Coordinate geometry

Summary: The development of coordinate geometry revolutionized mathematics, has a wide variety of applications, and is now widely used in many areas of mathematics.

The discovery that plane geometric configurations could be entirely described by real number pairs and two-variable equations revolutionized geometry and many other important fields of mathematics that emerged later, including real analysis, vectors, calculus, linear algebra, and matrix theory. Also referred to as “analytic geometry” or “Cartesian geometry,” named for the great philosopher and mathematician René Descartes, the subject of coordinate geometry is the study of geometry using the Cartesian coordinate system with algebraic operations. In twenty-first century classrooms, children in primary school begin to examine coordinate systems and create plots on graph paper.

The level of sophistication of knowledge builds through high school and college through the use of various coordinate systems including Cartesian, polar, and spherical systems and by representations in two- and three-dimensional geometry. Some calculus courses are titled “Calculus and Analytic Geometry.” Various coordinate system standards are in use in physics or mathematics, for surveyors, or at the state or company level. High school and college students learn to convert between some of these representations. Coordinate geometry has many applications and is used in every conceivable area of mathematics, science, and engineering to calculate precise locations and boundaries, distances and bearings from reference points, and to define graphs and curves using a point location, radius, and arc-lengths.

The fundamental building block of coordinate geometry is the Cartesian coordinate system, which includes an infinite collection of points on a plane determined by an ordered pair of numerical coordinates (x, y). The x-coordinate (called “abscissa”) represents the horizontal position, and the y-coordinate (called “ordinate”) represents the vertical position. These positions can be expressed as signed distances from the origin (0, 0), a point that is at the intersection of two perpendicular reference lines called the “coordinate axis” (see Figure 1).

Once points are determined by ordered pairs (x, y) on the coordinate plane, one can then obtain analytic formulas for various geometric quantities on the plane. For example, an application of the Pythagorean theorem then yields the distance between any two points (x₁, y₁) and (x₂, y₂) given by

Early Variations

Coordinate-like types of systems arose in cartography well before Descartes. Maps with grids date back to ancient times, including those by Dicaearchus of Messana and Eratosthenes of Cyrene. Claudius Ptolemy attempted to create coordinates of well-known places in the world, essentially their latitude and longitude, from spherical projections, although the astronomical and mathematical methods to accurately calculate these would not be completely developed until much later. Islamic Mathematicians in the medieval Islamic world, such as Abu Arrayhan Muhammad ibn Ahmad al-Biruni, who compared the work of Ptolemy and Abu Ja’far Muhammad ibn Musa Al-Khwarizmi, provided coordinates for more than 600 geographical locations. Al-Biruni also used rectangular coordinates to represent three-dimensional space as well as ideas that some consider as a precursor to polar coordinates. In the twenty-first century, the global positioning system calculates the coordinates of a user from a system of satellites.

Other aspects of coordinate geometry can also be found in various early contexts. Some have noted that the mathematical work of ancient Greek mathematician Menaechmus could be interpreted as one that used coordinates. However, there was no algebra in ancient Greece, and others have highlighted the challenge that mathematics historians face in judging historical works. Coordinate geometry is a natural leap for the historians but probably not for Menaechmus, critics assert. Graphing techniques were developed in the fourteenth century in publications of Nicole d’Oresme and a work titled De latitudinibus formarum (The Latitudes of Forms), which some attribute to d’Oresme. Others assert that this attribution is an error and that the author is unknown. These works may have influenced coordinate geometers.

Transformations of coordinate-like systems developed along with perspective drawing techniques of curves and shapes, like in the works of Leone Battista Alberti and Piero della Francesca. Polar coordinates were motivated through the work of mathematicians such as Bonaventura Cavalieri on spiral curves like the Archimedean spiral, named for Archimedes of Syracuse.

Development

Descartes and Pierre de Fermat are both credited with independently introducing coordinate geometry. They each introduced a type of single-axis system or ordinate geometry. Distances could be measured at a fixed angle to the reference line. In Fermat’s work, curves are generated as loci rather than by plotting points. Historian of science Michael Sean Mahoney noted: “There is connected with the system an intuitive sense of motion or flow wholly in keeping with the intuition which underlies the notion of an algebraic variable.” Descartes’ published work on coordinate geometry dates to 1637 in the appendix (La Géométrie) of a short book entitled Discourse on the Method. Descartes defined the five algebraic operations of addition, subtraction, multiplication, division, and extraction of square roots as geometric constructions on line segments and showed how these operations could be performed in the Euclidean plane by straightedge-and-compass constructions. He also developed geometric techniques for solving polynomial equations by intersecting curves, such as conic sections, with each other or with lines to obtain solutions algebraically. Coordinate geometry helps to classify conic sections, which are curves corresponding to the general quadratic equation

ax² + bxy + cy² + dx + ey + f = 0

where a, b, c, d, e, and f are constants and a, b, and c are not all zero. Coordinate geometry became useful in a wide variety of mathematical and physical situations. Sir Isaac Newton and others investigated various coordinate systems as well as how to convert between them. In the nineteenth century, Christof Gudermann investigated the sphere, and Julius Plücker published numerous volumes on analytic geometry.

Variations

In situations where there is no obvious origin or reference axes, mathematicians developed local coordinates or coordinate-free approaches. For instance, the Frenet–Serret frame is named for Jean Frédéric Frenet and Joseph Serret. It is a type of coordinate axis system for a curve in three-dimensional space and represents the twists and turns of a curve as three vectors that move along the curve. Jean-Gaston Darboux explored the analog for a surface.

Another example is “isothermal coordinates” on surfaces in the work of mathematicians, like Carl Friedrich Gauss. Engineer, mathematician, and physicist Gabriel Lamé is noted as the first to use the term in his 1833 work on heat transfer. August Möbius introduced barycentric coordinates, which utilizes notions related to the center of mass and the centroid of a triangle, and these coordinates can be found in computer graphics. Möbius’ work used both the position and magnitude.

Other mathematicians developed similar systems, including vectors, which allowed for compact notation. Hermann Grassmann and William Hamilton created the algebra of vectors. The development of vectors was especially useful when extending the geometry or physics to higher dimensions. A point (x, y) in the plane can also be represented by a vector as r = xî + yĵ where î and ĵ are unit vectors. Mathematicians including Jean-Victor Poncelet and Michel Chasles developed synthetic projective geometry, which focused on axioms instead of coordinates. Gregorio Ricci-Curbastro and Tullio Levi-Civita explored a coordinate-independent calculus, which led to the development of tensor analysis that later became important in general relativity. Bernhard Riemann’s work on geodesics and Riemannian geometry led to geodesic coordinates, which also became important in relativity.

Education

Coordinate geometry took on an increased prominence in schools in the nineteenth and twentieth centuries. One reason was the development and curricular use of graph paper. A patent for printed graph paper dates back to Dr. Buston in the late eighteenth century. Graph paper makes it easier to plot points and create curves, and it was found to be useful in surveying and civil engineering projects.

Mathematicians in the nineteenth century, like E. H. Moore, advocated the use of paper with “squared lines” in algebra classes. Coordinate geometry topics were also included in algebra textbooks and in textbooks devoted to the subject.

One notable textbook was published by Scottish mathematician Robert J. T. Bell in 1910. His treatise on coordinate geometry in three dimensions became a very successful textbook on the subject and was translated into numerous languages.

Bibliography

Boyer, Carl B. History of Analytic Geometry. New York: Dover Books on Mathematics, 2004.

Mahoney, Michael Sean. The Mathematical Career of Pierre de Fermat, 1601–1665. 2nd ed. Princeton, NJ: Princeton University Press, 1994.

Sasaki, C. Descartes’s Mathematical Thought. New York: Springer, 2010.