Curves

Summary: Curves have many different definitions and applications in various fields of mathematics.

Intuitively, a curve might be thought of as a path, like that of a curveball. A curve is viewed and defined in several ways depending on the branch of mathematics. A curve can be defined as the one-dimensional continuous trajectory of an object in space moving in time, the intersection of two surfaces in space, the image of the unit interval under a continuous function, or the graph of a solution of a polynomial equation. Each of these approaches captures the intuitive idea of a curve in their respective domains; the first is more physical, the second is geometric, the third is topological, and the last is an algebraic view of a curve. Curves can be used to create figures, model paths of motion, or express relationships between variables.

There are many types of curves that are the focus of classroom investigations, including yield curves, which are important to investors, and the normal distribution or bell-shaped curve. Felix Klein is noted to have said, “Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions.” In education, a “learning curve” is a phrase that is meant to informally capture the notion of the change in knowledge over time. Algebraic and geometric curves are also important in school. Children study lines and circles in primary and middle school. They investigate their lengths and areas. By high school and college, they learn about parametric equations of curves and the area under a curve. In order to enrich classroom learning, mathematicians and mathematics historians created the National Curve Bank Web site.

Early History of the Study of Curves

The Greeks initiated the study of curves and discovered numerous interesting curves. Apollonius of Perga studied conic sections as the intersections of a plane and a cone by changing the angle of intersection. Diocles of Carystus invented the cissoid curve and used it in his attempts to solve the problem of doubling the cube. Nicomedes invented the conchoid curve and used it in his attempts to solve the problems of doubling the cube and trisecting an angle. Some have noted that aspects in the design of the columns of the Parthenon may resemble a conchoid of Nicomedes, although others present different curves as the model. Canon of Samos invented the spiral that was eventually called the “Archimedean spiral.”

This curve was utilized by Archimedes of Syracuse as a method to attempt to trisect an angle and square the circle. The Greek view of curves was geometric, since Greek mathematics was, essentially, geometry-centered. Hence, their study of curves usually was through some elaborate and often ingenious methods of construction. Besides the lack of analytical tools, their insistence of having concrete or mechanical methods of construction, and—more importantly—their attempts to solve some important problems of antiquity that later were shown to be unsolvable by ruler and compass constructions are some of the factors contributing to the Greek concept of curves.

Mathematicians, philosophers, and others introduced and investigated the geometry of many interesting curves long after the ancient Greeks. For example, Nicholas of Cusa lived in the fifteenth century. He is noted as the first of many to explore the cycloid, which was eventually known as the path of a point on a wheel as the wheel rolls along a straight line.

Developments Since the Seventeenth Century

With the introduction of analytic geometry in the seventeenth century, the theory of curves received a new impetus—expressing curves by equations would make their study much easier compared to doing it via elaborate geometrical constructions. Analytic geometry enabled mathematicians to focus on the intrinsic features of curves; discover and investigate new curves; study curves in a more systematic way, leading to their classification into algebraic versus transcendental categories; and apply the results to various physical problems, such as the long-standing problem of determining the orbits of planets or solving the problem of a hanging chain, which was posed by Jacob Bernoulli. Gottfried Leibniz, Christiaan Huygens, and Jacob Bernoulli’s brother Johann Bernoulli responded to the elder Bernoulli’s challenge with the equation of the catenary. In the eighteenth century, Guido Grandi investigated rhodonea curves that resemble roses and what was later to be known as the Witch of Agnesi, named because of a mistranslation of the example in Maria Agnesi’s famous calculus textbook.

Beginning with the seventeenth century, smooth curves have been an intense subject of investigation leading to determination of various features. Smooth curves, like lines, circles, parabolas, spirals, and helices, possess properties that make them amenable to numerous applications besides lacking any jagged behavior. For example, younger students learn that a straight line is the shortest path between two points in the plane, and mathematicians in the seventeenth century wondered about an analog for surfaces. A geodesic curve is locally a minimizing path; as a result, it is important in advanced mathematics and physics classes. Leonhard Euler published differential equations for geodesics in 1732. Mathematicians also investigated the classification of smooth curves. One invariant is the length of a curve. In general, length does not distinguish two different curves. It turns out that two other invariants, called the “curvature” and “torsion,” work much better for this purpose. Broadly speaking, at any point on the curve, the curvature measures the deviance of the curve from being a straight line, and the torsion function measures the deviance of the curve from being a plane curve. Furthermore, the fundamental theorem of curves states that these invariants determine the curve, a result that is proved in twenty-first-century college differential geometry classrooms using the Frenet–Serret Formulas. These are named for Jean Frédéric Frenet and Joseph Serret, who independently discovered them in the nineteenth century.

With further investigations by prominent mathematicians, like Carl Friedrich Gauss, Gaspard Monge, Jean-Victor Poncelet, and their students, the theory of curves, particularly smooth curves, matured into an active field of research. The findings in the theory of curves not only enriched the realm of curve studies, they also contributed to the development of new ideas that ended up revolutionizing mathematics in the nineteenth century. Broadly speaking, the general definition of a curve is topological; namely, a curve is defined as a continuous map from an interval to a space. Curves can be algebraic (those defined via algebraic equations). For instance, a plane curve can also be expressed by an equation

F(x, y) = 0

and a space curve can be expressed by two equations

F(x, y, z) = 0 and G(x, y, z) = 0.

A curve is algebraic when its defining equations are algebraic—a polynomial in x and y (and z). The cardioid, a heart-shaped curve whose Cartesian equation is

(x² + y² - 2ax)² = 4a(x² + y²)

and the asteroid, whose Cartesian equation is

where a is a constant, are algebraic curves.

Before analytic geometry, each of these curves had been expressed using geometric investigations; for example, a circle turning around a circle that sweeps out the cardioid, or wheels turning within wheels that form the asteroid. Transcendental curves cannot be defined algebraically and include the brachistochrone curve, also known as the “curve of fastest descent”; very complicated looking fractal curves, such as the Koch snowflake, named for Helge von Koch, who explored the geometry in a 1904 paper; and paradoxical sounding space-filling curves, discovered by Giuseppe Peano in 1890. The last two types of curves can be extremely jagged curves with no smooth components.

An algebraic curve of the form y²=x³+ax+b,where a and b are real numbers, satisfying the relation 4a³+27b²≠0, is called an “elliptic curve.” Geometrically, this condition ensures that the curve does not have any cusps, self-intersections, or isolated points. On the points of elliptic curves (including the point at infinity), one can define an operation by three points sum to zero, if and only if they are collinear. This interesting feature of elliptic curves, besides being an important algebraic structure to be studied on its own, also has found some astonishing applications, such as in cryptography for developing elliptic curve-based public-key cryptosystems. Elliptic curves are also important in number theory; they are effective tools in integer factorization problems. They also turned up as an instrumental tool in the proof of Fermat’s Last Theorem, named for Pierre de Fermat.

Bibliography

Boyer, C. B. “Historical Stages in the Definition of Curves.” National Mathematics Magazine 19, no. 6 (1945).

Lockwood, E. H. A Book of Curves. New York; Cambridge University Press, 1963.

National Curve Bank. http://curvebank.calstatela.edu/home/home.htm.

O’Connor, John, and Edmund Robertson. “MacTutor: Famous Curves Index.” http://www-history.mcs.st-andrews.ac.uk/Curves/Curves.html.