
6006:
5954:
5945:
5894:
5885:
5831:
5773:
5713:
5707:
5662:
5656:
5609:
5560:
5532:-axis as pointing
5496:to the horizontal
5434:
5417:-plane) is called
5371:
5363:
5257:
5210:
5137:
5044:
4934:
4899:
4890:
4844:
4811:
4672:
4648:
4639:
4488:
4479:
4435:
4373:
4325:
4230:
4094:
4067:
4043:
4019:
3995:
3978:
3892:orthogonal vectors
3876:
3850:
3848:
3680:
3674:
3613:
3607:
3502:
3487:
3455:
3408:
3376:
3334:
3325:
3285:
3233:
3094:
3092:
2940:
2873:
2744:
2742:
2610:
2575:
2421:
2275:
2216:
2151:
2044:
1998:
1960:Euclidean distance
1910:
1702:coordinates in an
1564:
1519:
1497:
1448:
1257:
1255:
928:
626:
550:degrees of freedom
480:Frans van Schooten
305:
43:
7006:Analytic geometry
6993:
6992:
6941:Oblate spheroidal
6909:Three dimensional
6750:978-0-07-043316-8
6724:978-0-387-18430-2
6636:978-0-87220-567-3
6605:978-1-119-77798-4
6581:978-0-534-35188-5
6572:Modern Geometries
6518:978-0-13-805326-0
6511:. Prentice Hall.
6495:978-0-07-338315-6
6476:978-0-521-59787-6
6424:978-3-319-11079-0
6255:Consider the two
6199:978-0-321-38700-4
5244:In two dimensions
3879:{\displaystyle A}
2471:glide reflections
2447:Euclidean motions
2416:
2146:
1922:counter-clockwise
1461:Cartesian product
1451:Higher dimensions
855:| and |
742:computer graphics
506:polar coordinates
423:computer graphics
383:analytic geometry
141:coordinate system
7033:
6854:
6847:
6840:
6831:
6830:
6810:
6809:
6779:
6762:
6728:
6709:
6699:
6685:
6663:
6648:
6609:
6584:
6566:
6545:
6541:978-0470-88861-2
6530:(6th ed.).
6522:
6510:
6499:
6480:
6461:
6440:
6438:
6436:
6427:. Archived from
6390:
6385:
6379:
6373:
6367:
6358:
6352:
6346:
6340:
6334:
6328:
6323:
6317:
6316:
6314:
6312:
6298:
6292:
6291:
6289:
6287:
6273:
6260:
6253:
6247:
6241:
6235:
6230:
6224:
6218:
6212:
6211:
6183:
6177:
6168:
6162:
6161:
6159:
6157:
6142:
6072:
6060:
6046:
6024:
6022:
6021:
6016:
6011:
6010:
5975:
5963:
5961:
5960:
5955:
5950:
5949:
5914:
5903:
5901:
5900:
5895:
5890:
5889:
5854:
5840:
5838:
5837:
5832:
5827:
5816:
5805:
5794:
5782:
5780:
5779:
5774:
5722:
5720:
5719:
5714:
5712:
5711:
5683:
5671:
5669:
5668:
5663:
5661:
5660:
5632:
5618:
5616:
5615:
5610:
5605:
5594:
5583:
5569:
5567:
5566:
5561:
5559:
5385:along which the
5239:Axes conventions
5219:
5217:
5216:
5211:
5179:
5168:
5146:
5144:
5143:
5138:
5106:
5095:
5053:
5051:
5050:
5045:
5013:
5002:
4982:
4943:
4941:
4940:
4935:
4933:
4932:
4908:
4906:
4905:
4900:
4895:
4894:
4880:
4868:
4849:
4848:
4816:
4815:
4791:
4790:
4779:
4778:
4761:
4760:
4741:
4740:
4729:
4728:
4711:
4710:
4657:
4655:
4654:
4649:
4644:
4643:
4619:
4618:
4607:
4606:
4589:
4588:
4569:
4568:
4557:
4556:
4539:
4538:
4512:
4497:
4495:
4494:
4489:
4484:
4483:
4451:
4440:
4439:
4425:
4413:
4389:augmented matrix
4382:
4380:
4379:
4374:
4366:
4365:
4353:
4352:
4334:
4332:
4331:
4326:
4315:
4314:
4299:
4298:
4280:
4279:
4264:
4263:
4239:
4237:
4236:
4231:
4223:
4222:
4207:
4206:
4188:
4187:
4172:
4171:
4145:
4135:
4111:
4103:
4101:
4100:
4095:
4086:
4081:
4062:
4057:
4038:
4033:
4014:
4009:
3987:
3985:
3984:
3979:
3971:
3970:
3955:
3954:
3936:
3935:
3920:
3919:
3885:
3883:
3882:
3877:
3859:
3857:
3856:
3851:
3849:
3842:
3841:
3829:
3828:
3807:
3806:
3781:
3769:
3768:
3756:
3755:
3734:
3733:
3708:
3689:
3687:
3686:
3681:
3679:
3678:
3671:
3670:
3657:
3656:
3622:
3620:
3619:
3614:
3612:
3611:
3604:
3603:
3586:
3585:
3566:
3565:
3548:
3547:
3511:
3509:
3508:
3503:
3492:
3491:
3460:
3459:
3452:
3440:
3417:
3415:
3414:
3409:
3385:
3383:
3382:
3377:
3372:
3361:
3343:
3341:
3340:
3335:
3330:
3329:
3294:
3292:
3291:
3286:
3246:Glide reflection
3242:
3240:
3239:
3234:
3134:
3123:
3103:
3101:
3100:
3095:
3093:
3046:
2994:
2973:
2961:
2949:
2947:
2946:
2941:
2929:
2913:
2901:
2882:
2880:
2879:
2874:
2786:
2775:
2753:
2751:
2750:
2745:
2743:
2702:
2656:
2619:
2617:
2616:
2611:
2598:counterclockwise
2584:
2582:
2581:
2576:
2538:
2527:
2507:
2495:
2430:
2428:
2427:
2422:
2417:
2415:
2414:
2405:
2404:
2392:
2391:
2376:
2375:
2366:
2365:
2353:
2352:
2337:
2336:
2327:
2326:
2314:
2313:
2301:
2284:
2282:
2281:
2276:
2271:
2270:
2258:
2257:
2245:
2244:
2225:
2223:
2222:
2217:
2212:
2211:
2199:
2198:
2186:
2185:
2160:
2158:
2157:
2152:
2147:
2145:
2144:
2135:
2134:
2122:
2121:
2106:
2105:
2096:
2095:
2083:
2082:
2070:
2053:
2051:
2050:
2045:
2040:
2039:
2027:
2026:
2007:
2005:
2004:
1999:
1994:
1993:
1981:
1980:
1938:
1934:
1853:right-handedness
1794:image processing
1616:
1612:
1573:
1571:
1570:
1565:
1563:
1562:
1557:
1528:
1526:
1525:
1520:
1518:
1506:
1504:
1503:
1498:
1496:
1488:
1480:
1479:
1474:
1445:
1437:
1430:
1423:
1408:
1374:
1308:
1304:
1300:
1296:
1292:
1272:
1266:
1264:
1263:
1258:
1256:
1221:
1188:
1155:
1088:
1055:
1022:
970:coordinate plane
925:
921:
914:
907:
871:Three dimensions
860:
854:
849:-axis are |
836:
802:
798:
785:
784:
736:
732:
728:
724:
637:
635:
633:
632:
627:
576:
572:
462:Pierre de Fermat
391:complex analysis
357:
346:
340:
298:
278:
244:
238:
225:
219:
197:orthogonal basis
190:
166:coordinate lines
155:, which are the
134:
133:
130:
129:
126:
123:
120:
117:
114:
111:
108:
101:
93:
92:
89:
88:
85:
82:
79:
76:
73:
70:
67:
60:
40:
36:
32:
28:
7041:
7040:
7036:
7035:
7034:
7032:
7031:
7030:
6996:
6995:
6994:
6989:
6904:
6868:Two dimensional
6863:
6858:
6786:
6751:
6725:
6637:
6621:Descartes, René
6616:
6614:Further reading
6606:
6598:. p. 657.
6582:
6563:
6542:
6519:
6496:
6477:
6458:
6434:
6432:
6425:
6399:
6394:
6393:
6386:
6382:
6374:
6370:
6359:
6355:
6347:
6343:
6335:
6331:
6324:
6320:
6310:
6308:
6300:
6299:
6295:
6285:
6283:
6275:
6274:
6263:
6254:
6250:
6242:
6238:
6231:
6227:
6219:
6215:
6200:
6184:
6180:
6169:
6165:
6155:
6153:
6143:
6139:
6134:
6091:
6070:
6048:
6036:
6033:complex numbers
6005:
6004:
5998:
5997:
5991:
5990:
5980:
5979:
5971:
5969:
5966:
5965:
5944:
5943:
5937:
5936:
5930:
5929:
5919:
5918:
5910:
5908:
5905:
5904:
5884:
5883:
5877:
5876:
5870:
5869:
5859:
5858:
5850:
5848:
5845:
5844:
5823:
5812:
5801:
5790:
5788:
5785:
5784:
5750:
5747:
5746:
5706:
5705:
5699:
5698:
5688:
5687:
5679:
5677:
5674:
5673:
5655:
5654:
5648:
5647:
5637:
5636:
5628:
5626:
5623:
5622:
5601:
5590:
5579:
5577:
5574:
5573:
5555:
5553:
5550:
5549:
5542:
5438:right-hand rule
5355:
5324:right-hand rule
5254:right-hand rule
5246:
5241:
5235:Right-hand rule
5231:
5225:
5172:
5161:
5156:
5153:
5152:
5099:
5088:
5083:
5080:
5079:
5070:
5006:
4995:
4990:
4987:
4986:
4972:
4964:
4922:
4918:
4916:
4913:
4912:
4889:
4888:
4882:
4881:
4873:
4870:
4869:
4861:
4854:
4853:
4843:
4842:
4836:
4835:
4829:
4828:
4818:
4817:
4810:
4809:
4804:
4799:
4793:
4792:
4786:
4782:
4780:
4768:
4764:
4762:
4750:
4746:
4743:
4742:
4736:
4732:
4730:
4718:
4714:
4712:
4700:
4696:
4689:
4688:
4686:
4683:
4682:
4679:Euclidean plane
4664:
4638:
4637:
4632:
4627:
4621:
4620:
4614:
4610:
4608:
4596:
4592:
4590:
4578:
4574:
4571:
4570:
4564:
4560:
4558:
4546:
4542:
4540:
4528:
4524:
4517:
4516:
4505:
4503:
4500:
4499:
4478:
4477:
4471:
4470:
4464:
4463:
4453:
4452:
4444:
4434:
4433:
4427:
4426:
4418:
4415:
4414:
4406:
4399:
4398:
4396:
4393:
4392:
4361:
4357:
4348:
4344:
4342:
4339:
4338:
4304:
4300:
4288:
4284:
4269:
4265:
4253:
4249:
4247:
4244:
4243:
4212:
4208:
4196:
4192:
4177:
4173:
4161:
4157:
4155:
4152:
4151:
4148:rotation matrix
4141:
4138:identity matrix
4131:
4118:identity matrix
4107:
4082:
4071:
4058:
4047:
4034:
4023:
4010:
3999:
3993:
3990:
3989:
3960:
3956:
3944:
3940:
3925:
3921:
3909:
3905:
3903:
3900:
3899:
3871:
3868:
3867:
3847:
3846:
3837:
3833:
3818:
3814:
3796:
3792:
3782:
3774:
3771:
3770:
3764:
3760:
3745:
3741:
3723:
3719:
3709:
3701:
3697:
3695:
3692:
3691:
3673:
3672:
3666:
3662:
3659:
3658:
3652:
3648:
3641:
3640:
3632:
3629:
3628:
3606:
3605:
3593:
3589:
3587:
3575:
3571:
3568:
3567:
3555:
3551:
3549:
3537:
3533:
3526:
3525:
3517:
3514:
3513:
3486:
3485:
3479:
3478:
3468:
3467:
3454:
3453:
3445:
3442:
3441:
3433:
3426:
3425:
3423:
3420:
3419:
3391:
3388:
3387:
3365:
3354:
3349:
3346:
3345:
3324:
3323:
3317:
3316:
3306:
3305:
3303:
3300:
3299:
3268:
3265:
3264:
3257:
3248:
3127:
3116:
3111:
3108:
3107:
3091:
3090:
3047:
3039:
3036:
3035:
2995:
2987:
2983:
2981:
2978:
2977:
2963:
2951:
2935:
2932:
2931:
2919:
2903:
2891:
2888:
2779:
2768:
2763:
2760:
2759:
2741:
2740:
2703:
2695:
2692:
2691:
2657:
2649:
2645:
2643:
2640:
2639:
2605:
2602:
2601:
2590:
2531:
2520:
2515:
2512:
2511:
2497:
2485:
2479:
2455:Euclidean plane
2439:
2410:
2406:
2400:
2396:
2387:
2383:
2371:
2367:
2361:
2357:
2348:
2344:
2332:
2328:
2322:
2318:
2309:
2305:
2300:
2292:
2289:
2288:
2266:
2262:
2253:
2249:
2240:
2236:
2231:
2228:
2227:
2207:
2203:
2194:
2190:
2181:
2177:
2172:
2169:
2168:
2140:
2136:
2130:
2126:
2117:
2113:
2101:
2097:
2091:
2087:
2078:
2074:
2069:
2061:
2058:
2057:
2035:
2031:
2022:
2018:
2013:
2010:
2009:
1989:
1985:
1976:
1972:
1967:
1964:
1963:
1956:
1951:
1936:
1932:
1902:
1894:Main articles:
1892:
1849:right-hand rule
1802:display buffers
1742:, instead of a
1733:
1723:
1716:
1697:
1688:
1681:
1614:
1610:
1603:
1580:
1578:Generalizations
1558:
1553:
1552:
1550:
1547:
1546:
1531:Euclidean space
1514:
1512:
1509:
1508:
1492:
1484:
1475:
1470:
1469:
1467:
1464:
1463:
1453:
1443:
1432:
1425:
1418:
1394:
1379:right-hand rule
1372:
1306:
1302:
1298:
1294:
1274:
1270:
1254:
1253:
1220:
1187:
1154:
1121:
1120:
1087:
1054:
1021:
987:
985:
982:
981:
923:
916:
909:
902:
889:and axis lines
879:
873:
856:
850:
826:
800:
796:
783:Cartesian plane
782:
781:
777:Euclidean plane
734:
730:
726:
722:
656:
650:
606:
603:
602:
601:
598:linear function
574:
570:
535:
529:
522:
439:
397:, multivariate
348:
342:
336:
288:
257:
240:
234:
228:Euclidean space
221:
215:
201:Cartesian frame
188:
171:coordinate axes
105:
96:
95:
64:
55:
54:
38:
34:
30:
26:
17:
12:
11:
5:
7039:
7029:
7028:
7023:
7018:
7016:René Descartes
7013:
7008:
6991:
6990:
6988:
6987:
6985:
6983:
6978:
6973:
6968:
6963:
6958:
6953:
6948:
6943:
6938:
6933:
6928:
6923:
6918:
6912:
6910:
6906:
6905:
6903:
6902:
6897:
6892:
6887:
6877:
6871:
6869:
6865:
6864:
6857:
6856:
6849:
6842:
6834:
6828:
6827:
6822:
6816:
6811:
6792:
6785:
6784:External links
6782:
6781:
6780:
6763:
6749:
6729:
6723:
6710:
6686:
6649:
6635:
6615:
6612:
6611:
6610:
6604:
6585:
6580:
6567:
6561:
6546:
6540:
6523:
6517:
6500:
6494:
6481:
6475:
6462:
6456:
6441:
6431:on 27 May 2022
6423:
6398:
6395:
6392:
6391:
6388:Griffiths 1999
6380:
6368:
6353:
6341:
6329:
6318:
6306:planetmath.org
6293:
6261:
6248:
6236:
6233:Berlinski 2011
6225:
6213:
6198:
6178:
6163:
6136:
6135:
6133:
6130:
6129:
6128:
6123:
6118:
6113:
6108:
6102:
6097:
6090:
6087:
6067:imaginary unit
6014:
6009:
6003:
6000:
5999:
5996:
5993:
5992:
5989:
5986:
5985:
5983:
5978:
5974:
5953:
5948:
5942:
5939:
5938:
5935:
5932:
5931:
5928:
5925:
5924:
5922:
5917:
5913:
5893:
5888:
5882:
5879:
5878:
5875:
5872:
5871:
5868:
5865:
5864:
5862:
5857:
5853:
5830:
5826:
5822:
5819:
5815:
5811:
5808:
5804:
5800:
5797:
5793:
5772:
5769:
5766:
5763:
5760:
5757:
5754:
5738:standard basis
5710:
5704:
5701:
5700:
5697:
5694:
5693:
5691:
5686:
5682:
5659:
5653:
5650:
5649:
5646:
5643:
5642:
5640:
5635:
5631:
5608:
5604:
5600:
5597:
5593:
5589:
5586:
5582:
5558:
5541:
5538:
5354:
5351:
5340:left-hand rule
5245:
5242:
5227:Main article:
5224:
5221:
5209:
5206:
5203:
5200:
5197:
5194:
5191:
5188:
5185:
5182:
5178:
5175:
5171:
5167:
5164:
5160:
5136:
5133:
5130:
5127:
5124:
5121:
5118:
5115:
5112:
5109:
5105:
5102:
5098:
5094:
5091:
5087:
5069:
5066:
5043:
5040:
5037:
5034:
5031:
5028:
5025:
5022:
5019:
5016:
5012:
5009:
5005:
5001:
4998:
4994:
4963:
4960:
4931:
4928:
4925:
4921:
4898:
4893:
4887:
4884:
4883:
4879:
4876:
4872:
4871:
4867:
4864:
4860:
4859:
4857:
4852:
4847:
4841:
4838:
4837:
4834:
4831:
4830:
4827:
4824:
4823:
4821:
4814:
4808:
4805:
4803:
4800:
4798:
4795:
4794:
4789:
4785:
4781:
4777:
4774:
4771:
4767:
4763:
4759:
4756:
4753:
4749:
4745:
4744:
4739:
4735:
4731:
4727:
4724:
4721:
4717:
4713:
4709:
4706:
4703:
4699:
4695:
4694:
4692:
4663:
4660:
4647:
4642:
4636:
4633:
4631:
4628:
4626:
4623:
4622:
4617:
4613:
4609:
4605:
4602:
4599:
4595:
4591:
4587:
4584:
4581:
4577:
4573:
4572:
4567:
4563:
4559:
4555:
4552:
4549:
4545:
4541:
4537:
4534:
4531:
4527:
4523:
4522:
4520:
4515:
4511:
4508:
4487:
4482:
4476:
4473:
4472:
4469:
4466:
4465:
4462:
4459:
4458:
4456:
4450:
4447:
4443:
4438:
4432:
4429:
4428:
4424:
4421:
4417:
4416:
4412:
4409:
4405:
4404:
4402:
4372:
4369:
4364:
4360:
4356:
4351:
4347:
4324:
4321:
4318:
4313:
4310:
4307:
4303:
4297:
4294:
4291:
4287:
4283:
4278:
4275:
4272:
4268:
4262:
4259:
4256:
4252:
4229:
4226:
4221:
4218:
4215:
4211:
4205:
4202:
4199:
4195:
4191:
4186:
4183:
4180:
4176:
4170:
4167:
4164:
4160:
4129:if and only if
4093:
4090:
4085:
4080:
4077:
4074:
4070:
4066:
4061:
4056:
4053:
4050:
4046:
4042:
4037:
4032:
4029:
4026:
4022:
4018:
4013:
4008:
4005:
4002:
3998:
3977:
3974:
3969:
3966:
3963:
3959:
3953:
3950:
3947:
3943:
3939:
3934:
3931:
3928:
3924:
3918:
3915:
3912:
3908:
3896:Euclidean norm
3875:
3845:
3840:
3836:
3832:
3827:
3824:
3821:
3817:
3813:
3810:
3805:
3802:
3799:
3795:
3791:
3788:
3785:
3783:
3780:
3777:
3773:
3772:
3767:
3763:
3759:
3754:
3751:
3748:
3744:
3740:
3737:
3732:
3729:
3726:
3722:
3718:
3715:
3712:
3710:
3707:
3704:
3700:
3699:
3677:
3669:
3665:
3661:
3660:
3655:
3651:
3647:
3646:
3644:
3639:
3636:
3610:
3602:
3599:
3596:
3592:
3588:
3584:
3581:
3578:
3574:
3570:
3569:
3564:
3561:
3558:
3554:
3550:
3546:
3543:
3540:
3536:
3532:
3531:
3529:
3524:
3521:
3501:
3498:
3495:
3490:
3484:
3481:
3480:
3477:
3474:
3473:
3471:
3466:
3463:
3458:
3451:
3448:
3444:
3443:
3439:
3436:
3432:
3431:
3429:
3407:
3404:
3401:
3398:
3395:
3375:
3371:
3368:
3364:
3360:
3357:
3353:
3333:
3328:
3322:
3319:
3318:
3315:
3312:
3311:
3309:
3284:
3281:
3278:
3275:
3272:
3256:
3253:
3247:
3244:
3232:
3229:
3226:
3222:
3219:
3216:
3213:
3210:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3186:
3183:
3179:
3176:
3173:
3170:
3167:
3164:
3161:
3158:
3155:
3152:
3149:
3146:
3143:
3140:
3137:
3133:
3130:
3126:
3122:
3119:
3115:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3059:
3056:
3053:
3050:
3048:
3045:
3042:
3038:
3037:
3034:
3031:
3028:
3025:
3022:
3019:
3016:
3013:
3010:
3007:
3004:
3001:
2998:
2996:
2993:
2990:
2986:
2985:
2939:
2887:
2884:
2872:
2869:
2866:
2862:
2859:
2856:
2853:
2850:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2825:
2822:
2819:
2816:
2813:
2810:
2807:
2804:
2801:
2798:
2795:
2792:
2789:
2785:
2782:
2778:
2774:
2771:
2767:
2739:
2736:
2733:
2730:
2727:
2724:
2721:
2718:
2715:
2712:
2709:
2706:
2704:
2701:
2698:
2694:
2693:
2690:
2687:
2684:
2681:
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2658:
2655:
2652:
2648:
2647:
2609:
2589:
2586:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2547:
2544:
2541:
2537:
2534:
2530:
2526:
2523:
2519:
2478:
2475:
2438:
2435:
2420:
2413:
2409:
2403:
2399:
2395:
2390:
2386:
2382:
2379:
2374:
2370:
2364:
2360:
2356:
2351:
2347:
2343:
2340:
2335:
2331:
2325:
2321:
2317:
2312:
2308:
2304:
2299:
2296:
2274:
2269:
2265:
2261:
2256:
2252:
2248:
2243:
2239:
2235:
2215:
2210:
2206:
2202:
2197:
2193:
2189:
2184:
2180:
2176:
2150:
2143:
2139:
2133:
2129:
2125:
2120:
2116:
2112:
2109:
2104:
2100:
2094:
2090:
2086:
2081:
2077:
2073:
2068:
2065:
2043:
2038:
2034:
2030:
2025:
2021:
2017:
1997:
1992:
1988:
1984:
1979:
1975:
1971:
1955:
1952:
1950:
1947:
1918:Roman numerals
1891:
1888:
1728:
1721:
1714:
1693:
1686:
1679:
1602:
1599:
1579:
1576:
1561:
1556:
1517:
1495:
1491:
1487:
1483:
1478:
1473:
1452:
1449:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
999:
996:
993:
990:
989:
872:
869:
863:absolute value
820:first quadrant
805:unit hyperbola
676:unit of length
649:
648:Two dimensions
646:
625:
622:
619:
616:
613:
610:
588:, for example
548:There are two
531:Main article:
528:
525:
521:
518:
454:René Descartes
441:The adjective
438:
435:
387:linear algebra
309:René Descartes
303:is the radius.
15:
9:
6:
4:
3:
2:
7038:
7027:
7024:
7022:
7019:
7017:
7014:
7012:
7009:
7007:
7004:
7003:
7001:
6986:
6984:
6982:
6979:
6977:
6974:
6972:
6969:
6967:
6964:
6962:
6959:
6957:
6954:
6952:
6949:
6947:
6944:
6942:
6939:
6937:
6934:
6932:
6929:
6927:
6924:
6922:
6919:
6917:
6914:
6913:
6911:
6907:
6901:
6898:
6896:
6893:
6891:
6888:
6885:
6881:
6878:
6876:
6873:
6872:
6870:
6866:
6862:
6855:
6850:
6848:
6843:
6841:
6836:
6835:
6832:
6826:
6823:
6820:
6817:
6815:
6812:
6807:
6806:
6801:
6798:
6793:
6791:
6788:
6787:
6777:
6773:
6769:
6764:
6760:
6756:
6752:
6746:
6742:
6738:
6734:
6730:
6726:
6720:
6716:
6711:
6707:
6703:
6698:
6697:
6691:
6687:
6683:
6679:
6675:
6671:
6667:
6662:
6661:
6655:
6650:
6646:
6642:
6638:
6632:
6628:
6627:
6622:
6618:
6617:
6607:
6601:
6597:
6593:
6592:
6586:
6583:
6577:
6573:
6568:
6564:
6562:9781317568216
6558:
6555:. Routledge.
6554:
6553:
6547:
6543:
6537:
6533:
6529:
6524:
6520:
6514:
6509:
6508:
6501:
6497:
6491:
6487:
6482:
6478:
6472:
6468:
6463:
6459:
6457:9780307789730
6453:
6449:
6448:
6442:
6430:
6426:
6420:
6416:
6412:
6408:
6407:
6401:
6400:
6389:
6384:
6377:
6372:
6366:
6362:
6357:
6350:
6345:
6338:
6333:
6327:
6322:
6307:
6303:
6297:
6282:
6278:
6272:
6270:
6268:
6266:
6258:
6252:
6245:
6240:
6234:
6229:
6222:
6217:
6209:
6205:
6201:
6195:
6191:
6190:
6182:
6176:
6172:
6167:
6152:
6148:
6141:
6137:
6127:
6124:
6122:
6119:
6117:
6114:
6112:
6109:
6106:
6105:Jones diagram
6103:
6101:
6098:
6096:
6093:
6092:
6086:
6084:
6080:
6076:
6068:
6064:
6059:
6055:
6051:
6044:
6040:
6034:
6030:
6025:
6012:
6007:
6001:
5994:
5987:
5981:
5976:
5951:
5946:
5940:
5933:
5926:
5920:
5915:
5891:
5886:
5880:
5873:
5866:
5860:
5855:
5841:
5828:
5820:
5817:
5809:
5806:
5798:
5795:
5767:
5764:
5761:
5758:
5755:
5744:
5740:
5739:
5734:
5730:
5726:
5708:
5702:
5695:
5689:
5684:
5657:
5651:
5644:
5638:
5633:
5619:
5606:
5598:
5595:
5587:
5584:
5571:
5547:
5537:
5535:
5531:
5527:
5523:
5517:
5515:
5511:
5507:
5504:-axis to the
5503:
5499:
5495:
5491:
5487:
5481:
5479:
5475:
5471:
5467:
5463:
5459:
5455:
5451:
5447:
5446:middle finger
5443:
5439:
5430:
5426:
5424:
5420:
5416:
5412:
5408:
5404:
5400:
5396:
5392:
5388:
5384:
5380:
5376:
5367:
5359:
5350:
5346:
5343:
5341:
5336:
5334:
5331:-axis to the
5330:
5326:
5325:
5319:
5318:orientation.
5317:
5313:
5309:
5305:
5301:
5297:
5293:
5288:
5286:
5282:
5278:
5274:
5273:perpendicular
5270:
5266:
5262:
5255:
5250:
5240:
5236:
5230:
5229:Orientability
5220:
5204:
5201:
5198:
5195:
5192:
5189:
5183:
5176:
5173:
5169:
5165:
5162:
5150:
5147:
5131:
5128:
5125:
5122:
5119:
5116:
5110:
5103:
5100:
5096:
5092:
5089:
5077:
5075:
5065:
5063:
5059:
5054:
5041:
5035:
5032:
5029:
5026:
5023:
5017:
5010:
5007:
5003:
4999:
4996:
4984:
4980:
4976:
4970:
4959:
4956:
4954:
4949:
4947:
4929:
4926:
4923:
4919:
4909:
4896:
4891:
4885:
4877:
4874:
4865:
4862:
4855:
4850:
4845:
4839:
4832:
4825:
4819:
4812:
4806:
4801:
4796:
4787:
4783:
4775:
4772:
4769:
4765:
4757:
4754:
4751:
4747:
4737:
4733:
4725:
4722:
4719:
4715:
4707:
4704:
4701:
4697:
4690:
4680:
4676:
4668:
4659:
4645:
4640:
4634:
4629:
4624:
4615:
4611:
4603:
4600:
4597:
4593:
4585:
4582:
4579:
4575:
4565:
4561:
4553:
4550:
4547:
4543:
4535:
4532:
4529:
4525:
4518:
4513:
4509:
4506:
4485:
4480:
4474:
4467:
4460:
4454:
4448:
4445:
4441:
4436:
4430:
4422:
4419:
4410:
4407:
4400:
4390:
4386:
4370:
4367:
4362:
4358:
4354:
4349:
4345:
4335:
4322:
4319:
4316:
4311:
4308:
4305:
4301:
4295:
4292:
4289:
4285:
4281:
4276:
4273:
4270:
4266:
4260:
4257:
4254:
4250:
4240:
4227:
4224:
4219:
4216:
4213:
4209:
4203:
4200:
4197:
4193:
4189:
4184:
4181:
4178:
4174:
4168:
4165:
4162:
4158:
4149:
4144:
4139:
4134:
4130:
4125:
4123:
4119:
4115:
4110:
4104:
4091:
4088:
4083:
4078:
4075:
4072:
4068:
4064:
4059:
4054:
4051:
4048:
4044:
4040:
4035:
4030:
4027:
4024:
4020:
4016:
4011:
4006:
4003:
4000:
3996:
3975:
3972:
3967:
3964:
3961:
3957:
3951:
3948:
3945:
3941:
3937:
3932:
3929:
3926:
3922:
3916:
3913:
3910:
3906:
3897:
3893:
3889:
3873:
3865:
3860:
3843:
3838:
3834:
3830:
3825:
3822:
3819:
3815:
3811:
3808:
3803:
3800:
3797:
3793:
3789:
3786:
3784:
3778:
3775:
3765:
3761:
3757:
3752:
3749:
3746:
3742:
3738:
3735:
3730:
3727:
3724:
3720:
3716:
3713:
3711:
3705:
3702:
3675:
3667:
3663:
3653:
3649:
3642:
3637:
3634:
3626:
3608:
3600:
3597:
3594:
3590:
3582:
3579:
3576:
3572:
3562:
3559:
3556:
3552:
3544:
3541:
3538:
3534:
3527:
3522:
3519:
3499:
3496:
3493:
3488:
3482:
3475:
3469:
3464:
3461:
3456:
3449:
3446:
3437:
3434:
3427:
3402:
3399:
3396:
3369:
3366:
3362:
3358:
3355:
3331:
3326:
3320:
3313:
3307:
3298:
3297:column matrix
3279:
3276:
3273:
3262:
3252:
3243:
3230:
3220:
3217:
3214:
3211:
3208:
3205:
3202:
3199:
3196:
3193:
3190:
3184:
3177:
3174:
3171:
3168:
3165:
3162:
3159:
3156:
3153:
3150:
3147:
3138:
3131:
3128:
3124:
3120:
3117:
3104:
3087:
3084:
3081:
3078:
3075:
3072:
3069:
3066:
3063:
3060:
3057:
3054:
3051:
3049:
3043:
3040:
3032:
3029:
3026:
3023:
3020:
3017:
3014:
3011:
3008:
3005:
3002:
2999:
2997:
2991:
2988:
2975:
2971:
2967:
2959:
2955:
2937:
2927:
2923:
2917:
2911:
2907:
2899:
2895:
2883:
2870:
2860:
2857:
2854:
2851:
2848:
2845:
2842:
2839:
2836:
2830:
2823:
2820:
2817:
2814:
2811:
2808:
2805:
2802:
2799:
2790:
2783:
2780:
2776:
2772:
2769:
2757:
2754:
2737:
2734:
2731:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2707:
2705:
2699:
2696:
2688:
2685:
2682:
2679:
2676:
2673:
2670:
2667:
2664:
2661:
2659:
2653:
2650:
2637:
2635:
2631:
2627:
2623:
2607:
2599:
2595:
2585:
2572:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2542:
2535:
2532:
2528:
2524:
2521:
2509:
2505:
2501:
2493:
2489:
2483:
2474:
2472:
2468:
2464:
2460:
2456:
2452:
2448:
2444:
2434:
2431:
2418:
2411:
2401:
2397:
2393:
2388:
2384:
2377:
2372:
2362:
2358:
2354:
2349:
2345:
2338:
2333:
2323:
2319:
2315:
2310:
2306:
2297:
2294:
2286:
2267:
2263:
2259:
2254:
2250:
2246:
2241:
2237:
2208:
2204:
2200:
2195:
2191:
2187:
2182:
2178:
2166:
2161:
2148:
2141:
2131:
2127:
2123:
2118:
2114:
2107:
2102:
2092:
2088:
2084:
2079:
2075:
2066:
2063:
2055:
2036:
2032:
2028:
2023:
2019:
1990:
1986:
1982:
1977:
1973:
1961:
1946:
1944:
1943:
1930:
1925:
1923:
1919:
1915:
1906:
1901:
1897:
1887:
1885:
1881:
1877:
1873:
1869:
1865:
1861:
1856:
1854:
1850:
1846:
1842:
1838:
1834:
1830:
1826:
1822:
1821:3D projection
1818:
1814:
1810:
1805:
1803:
1799:
1795:
1790:
1788:
1784:
1780:
1776:
1772:
1768:
1764:
1760:
1756:
1751:
1749:
1745:
1741:
1737:
1731:
1727:
1720:
1713:
1709:
1705:
1701:
1696:
1692:
1685:
1678:
1673:
1671:
1667:
1663:
1659:
1655:
1651:
1647:
1642:
1640:
1636:
1632:
1628:
1624:
1620:
1608:
1598:
1596:
1592:
1591:
1586:
1575:
1559:
1544:
1540:
1536:
1533:of dimension
1532:
1489:
1481:
1476:
1462:
1458:
1441:
1435:
1428:
1421:
1416:
1412:
1406:
1402:
1398:
1392:
1387:
1383:
1381:
1380:
1370:
1366:
1361:
1359:
1355:
1351:
1347:
1343:
1339:
1335:
1331:
1327:
1323:
1319:
1315:
1310:
1290:
1286:
1282:
1278:
1267:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1127:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1060:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
979:
977:
976:
971:
966:
964:
960:
955:
953:
949:
945:
941:
937:
933:
919:
912:
905:
900:
896:
892:
888:
883:
878:
868:
866:
865:of a number.
864:
859:
853:
848:
844:
840:
834:
830:
823:
821:
817:
813:
808:
807:, and so on.
806:
794:
790:
786:
778:
773:
771:
767:
763:
759:
755:
751:
747:
743:
738:
720:
716:
712:
711:
706:
705:
699:
697:
693:
689:
685:
681:
677:
673:
672:perpendicular
669:
665:
661:
655:
645:
643:
641:
623:
620:
617:
614:
608:
599:
595:
591:
587:
583:
578:
567:
563:
559:
555:
551:
546:
544:
540:
534:
527:One dimension
524:
517:
515:
511:
507:
502:
500:
499:vector spaces
496:
492:
488:
483:
481:
477:
476:
469:
467:
466:Nicole Oresme
463:
459:
455:
452:
448:
447:mathematician
444:
434:
432:
428:
424:
420:
416:
412:
408:
404:
400:
396:
392:
388:
384:
379:
377:
373:
369:
365:
361:
355:
351:
345:
339:
334:
330:
326:
322:
318:
314:
310:
302:
296:
292:
286:
282:
277:
273:
269:
265:
261:
254:
250:
248:
243:
237:
233:
229:
226:-dimensional
224:
218:
213:
209:
204:
202:
198:
195:represent an
194:
186:
185:
180:
176:
172:
168:
167:
162:
161:perpendicular
158:
154:
150:
146:
142:
138:
132:
99:
91:
58:
52:
48:
23:
19:
6936:Paraboloidal
6915:
6874:
6803:
6767:
6740:
6714:
6695:
6659:
6625:
6590:
6571:
6551:
6527:
6506:
6485:
6466:
6446:
6433:. Retrieved
6429:the original
6405:
6383:
6371:
6356:
6344:
6332:
6321:
6309:. Retrieved
6305:
6296:
6284:. Retrieved
6280:
6251:
6239:
6228:
6216:
6188:
6181:
6166:
6154:. Retrieved
6150:
6140:
6121:Regular grid
6078:
6074:
6062:
6057:
6053:
6049:
6042:
6038:
6028:
6027:There is no
6026:
5842:
5736:
5732:
5728:
5725:unit vectors
5620:
5572:
5543:
5533:
5529:
5518:
5513:
5509:
5505:
5501:
5497:
5493:
5489:
5485:
5482:
5477:
5473:
5469:
5466:right-handed
5465:
5461:
5457:
5453:
5442:index finger
5435:
5422:
5419:right-handed
5418:
5414:
5410:
5406:
5402:
5398:
5394:
5390:
5386:
5378:
5374:
5372:
5347:
5344:
5339:
5337:
5332:
5328:
5322:
5320:
5316:right-handed
5315:
5311:
5307:
5303:
5299:
5295:
5291:
5289:
5284:
5280:
5276:
5268:
5264:
5260:
5258:
5151:
5148:
5078:
5071:
5061:
5057:
5055:
4985:
4978:
4974:
4968:
4965:
4957:
4950:
4910:
4673:
4336:
4241:
4142:
4132:
4126:
4108:
4105:
3861:
3258:
3249:
3105:
2976:
2969:
2965:
2957:
2953:
2925:
2921:
2909:
2905:
2897:
2893:
2889:
2758:
2755:
2638:
2633:
2629:
2625:
2621:
2591:
2510:
2503:
2499:
2491:
2487:
2480:
2459:translations
2446:
2440:
2432:
2287:
2162:
2056:
1957:
1940:
1928:
1926:
1913:
1911:
1883:
1879:
1875:
1874:. The words
1871:
1867:
1863:
1859:
1857:
1840:
1836:
1832:
1828:
1827:) shows the
1816:
1812:
1808:
1806:
1797:
1791:
1786:
1782:
1778:
1774:
1770:
1752:
1729:
1725:
1718:
1711:
1707:
1703:
1699:
1694:
1690:
1683:
1676:
1674:
1669:
1665:
1661:
1657:
1653:
1648:varies with
1643:
1638:
1634:
1630:
1626:
1622:
1618:
1604:
1595:affine plane
1588:
1581:
1542:
1534:
1457:real numbers
1454:
1439:
1433:
1426:
1419:
1414:
1410:
1404:
1400:
1396:
1376:
1368:
1364:
1362:
1357:
1356:-plane, and
1353:
1349:
1345:
1341:
1337:
1333:
1329:
1325:
1321:
1317:
1313:
1311:
1288:
1284:
1280:
1276:
1271:(3, −2.5, 1)
1268:
980:
973:
969:
967:
962:
958:
956:
951:
947:
943:
939:
935:
931:
929:
917:
910:
903:
898:
894:
890:
886:
867:
857:
851:
846:
842:
832:
828:
824:
819:
815:
812:right angles
809:
780:
774:
769:
765:
761:
757:
753:
749:
745:
739:
718:
714:
708:
702:
700:
695:
691:
687:
683:
679:
668:ordered pair
663:
659:
657:
644:
579:
561:
554:real numbers
547:
542:
536:
523:
503:
491:Isaac Newton
484:
475:La Géométrie
473:
470:
442:
440:
403:group theory
380:
368:tangent line
353:
349:
343:
337:
306:
300:
294:
290:
284:
280:
275:
271:
267:
263:
259:
241:
235:
222:
216:
211:
205:
200:
182:
178:
174:
170:
164:
152:
149:real numbers
50:
44:
35:(−1.5, −2.5)
18:
6966:Bispherical
6951:Ellipsoidal
6921:Cylindrical
6246:, p. 1
6221:Burton 2011
6083:quaternions
6073:, so it is
5524:cube and a
5510:in front of
5464:-axes in a
4953:composition
3344:The result
2482:Translating
2477:Translation
2467:reflections
1845:perspective
1607:parentheses
1541:(lists) of
1344:-axis, and
837:, then its
793:unit square
789:unit circle
590:translation
543:number line
539:affine line
533:Number line
520:Description
458:Netherlands
451:philosopher
419:engineering
376:derivatives
247:hyperplanes
177:(plural of
153:coordinates
7000:Categories
6737:Feshbach H
6690:Margenau H
6363:, p.
6337:Smart 1998
6244:Axler 2015
5731:-axis and
5401:- and the
5285:handedness
5233:See also:
4946:orthogonal
4112:times its
3888:orthogonal
2916:reflection
2886:Reflection
1759:horizontal
1698:) for the
1585:hyperplane
768:-axis and
723:(3, −10.5)
640:affine map
429:and other
193:directions
41:in purple.
29:in green,
6931:Parabolic
6926:Spherical
6916:Cartesian
6890:Parabolic
6884:Log-polar
6875:Cartesian
6805:MathWorld
6652:Korn GA,
6645:488633510
6339:, Chap. 2
6311:25 August
6223:, p. 374.
6132:Citations
5440:. If the
5373:Once the
4320:−
4282:−
4190:−
4114:transpose
3623:is a 2×2
3221:θ
3215:
3206:−
3203:θ
3197:
3178:θ
3172:
3160:θ
3154:
3085:θ
3079:
3070:−
3067:θ
3061:
3033:θ
3027:
3015:θ
3009:
2938:θ
2861:θ
2858:
2846:θ
2843:
2824:θ
2821:
2812:−
2809:θ
2806:
2735:θ
2732:
2720:θ
2717:
2689:θ
2686:
2677:−
2674:θ
2671:
2636:), where
2608:θ
2596:a figure
2463:rotations
2451:bijective
2449:are the (
2394:−
2355:−
2316:−
2124:−
2085:−
1914:quadrants
1884:applicate
1872:applicate
1748:subscript
1615:(3, 5, 7)
1490:×
1444:(1, −1, 1
1373:(0, 0, 1)
1322:applicate
1307:(0, 0, 1)
1303:(0, 1, 0)
1299:(1, 0, 0)
1295:(0, 0, 0)
1245:−
1236:−
1227:−
1203:−
1194:−
1179:−
1161:−
1146:−
1137:−
1112:−
1070:−
1028:−
924:(2, 3, 4)
841:from the
839:distances
816:quadrants
814:, called
612:↦
510:spherical
443:Cartesian
411:astronomy
372:integrals
364:perimeter
325:equations
232:dimension
6981:6-sphere
6961:Toroidal
6900:Elliptic
6776:67-25285
6759:52-11515
6739:(1953).
6733:Morse PM
6706:55-10911
6682:19959906
6674:59-14456
6656:(1961).
6623:(2001).
6467:Geometry
6435:17 April
6351:, pg. 49
6286:6 August
6208:71006826
6156:6 August
6089:See also
6061:. Here,
5494:parallel
5423:positive
5312:standard
5308:positive
5177:′
5166:′
5104:′
5093:′
5068:Shearing
5011:′
5000:′
4878:′
4866:′
4510:′
4449:′
4423:′
4411:′
4385:composed
3779:′
3706:′
3450:′
3438:′
3370:′
3359:′
3132:′
3121:′
3044:′
2992:′
2974:, where
2784:′
2773:′
2700:′
2654:′
2588:Rotation
2536:′
2525:′
1880:ordinate
1876:abscissa
1789:-axis).
1767:vertical
1763:ordinate
1755:abscissa
1646:pressure
1507:, where
1369:altitude
1360:-plane.
1352:-plane,
1318:ordinate
1314:abscissa
710:ordinate
707:and the
704:abscissa
566:oriented
487:calculus
399:calculus
366:and the
317:calculus
230:for any
187:and has
173:or just
47:geometry
33:in red,
6976:Conical
6895:Bipolar
6654:Korn TM
6065:is the
6029:natural
5743:versors
5534:towards
5526:concave
5516:-axis.
5486:towards
5460:-, and
5275:to the
4962:Scaling
4677:of the
4498:where
4136:is the
4116:is the
1942:orthant
1937:(− + −)
1933:(+ + +)
1929:octants
1777:-, and
1724:, ...,
1689:, ...,
1672:, etc.
1611:(10, 5)
1340:-axis,
975:octants
803:), the
594:scaling
437:History
415:physics
313:algebra
151:called
135:) in a
31:(−3, 1)
6774:
6757:
6747:
6721:
6704:
6680:
6672:
6643:
6633:
6602:
6578:
6559:
6538:
6515:
6492:
6473:
6454:
6421:
6206:
6196:
6173:, See
6071:(0, 1)
5843:where
5621:where
5546:vector
5522:convex
5377:- and
3625:matrix
3512:where
3106:Thus:
2756:Thus:
2594:rotate
1831:- and
1746:, the
1744:record
1670:t-axis
1668:, the
1666:y-axis
1664:, the
1662:x-axis
1539:tuples
1409:. The
1365:height
1332:, and
1305:, and
936:origin
915:, and
801:(1, 1)
797:(0, 0)
735:(0, 1)
731:(1, 0)
727:(0, 0)
719:origin
662:or an
562:origin
362:, the
358:; the
329:circle
321:curves
279:where
189:(0, 0)
184:origin
157:signed
39:(0, 0)
27:(2, 3)
6880:Polar
6666:55–79
5450:thumb
5411:above
4971:. If
4146:is a
1740:array
922:, or
756:, or
266:) + (
145:point
139:is a
137:plane
6772:LCCN
6755:LCCN
6745:ISBN
6719:ISBN
6702:LCCN
6678:OCLC
6670:LCCN
6641:OCLC
6631:ISBN
6600:ISBN
6576:ISBN
6557:ISBN
6536:ISBN
6513:ISBN
6490:ISBN
6471:ISBN
6452:ISBN
6437:2022
6419:ISBN
6313:2024
6288:2017
6257:rays
6204:OCLC
6194:ISBN
6175:here
6158:2017
5964:and
5723:are
5672:and
5512:the
5490:away
5413:the
5383:line
5252:The
5237:and
3988:and
3627:and
3259:All
2469:and
2441:The
2226:and
2008:and
1958:The
1898:and
1882:and
1862:and
1656:and
1650:time
1436:= −1
1389:The
1377:the
1320:and
932:axes
897:and
799:and
760:and
752:and
733:and
512:and
493:and
449:and
374:and
360:area
341:and
315:and
299:and
283:and
274:) =
179:axis
175:axes
49:, a
6411:doi
6365:657
6075:not
5456:-,
5421:or
5310:or
5056:If
4944:is
3894:of
3886:is
3212:cos
3194:sin
3169:sin
3151:cos
3076:cos
3058:sin
3024:sin
3006:cos
2924:, −
2890:If
2855:cos
2840:sin
2818:sin
2803:cos
2729:cos
2714:sin
2683:sin
2668:cos
2592:To
2445:or
2285:is
2054:is
1935:or
1823:or
1773:-,
1613:or
1597:).
1429:= 1
1422:= 1
1367:or
1291:/2)
1273:or
920:= 4
913:= 3
906:= 2
713:of
690:of
670:of
573:or
537:An
489:by
356:= 4
333:set
110:ɑːr
69:ɑːr
45:In
7002::
6802:.
6753:.
6735:,
6676:.
6668:.
6639:.
6594:.
6534:.
6417:.
6304:.
6279:.
6264:^
6202:.
6149:.
6085:.
6058:iy
6056:+
6052:=
6041:,
5498:xy
5425:.
5415:xy
5407:xy
5391:xy
5072:A
4977:,
4948:.
4323:1.
4228:1.
4124:.
4092:1.
2972:′)
2968:′,
2956:,
2908:,
2904:(−
2896:,
2634:y'
2630:x'
2502:,
2490:,
2473:.
2465:,
2461:,
1878:,
1809:xy
1804:.
1732:−1
1717:,
1682:,
1637:,
1633:,
1625:,
1574:.
1446:).
1403:,
1399:,
1382:.
1358:xz
1354:yz
1350:xy
1328:,
1316:,
1309:.
1301:,
1287:,
1283:+
1279:,
908:,
893:,
831:,
822:.
775:A
737:.
580:A
501:.
433:.
425:,
417:,
413:,
401:,
393:,
389:,
352:+
293:,
270:−
262:−
249:.
203:.
169:,
119:iː
100::
98:US
94:,
81:zj
78:iː
59::
57:UK
6886:)
6882:(
6853:e
6846:t
6839:v
6808:.
6778:.
6761:.
6727:.
6708:.
6684:.
6647:.
6608:.
6565:.
6544:.
6521:.
6498:.
6479:.
6460:.
6439:.
6413::
6315:.
6290:.
6210:.
6160:.
6079:x
6063:i
6054:x
6050:z
6045:)
6043:y
6039:x
6037:(
6013:.
6008:)
6002:1
5995:0
5988:0
5982:(
5977:=
5973:k
5952:,
5947:)
5941:0
5934:1
5927:0
5921:(
5916:=
5912:j
5892:,
5887:)
5881:0
5874:0
5867:1
5861:(
5856:=
5852:i
5829:,
5825:k
5821:z
5818:+
5814:j
5810:y
5807:+
5803:i
5799:x
5796:=
5792:r
5771:)
5768:z
5765:,
5762:y
5759:,
5756:x
5753:(
5733:y
5729:x
5709:)
5703:1
5696:0
5690:(
5685:=
5681:j
5658:)
5652:0
5645:1
5639:(
5634:=
5630:i
5607:,
5603:j
5599:y
5596:+
5592:i
5588:x
5585:=
5581:r
5557:r
5530:x
5514:z
5506:y
5502:x
5478:z
5474:y
5470:x
5462:z
5458:y
5454:x
5403:y
5399:x
5395:z
5387:z
5379:y
5375:x
5333:y
5329:x
5304:y
5300:x
5296:y
5292:x
5281:x
5277:x
5269:y
5265:y
5261:x
5208:)
5205:y
5202:+
5199:s
5196:x
5193:,
5190:x
5187:(
5184:=
5181:)
5174:y
5170:,
5163:x
5159:(
5135:)
5132:y
5129:,
5126:s
5123:y
5120:+
5117:x
5114:(
5111:=
5108:)
5101:y
5097:,
5090:x
5086:(
5062:m
5058:m
5042:.
5039:)
5036:y
5033:m
5030:,
5027:x
5024:m
5021:(
5018:=
5015:)
5008:y
5004:,
4997:x
4993:(
4981:)
4979:y
4975:x
4973:(
4969:m
4930:j
4927:,
4924:i
4920:A
4897:.
4892:)
4886:1
4875:y
4863:x
4856:(
4851:=
4846:)
4840:1
4833:y
4826:x
4820:(
4813:)
4807:1
4802:0
4797:0
4788:2
4784:b
4776:2
4773:,
4770:2
4766:A
4758:2
4755:,
4752:1
4748:A
4738:1
4734:b
4726:1
4723:,
4720:2
4716:A
4708:1
4705:,
4702:1
4698:A
4691:(
4646:.
4641:)
4635:1
4630:0
4625:0
4616:2
4612:b
4604:2
4601:,
4598:2
4594:A
4586:1
4583:,
4580:2
4576:A
4566:1
4562:b
4554:2
4551:,
4548:1
4544:A
4536:1
4533:,
4530:1
4526:A
4519:(
4514:=
4507:A
4486:,
4481:)
4475:1
4468:y
4461:x
4455:(
4446:A
4442:=
4437:)
4431:1
4420:y
4408:x
4401:(
4371:0
4368:=
4363:2
4359:b
4355:=
4350:1
4346:b
4317:=
4312:2
4309:,
4306:1
4302:A
4296:1
4293:,
4290:2
4286:A
4277:2
4274:,
4271:2
4267:A
4261:1
4258:,
4255:1
4251:A
4225:=
4220:2
4217:,
4214:1
4210:A
4204:1
4201:,
4198:2
4194:A
4185:2
4182:,
4179:2
4175:A
4169:1
4166:,
4163:1
4159:A
4143:A
4133:A
4109:A
4089:=
4084:2
4079:2
4076:,
4073:2
4069:A
4065:+
4060:2
4055:2
4052:,
4049:1
4045:A
4041:=
4036:2
4031:1
4028:,
4025:2
4021:A
4017:+
4012:2
4007:1
4004:,
4001:1
3997:A
3976:0
3973:=
3968:2
3965:,
3962:2
3958:A
3952:1
3949:,
3946:2
3942:A
3938:+
3933:2
3930:,
3927:1
3923:A
3917:1
3914:,
3911:1
3907:A
3874:A
3844:.
3839:2
3835:b
3831:+
3826:2
3823:,
3820:2
3816:A
3812:y
3809:+
3804:1
3801:,
3798:2
3794:A
3790:x
3787:=
3776:y
3766:1
3762:b
3758:+
3753:1
3750:,
3747:1
3743:A
3739:y
3736:+
3731:1
3728:,
3725:1
3721:A
3717:x
3714:=
3703:x
3676:)
3668:2
3664:b
3654:1
3650:b
3643:(
3638:=
3635:b
3609:)
3601:2
3598:,
3595:2
3591:A
3583:1
3580:,
3577:2
3573:A
3563:2
3560:,
3557:1
3553:A
3545:1
3542:,
3539:1
3535:A
3528:(
3523:=
3520:A
3500:,
3497:b
3494:+
3489:)
3483:y
3476:x
3470:(
3465:A
3462:=
3457:)
3447:y
3435:x
3428:(
3406:)
3403:y
3400:,
3397:x
3394:(
3374:)
3367:y
3363:,
3356:x
3352:(
3332:.
3327:)
3321:y
3314:x
3308:(
3283:)
3280:y
3277:,
3274:x
3271:(
3231:.
3228:)
3225:)
3218:2
3209:y
3200:2
3191:x
3188:(
3185:,
3182:)
3175:2
3166:y
3163:+
3157:2
3148:x
3145:(
3142:(
3139:=
3136:)
3129:y
3125:,
3118:x
3114:(
3088:.
3082:2
3073:y
3064:2
3055:x
3052:=
3041:y
3030:2
3021:y
3018:+
3012:2
3003:x
3000:=
2989:x
2970:y
2966:x
2964:(
2960:)
2958:y
2954:x
2952:(
2928:)
2926:y
2922:x
2920:(
2912:)
2910:y
2906:x
2900:)
2898:y
2894:x
2892:(
2871:.
2868:)
2865:)
2852:y
2849:+
2837:x
2834:(
2831:,
2828:)
2815:y
2800:x
2797:(
2794:(
2791:=
2788:)
2781:y
2777:,
2770:x
2766:(
2738:.
2726:y
2723:+
2711:x
2708:=
2697:y
2680:y
2665:x
2662:=
2651:x
2632:,
2626:y
2624:,
2622:x
2573:.
2570:)
2567:b
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