analytic geometry - Encyclopædia Britannica

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analytic geometry

Encyclopædia Britannica

analytic geometry, also called coordinate geometry [55]

, mathematical subject in which

algebraic symbolism and methods are used to represent and solve problems in geometry.

The importance of analytic geometry is that it establishes a correspondence between

geometric curves and algebraic equations[56]

. This correspondence makes it possible to

reformulate problems in geometry as equivalent problems in algebra[57]

, and vice versa;

the methods of either subject can then be used to solve problems in the other. For

example, computers create animations for display in games and films by manipulating

algebraic equations.

Elementary analytic geometry

Apollonius of Perga[58]

(c. 262–190 bc), known by his contemporaries as the “Great

Geometer,” foreshadowed the development of analytic geometry by more than 1,800

years with his book Conics[59]

. He defined a conic as the intersection of a cone and a

plane (see figure). Using Euclid’s results on similar triangles and on secants of circles, he

found a relation satisfied by the distances from any point P of a conic to two

perpendicular lines, the major axis of the conic and the tangent at an endpoint of the axis.

These distances correspond to coordinates of P, and the relation between these

coordinates corresponds to a quadratic equation[60]

of the conic. Apollonius used this

relation to deduce fundamental properties of conics[61]

. See conic section[62]

.

Further development of coordinate systems (see figure) in mathematics[63]

emerged only

after algebra[64]

had matured under Islamic and Indian mathematicians. (See

mathematics: The Islamic world (8th–15th centuries) and mathematics, South Asian[65]

.)

At the end of the 16th century, the French mathematician François Viète[66]

introduced

the first systematic algebraic notation, using letters to represent known and unknown

numerical quantities, and he developed powerful general methods for working with

algebraic expressions and solving algebraic equations. With the power of algebraic

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notation, mathematicians were no longer completely dependent upon geometric figures

and geometric intuition to solve problems. The more daring began to leave behind the

standard geometric way of thinking in which linear (first power) variables corresponded

to lengths, squares (second power) to areas, and cubics (third power) to volumes, with

higher powers lacking “physical” interpretation. Two Frenchmen, the mathematician-

philosopher René Descartes

[67]

and the lawyer-mathematician Pierre de Fermat

[68]

,were among the first to take this daring step.

Descartes[69]

and Fermat[70]

independently founded analytic geometry in the 1630s by

adapting Viète’s algebra to the study of geometric loci. They moved decisively beyond

Viète by using letters to represent distances that are variable instead of fixed. Descartes

used equations to study curves[71]

defined geometrically, and he stressed the need to

consider general algebraic curves—graphs of polynomial equations in x and y of all

degrees. He demonstrated his method on a classical problem: finding all points P such

that the product of the distances from P to certain lines equals the product of the distances

to other lines. See geometry: Cartesian geometry.

Fermat emphasized that any relation between x and y coordinates determines a curve[72]

(see figure). Using this idea, he recast Apollonius’s arguments in algebraic terms and

restored lost work. Fermat indicated that any quadratic equation[73]

in x and y can be put

into the standard form of one of the conic sections.

Fermat did not publish his work, and Descartes deliberately made his hard to read in

order to discourage “dabblers.” Their ideas gained general acceptance only through the

efforts of other mathematicians in the latter half of the 17th century. In particular, the

Dutch mathematician Frans van Schooten translated Descartes’s writings from French to

Latin. He added vital explanatory material, as did the French lawyer Florimond de

Beaune, and the Dutch mathematician Johan de Witt. In England, the mathematician John

Wallis[74]

popularized analytic geometry, using equations to define conics and derive

their properties. He used negative coordinates freely, although it was Isaac Newton[75]

who unequivocally used two (oblique) axes to divide the plane into four quadrants, as

shown in the figure.

Analytic geometry had its greatest impact on mathematics via calculus[76]

. Without

access to the power of analytic geometry, classical Greek mathematicians such as

Archimedes[77]

(c. 285–212/211 bc) solved special cases of the basic problems of

calculus: finding tangents and extreme points (differential calculus) and arc lengths,


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areas, and volumes (integral calculus). Renaissance mathematicians were led back to

these problems by the needs of astronomy, optics, navigation, warfare, and commerce.

They naturally sought to use the power of algebra to define and analyze a growing range

of curves.

Fermat developed an algebraic algorithm for finding the tangent to an algebraic curve

[78]

at a point by finding a line that has a double intersection with the curve at the point—in

essence, inventing differential calculus. Descartes introduced a similar but more

complicated algorithm using a circle. Fermat computed areas under the curves y = axk for

all rational numbers k ≠ −1 by summing areas of inscribed and circumscribed rectangles.

(See exhaustion, method of [79]

.) For the rest of the 17th century, the groundwork for

calculus was continued by many mathematicians, including the Frenchman Gilles

Personne de Roberval[80]

, the Italian Bonaventura Cavalieri[81]

, and the Britons James

Gregory[82]

, John Wallis, and Isaac Barrow[83]

.

Newton[84]

and the German Gottfried Leibniz[85]

revolutionized mathematics at the end

of the 17th century by independently demonstrating the power of calculus. Both men

used coordinates to develop notations that expressed the ideas of calculus in full

generality and led naturally to differentiation rules and the fundamental theorem[86]

of

calculus (connecting differential and integral calculus). See analysis[87]

.

Newton demonstrated the importance of analytic methods in geometry, apart from their

role in calculus, when he asserted that any cubic—or, algebraic curve of degree

three—has one of four standard

equations, xy2 + ey = ax3 + bx2 + cx + d , xy = ax3 + bx2 + cx + d , y2 = ax3 + bx2 + cx + d , y = ax3 + bx2 + cx + d ,

for suitable coordinate axes. The Scottish mathematician James Stirling[88]

proved this

assertion in 1717, possibly with Newton’s aid. Newton divided cubics into 72 species, a

total later corrected to 78.

Newton also showed how to express an algebraic curve near the origin in terms of the

fractional power series y = a1 x1/k + a2 x2/k + … for a positive integer[89]

k .

Mathematicians have since used this technique to study algebraic curves of all degrees.

Analytic geometry of three and more dimensions

Although both Descartes and Fermat suggested using three coordinates to study curves


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and surfaces in space, three-dimensional analytic geometry developed slowly until about

1730, when the Swiss mathematicians Leonhard Euler[90]

and Jakob Hermann and the

French mathematician Alexis Clairaut[91]

produced general equations for cylinders,

cones, and surfaces of revolution. For example, Euler and Hermann showed that the

equation[92]

f ( z) = x2 + y2 gives the surface[93]

that is produced by revolving the curve

f ( z) = x2 about the z-axis (see the figure, which shows the elliptic paraboloid z = x2 + y2).

Newton made the remarkable claim that all plane cubics arise from those in his third

standard form by projection between planes. This was proved independently in 1731 by

Clairaut and the French mathematician François Nicole. Clairaut obtained all the cubics

in Newton’s four standard forms as sections of the cubical

cone zy2 = ax3 + bx2 z + cxz2 + dz3 consisting of the lines in space that join the origin

(0, 0, 0) to the points on the third standard cubic in the plane z = 1.

In 1748 Euler used equations for rotations and translations in space to transform the

general quadric surfaceax2 + by2 + cz2 + dxy + exz + fyz + gx + hy + iz + j = 0 so that its

principal axes coincide with the coordinate axes. Euler and the French mathematicians

Joseph-Louis Lagrange[94]

and Gaspard Monge[95]

made analytic geometry

independent of synthetic (nonanalytic) geometry.

Vector analysis[96]

In Euclidean space[97]

of any dimension, vectors[98]

—directed line segments—can be

specified by coordinates. An n-tuple (a1, …, an) represents the vector in n-dimensional

space that projects onto the real numbers a1, …, an on the coordinate axes.

In 1843 the Irish mathematician-astronomer William Rowan Hamilton[99]

represented

four-dimensional vectors algebraically and invented the quaternions[100]

, the first

noncommutative algebra to be extensively studied. Multiplying quaternions with one

coordinate zero led Hamilton to discover fundamental operations on vectors.

Nevertheless, mathematical physicists found the notation used in vector analysis [101]

more flexible—in particular, it is readily extendable to infinite-dimensional spaces. The

quaternions remained of interest algebraically and were incorporated in the 1960s into

certain new particle physics[102]

models.


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Projections

As readily available computing power grew exponentially in the last decades of the 20th

century, computer animation[103]

and computer-aided design[104]

became ubiquitous.

These applications are based on three-dimensional analytic geometry. Coordinates are

used to determine the edges or parametric curves that form boundaries of the surfaces of

virtual objects. Vector analysis is used to model lighting and determine realistic shadings

of surfaces.

As early as 1850, Julius Plücker[105]

had united analytic and projective geometry [106]

by introducing homogeneous coordinates that represent points in the Euclidean plane (see

Euclidean geometry [107]

) and at infinity in a uniform way as triples. Projective

transformations, which are invertible linear changes of homogeneous coordinates, are

given by matrix[108]

multiplication. This lets computer graphics programs efficiently

change the shape or the view of pictured objects and project them from three-dimensional

virtual space to the two-dimensional viewing screen.

Robert Alan BixHarry Joseph D’Souza

Additional Reading

Carl B. Boyer, History of Analytic Geometry (1956, reissued 1988), traces the early

development of analytic geometry. Julian Lowell Coolidge, A History of Geometrical

Methods (1940, reissued 1963), provides proofs of important results in the history of

analytic geometry. Gordon Fuller and Dalton Tarwater, Analytic Geometry, 7th ed.

(1992, reissued 1994), is a classic introduction to the subject. Robert Bix, Conics and Cubics: A Concrete Introduction to Algebraic Curves (1998), provides a transition from

analytic to algebraic geometry.

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