# Cartesian coordinate system

A Cartesian coordinate system specifies the location of points by ordered numbers giving their distances from fixed perpendicular reference lines called axes. In the plane, a point is addressed by a pair of numbers (x, y); in space, by a triple (x, y, z). The system is named after [René Descartes](https://www.edgepedia.org/rene-descartes), whose Latin name Renatus Cartesius is the origin of the word "Cartesian".<sup>[1](https://encyclopediaofmath.org/wiki/Cartesian_coordinates)</sup><sup> • </sup><sup>[2](https://academic.sun.ac.za/mathed/shoma/MATUNIT12_02.htm)</sup>

| Key fact | Detail |
|---|---|
| Definition | Location of points on a plane by their distances from two fixed perpendicular straight lines (axes)<sup>[1](https://encyclopediaofmath.org/wiki/Cartesian_coordinates)</sup> |
| Naming | From René Descartes (1596–1650), Latin Renatius Cartesius; analytical geometry is also called Cartesian or coordinate geometry<sup>[2](https://academic.sun.ac.za/mathed/shoma/MATUNIT12_02.htm)</sup> |
| Systematic development | René Descartes and Pierre de Fermat developed the idea systematically; in their formulations distances could only be positive or zero<sup>[1](https://encyclopediaofmath.org/wiki/Cartesian_coordinates)</sup> |
| Negative coordinates | The idea of assigning negative values to one or both distances was proposed by Isaac Newton<sup>[1](https://encyclopediaofmath.org/wiki/Cartesian_coordinates)</sup> |
| The word "coordinates" | Gottfried Leibniz was the first to name these distances "coordinates"<sup>[1](https://encyclopediaofmath.org/wiki/Cartesian_coordinates)</sup> |
| Distance formula | The distance between P(x₁, y₁) and Q(x₂, y₂) is \( d(PQ) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \), the metric of the system, from the Pythagorean theorem<sup>[3](https://math.bu.edu/people/if/calculus_one.pdf)</sup> |
| Earth applications | The Earth-Centered Earth-Fixed (ECEF) frame is a geocentric right-handed Cartesian system with the X-Y plane coplanar with the equator, the Z-axis through the North Pole, and the X-axis through the intersection of the equator and prime meridian<sup>[4](https://ntrs.nasa.gov/api/citations/20205002898/downloads/NASA-TM-2020-5002898%20corrected.pdf)</sup> |

## How the plane system works

The Cartesian system in the plane consists of two orthogonal axes, labeled x and y, sharing a common origin. To arrive at P(x, y), one covers a distance of |x| on the x-axis — moving rightward when x > 0, leftward when x < 0 — then makes a 90-degree turn and covers a distance of |y| in a direction parallel to the y-axis, upward when y > 0 and downward when y < 0. The point is fixed by the intersection of two perpendicular lines, one parallel to the x-axis passing through y on the y-axis, and one parallel to the y-axis passing through x on the x-axis.<sup>[3](https://math.bu.edu/people/if/calculus_one.pdf)</sup>

Because the axes are orthogonal, the distance between two points follows from the theorem of [Pythagoras](https://www.edgepedia.org/pythagoras) as \( d(PQ) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \), which is described as the metric of the system.<sup>[3](https://math.bu.edu/people/if/calculus_one.pdf)</sup> A distinctive property noted in multivariable calculus teaching is that two points in Cartesian coordinates are equal only if they are equal coordinate by coordinate; every point has exactly one representation.<sup>[5](https://opencw.aprende.org/resources/res-18-007-calculus-revisited-multivariable-calculus-fall-2011/part-ii/lecture-3-polar-coordinates/)</sup><sup> • </sup><sup>[6](https://www.varsitytutors.com/practice/lessons/ap-precalculus/trigonometry-and-polar-coordinates)</sup>

## History

According to the Encyclopedia of Mathematics, the basic notion—locating a point via its distances from fixed perpendicular lines—appears as early as the writings of [Archimedes](https://www.edgepedia.org/archimedes) and Apollonius over two millennia back, and even in ancient Egyptian texts. The Greek mathematician Menaechmus (380–320 BCE) proved theorems using a method very close to using coordinates, and it has sometimes been maintained that he introduced coordinate geometry.<sup>[1](https://encyclopediaofmath.org/wiki/Cartesian_coordinates)</sup><sup> • </sup><sup>[7](https://amsi.org.au/ESA_Senior_Years/SeniorTopic1/1b/1b_4history.html)</sup>

**Descartes and Fermat.** Coordinate geometry has traditionally been attributed to René Descartes (1596–1650) and [Pierre de Fermat](https://www.edgepedia.org/pierre-de-fermat) (1601–1665), who independently provided the beginning of the subject as it is known today.<sup>[7](https://amsi.org.au/ESA_Senior_Years/SeniorTopic1/1b/1b_4history.html)</sup> Descartes published a work explaining how to use coordinates for finding points in space, and presented a geometric interpretation of a mathematical function, although [Nicole Oresme](https://www.edgepedia.org/nicole-oresme) had already made graphs of varying quantities in the fourteenth century; Fermat did much the same work independently but did not publish it.<sup>[8](https://www.papertrell.com/apps/preview/The-Handy-Math-Answer-Book/Handy%20Answer%20book/Who-developed-analytic-geometry/001137022/content/SC/52cb008a82fad14abfa5c2e0_default.html)</sup><sup> • </sup><sup>[12](https://mathshistory.st-andrews.ac.uk/Biographies/Oresme/)</sup> The concept of a rectangular coordinate system is attributed to Descartes in 1637, the year of his *Discours de la Méthode*.<sup>[9](https://spsweb.fltops.jpl.nasa.gov/portaldataops/mpg/MPG_Docs/MPG%20Book/Release/Chapter3-Coordinate%20%26%20Reference%20Systems.pdf)</sup><sup> • </sup><sup>[2](https://academic.sun.ac.za/mathed/shoma/MATUNIT12_02.htm)</sup>

In the formulations of Descartes and Fermat the distances from the axes could only be positive or zero. [Isaac Newton](https://www.edgepedia.org/isaac-newton) proposed assigning negative values to one or both distances, and Gottfried Leibniz was the first to call these distances "coordinates".<sup>[1](https://encyclopediaofmath.org/wiki/Cartesian_coordinates)</sup>

Descartes also drew a practical consequence from the algebraic treatment of geometry: he demonstrated that whenever the analytic form of a geometric construction calls only for addition, subtraction, multiplication, division, and taking square roots, ruler and compass will suffice to carry it out.<sup>[2](https://academic.sun.ac.za/mathed/shoma/MATUNIT12_02.htm)</sup>

## Cartesian and other coordinate systems

Cartesian coordinates are one of several ways of locating points. The rectangular (Cartesian), spherical (latitude-longitude-radius), and cylindrical systems are the coordinate systems used predominantly in spacecraft mission planning, and two-dimensional systems include rectangular, polar, azimuth/elevation, and right-ascension/declination coordinates.<sup>[9](https://spsweb.fltops.jpl.nasa.gov/portaldataops/mpg/MPG_Docs/MPG%20Book/Release/Chapter3-Coordinate%20%26%20Reference%20Systems.pdf)</sup> Transformations between systems may require both translation and rotation; the Cartesian-to-spherical transformation and rotations using Eulerian angles are standard tools in applications such as antenna pattern analysis.<sup>[10](https://descanso.jpl.nasa.gov/monograph/series4/Mono4_Ch1.pdf)</sup>

The choice of system is a matter of fit to the problem. In polar coordinates, a point is given by a distance r from the origin and an angle θ from the positive x-axis; circles through the origin take the simple forms r = a cos θ or r = a sin θ, and spirals such as r = aθ are naturally expressed, while the corresponding Cartesian equations are more complex. Lines, by contrast, are simple in Cartesian form (y = mx + b) and more complex in polar form. A further difference is uniqueness: every point has exactly one Cartesian representation, while a polar point has infinitely many, since 2π may be added to θ and r may be negated.<sup>[6](https://www.varsitytutors.com/practice/lessons/ap-precalculus/trigonometry-and-polar-coordinates)</sup>

## Applications

Cartesian frames underpin practical positioning systems. In the [World Geodetic System](https://www.edgepedia.org/world-geodetic-system) 1984 (WGS84) standard, the Earth-Centered Earth-Fixed (ECEF) frame is a geocentric right-handed Cartesian coordinate system whose X-Y plane is coplanar with the Earth's equatorial plane, whose Z-axis passes through the [North Pole](https://www.edgepedia.org/north-pole), and whose X-axis passes through the intersection of the equator and the prime meridian (0° latitude, 0° longitude). It is used alongside the spherical Latitude-Longitude-Altitude frame, and source code for the reference-frame transformations is publicly available.<sup>[4](https://ntrs.nasa.gov/api/citations/20205002898/downloads/NASA-TM-2020-5002898%20corrected.pdf)</sup> Converting spherical coordinates into Cartesian coordinates is likewise described as an important mathematical operation in computer science, with hardware implementations on FPGAs.<sup>[11](https://arxiv.org/abs/2407.18255)</sup>

## References

1. [Cartesian coordinates, Encyclopedia of Mathematics (M.I. Voitsekhovskii, originator)](https://encyclopediaofmath.org/wiki/Cartesian_coordinates)
2. [Maths Unit 12: The History of Analytical Geometry, Stellenbosch University](https://academic.sun.ac.za/mathed/shoma/MATUNIT12_02.htm)
3. [Calculus One (Boston University course text), sections 2.3–2.4](https://math.bu.edu/people/if/calculus_one.pdf)
4. [NASA Technical Memorandum 2020-5002898: Reference Frames](https://ntrs.nasa.gov/api/citations/20205002898/downloads/NASA-TM-2020-5002898%20corrected.pdf)
5. [Lecture 3: Polar Coordinates, MIT RES.18-007 Calculus Revisited: Multivariable Calculus](https://opencw.aprende.org/resources/res-18-007-calculus-revisited-multivariable-calculus-fall-2011/part-ii/lecture-3-polar-coordinates/)
6. [Trigonometry and Polar Coordinates, Varsity Tutors AP Precalculus](https://www.varsitytutors.com/practice/lessons/ap-precalculus/trigonometry-and-polar-coordinates)
7. [History and applications, AMSI Senior Years](https://amsi.org.au/ESA_Senior_Years/SeniorTopic1/1b/1b_4history.html)
8. [Who developed analytic geometry?, The Handy Math Answer Book](https://www.papertrell.com/apps/preview/The-Handy-Math-Answer-Book/Handy%20Answer%20book/Who-developed-analytic-geometry/001137022/content/SC/52cb008a82fad14abfa5c2e0_default.html)
9. [Chapter 3: Coordinate & Reference Systems, JPL Mission Planning Geometry](https://spsweb.fltops.jpl.nasa.gov/portaldataops/mpg/MPG_Docs/MPG%20Book/Release/Chapter3-Coordinate%20%26%20Reference%20Systems.pdf)
10. [JPL DESCANSO Monograph Series 4, Chapter 1: Coordinate Transformations](https://descanso.jpl.nasa.gov/monograph/series4/Mono4_Ch1.pdf)
11. [Flexible and Cost-Effective Spherical to Cartesian Coordinate Conversion Using 3-D CORDIC Algorithm on FPGA, arXiv:2407.18255](https://arxiv.org/abs/2407.18255)
12. [Nicholas Oresme - Biography](https://mathshistory.st-andrews.ac.uk/Biographies/Oresme/)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Geometry and topology › Analytic and coordinate geometry*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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