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# Advances in the Study of Curves and Surfaces

## Overview

Eighteenth-century mathematicians enjoyed a vastly expanded set of techniques that could be applied to the study of curves and surfaces and a vastly expanded set of reasons to study them. Problems of projectile and planetary motion required a renewed understanding of the conic sections. Problems from engineering and the need for accurate maps of Earth's curved surface drew special attention to the general problems of representing curves and surfaces by equations. The researches of Jacob Hermann, Leonhard Euler, Gaspard Monge, and others would lead to the new disciplines of descriptive and differential geometry.

## Background

The ancient Greeks had a good understanding of those curves generated by the conic sections—the hyperbola, parabola, ellipse, and circle—but from a geometrical perspective only. With the invention of analytic geometry by the French mathematician and philosopher René Descartes (1596-1650) and French mathematician Pierre de Fermat (1601-1665), such curves were all understood to be described by algebraic equations. The study of curves and curved surfaces received new impetus in the eighteenth century from both science and technology. Given the laws of motion and gravitation of Sir Isaac Newton (1642-1727), it was now possible to calculate the trajectory of projectiles and planets with accuracy. Practical questions also arose about the shapes of freely hanging chains and beams under stress, the answers to which were important in the construction of buildings and bridges.

One significant innovation was the introduction of polar coordinates. In Cartesian coordinates the position of a point was specified by giving its distance from two, usually perpendicular, lines. In polar coordinates, the location of a point is specified by its distance from a single reference point and an angle of rotation measured with respect to a fixed line passing through the reference point. Polar coordinates provided a more natural description of curves like the circle and the ellipse, and were especially suited to describing the motion of particles around a center of gravitational attraction. The polar coordinate system had been introduced in 1671 by Newton, but as that work was not published in English until 1736, credit for the invention of polar coordinates is usually assigned to the Swiss mathematician Jakob Bernoulli (1654-1705), who published a paper on the subject in 1691. Bernoulli introduced a number of important curves, including the figure eight-shaped lemniscate and the logarithmic spiral. The general utility of polar coordinates was demonstrated by Swiss mathematician Jacob Hermann (1678-1729), who also gave the general relationship between Cartesian and polar coordinates. A second important innovation involved the parametric representation of curves. This method was introduced by the Swiss mathematician Leonhard Euler (1707-1783) in 1787 and allowed the specification of a curve given by a relation between two variables x and y, to be stated by separate relations between x and y and a third variable, say t. If x and y described the trajectory of a projectile, then t might represent the time.

While an equation in two variables would generally describe a curve in the plane, a single equation in three dimensions would define a surface in three-dimensional space. Three-dimensional Cartesian coordinates were essentially introduced in their modern form by Swiss mathematician Johann Bernoulli (1667-1748), brother of Jakob. With equations in three variables representing a surface, a question naturally arose as to whether two equations might represent the same surface, only with different coordinate axes. In 1848 Euler presented a general analysis of the second-degree algebraic equation in three variables: in which he established a general procedure for rotating the axes of one three-dimensional coordinate system into those of another by a series of three rotations. The angles of the three rotations would come to be known as the Euler angles. By finding axes that simplified the equation through the elimination of one or more terms, Euler was able to classify all such surfaces as belonging to one of six special cases: cones, cylinders, ellipsoids, hyperboloids, hyperbolic paraboloids, and parabolic cylinders.

One of the most important figures in eighteenth-century geometry was the French engineer, scientist, and mathematician Gaspard Monge (1746-1818). The son of a merchant, Monge received some education at a local religious school, and at the age of 16 he was allowed to attend classes at Ecole Royale du Genie at Mezieres, a select military school that emphasized science and engineering. His admission to the school had come through the intervention of a military officer who was impressed by the quality of a map made by Monge using surveying instruments he had made himself. Within a few years Monge was appointed to the staff of the school, but because of his family's low social status, he was at first restricted to drafting and working as a technician. Assigned to design a fortification, Monge discovered a way to replace much of the tedious arithmetic work that had gone into calculating lines of fire and lines of sight with a much more direct geometrical method. He was appointed a professor in 1768 so that he could teach his methods to military engineers. In the process he developed the basic notions of descriptive geometry, the representation of three-dimensional objects through a combination of top and side projections that is now the basis of mechanical drawing. Monge's techniques would be treated as a military secret by France until 1794.

In 1771 Monge presented a number of memoirs to the Academy of Sciences in Paris. One of these concerned the geometric representation of the solutions to partial differential equations. While an algebraic equation in three variables would describe a single surface, a partial differential equation, involving three variables and their rates of change with respect to each other, would describe a family of related surfaces. For example, all of the spheres with a specified radius and centers in the xy plane would be allowed solutions to just one such equation, with the x and y coordinates of their centers serving to distinguish one sphere from another. Monge introduced additional geometrical concepts, such as that of the characteristic curve and characteristic cone, that allowed one in a sense to visualize all the solutions at once.

In 1775 Monge turned to the theory of developable surfaces. A developable surface is one that can be "flattened out" into a plane in such a way that distances between points are accurately represented. Euler had established the basic ideas of this field three years earlier. Interest in it was motivated by the fundamental problem of mapmakers, that of displaying the features of the Earth's surface on a flat piece of paper in such a way that the distance between points is accurately represented. The sphere is clearly not a developable surface, but there was considerable interest in finding developable surfaces that approximated portions of a sphere.

Like many of the thinkers of his time, Monge did not confine himself to mathematics but also was interested in physics and chemistry. He was the first scientist to produce water by burning hydrogen gas and he worked on the storage of hydrogen for use in balloons.

In 1794, following the French Revolution, Monge became a member of the Commission of Public Works, set up by the government to establish an institution of higher education for engineers. The school would become the Ecole Polytechnique, which would attract many of the best mathematical minds of the time to its faculty. There Monge himself taught in two different areas. The first was the descriptive geometry he had earlier invented. From his lecture notes it is known that he addressed a number of fundamental mathematical problems, such as determining the curve defined by the intersection of two surfaces.

Monge also taught a course in what would come to be known as differential geometry, the application of the calculus to curves and surfaces in three dimensions. This was in part an outgrowth of his earlier work on characteristic curves of partial differential equations and on developable surfaces. His 1795 text on the applications of analysis to geometry was the first major work in the area.

## Impact

Euler's investigation of the behavior of curves under rotation would have important implications for subsequent studies of the physics of rotating bodies. The Euler angles, in particular, form the basis of the theory of rotating bodies in physics, a theory that includes the rotational behavior of planets, spacecraft, gyroscopes, molecules, and atoms. The study of the behavior of curves and surfaces under coordinate transformations would, in the late nineteenth century, be turned into a study of "algebraic invariants," properties of an algebraic expression that would not change under certain types of coordinate transformation. This area of mathematics would combine with other approaches to the study of mathematical transformations to produce the modern mathematical theory of groups, which provides the most general mathematical framework for the description of symmetry and has turned into a powerful tool in elementary particle physics.

Both technical understanding of the behavior of curves and surfaces under rotations and the descriptive geometry of Monge are, of course, very important in the new field of computer graphics, and computer-aided design (CAD). In modern engineering practice the design of any high performance device, say a new jet engine, is in its initial stages almost entirely done using computer visualization software. Only once the components and their function are simulated and viewed from a great many vantage points are the first prototypes built.

DONALD R. FRANCESCHETTI