Like space curve theory, the establishment of surface theory is also a rather long process. The surface theory begins with the study of geodesics on the surface (the earth). 1697 johann bernoulli q: how to find the shortest arc between two points on a convex surface? 1698, he wrote to Leibniz that the compact plane (compact circular plane) of any point of geodesic is perpendicular to the surface of that point. In the same year, his brother James Bernoulli solved the geodesic problems on cylinders, cones and rotating surfaces. Thirty years later, johann bernoulli used his brother's method to find geodesics of other surfaces, but James Bernoulli's method had limitations.

In 1728, Euler gave the differential equation of geodesic on the surface by the method he introduced in the variational method, and in 1732, Jacob Herman also calculated the geodesic on some special surfaces. Clairaux fully discussed geodesics on the surface of revolution in his works 1733 and 1739, and proved that the sine of the angle between geodesic and any meridian passing through geodesic is inversely proportional to the vertical distance from the intersection point to the rotation axis. He also proved that if a plane passes through any point m on the surface of revolution and is perpendicular to the surface and the meridian plane passing through the point m, the radius of curvature of the intersection of the plane and the surface at the point m is equal to the length of the normal between the point m and the rotation axis. Although he used analytical methods, he didn't have the idea of variational method.

1760, Euler published "Research on Curves on Surfaces", in which the surface theory was established, which was a milestone in the development history of differential geometry. He expressed the surface as z=f(x, y) and introduced modern standard symbols. First, he found the expression of the curvature radius of any plane section of a surface, and then applied the result to the normal section. He defined the normal line perpendicular to the xy plane as the main normal line, and obtained the radius of curvature of the normal line. He wanted to find the maximum curvature and minimum curvature of all normals at a point on the curve, and found that there are two roots with a difference of 90, that is, the two normal planes are perpendicular to each other. We call these two curvatures principal curvatures κ 1 and κ2. According to Euler's results, the curvature κ of any normal section line that forms an angle α with any normal section line where the principal curvature is located is: κ = κ 1cos 2α+κ 2sin 2α, and this result is called euler theorem. 、

Jean Baptiste Marie (J.B.M. Meusnier de la Place, translated in Chinese as Mesny, 1754- 1793), a student from gaspard monge, got the same result in a more subtle way in 1776. Mesny and lavoisier studied fluid dynamics and chemistry together. He dealt with the curvature of the illegal section (Euler has a complicated expression), which is called Mesny theorem: the curvature of the plane section of a surface at point P is the sine of the angle between the original plane and the tangent plane at point P divided by the curvature of the normal section of the same tangent at point P, and he proved that two surfaces with equal principal curvatures only have a plane and a sphere. His paper made many intuitive results in the18th century.

The demand of drawing maps has developed a main field of surface theory: the study of developable surfaces, that is, surfaces that are flat on a plane without deformation and whose shape is close to a sphere. Euler was the first person to study this problem. /kloc-the curved surface of the 0/8th century is considered as the boundary of a solid, so he thinks that a solid surface can be flattened on a plane. He introduced the parametric representation of the surface, trying to find out what conditions the function meets to make the surface expand on the plane. He derived the necessary and sufficient conditions for developability, and the equation is equivalent to the line element on the surface and the line element on the plane.

Then Euler studied the relationship between space curve and developable surface, and proved that any tangent family of space curve fills or constitutes developable surface. He tried to prove that every developable surface is a ruled surface (a surface generated by linear motion), and the inverse theorem was also established, but failed (in fact, the inverse theorem was not established).

Gaspard monge independently studied the topic of developable surfaces. He combines analytic method with geometric method, and is the second representative figure in the field of integrated geometry after Descartes. Gaspard monge's work in descriptive geometry (for architecture), analytic geometry, differential geometry, ordinary differential equations and partial differential equations won Lagrange's admiration and envy. He also made many contributions in physics, chemistry (he worked with Mesny and lavoisier), metallurgy and mechanics. He saw the demand for science in industrial development and advocated industrialization to improve people's livelihood. Perhaps because he was born in poverty and understood the sufferings of the bottom, he was enthusiastic about social affairs. After the French Revolution (probably around 1792), he served as the minister of the navy and a member of the public health committee in the government, and made friends with Napoleon, who has not yet become famous, but he can't remember it himself. Later, as the revolution intensified, gaspard monge was almost killed by the masses and saved by Napoleon. He designs weapons and guides government officials with technical ideas. He is a supporter of Bonaparte (but after reading Baidu Encyclopedia, I don't think he worships Napoleon). Later, the Bourbon Dynasty was restored, which made the future of genius bleak. Gaspard monge helped organize many technical schools and set up the Institute of Geometry (the descriptive geometry he founded was required to sign a confidentiality agreement because it was so powerful that it was not publicly taught in Paris until many years later). He is a great teacher, and at least 65,438+02 students are famous figures in the early 65,438+09 century.

Gaspard monge's contribution to three-dimensional differential geometry far exceeds Euler's. 1795 published a paper, which systematized and expanded the past achievements, put forward some new important results, and translated the properties of curves and surfaces into the language of partial differential equations. When seeking to analyze the correspondence with geometry, he realized that a family of surfaces with the same geometric properties or defined by the same generation method should satisfy a partial differential equation.

Gaspard monge's first important work is about the developable surface of hyperbolic curve, and studies the space curve and its related surfaces. He thinks that space curve is the projection of two intersecting straight lines or two mutually perpendicular planes in space. He called the extreme position where the normal plane intersects with the adjacent normal plane the polar axis. When moving along a curve, the envelope of the normal plane is an developable surface, which is called extremely developable surface. In order to solve the equation of extremely developable surface, he gave the normal plane equation, and then gave the rule of finding the envelope of single parameter plane family, which is still applicable to single parameter surface family.

Gaspard monge also studied the ridgeline of developable surfaces, which is formed by a set of straight lines that generate surfaces. Ridge divides a developable surface into two leaves, just as a sharp point divides a plane curve into two parts. Gaspard monge got the ridgeline equation. On the extremely developable surface, the ridge line is the locus of the curvature center of the original space curve.

1775, gaspard monge published a paper on developable surfaces in the theory of shadow and semi-shadow, which intuitively explained that developable surfaces are ruled surfaces (not the other way around). On the ruled surface, two adjacent straight lines are * * * points or parallel, and any developable surface is equivalent to the surface generated by the tangent of the space curve. In this paper, he gave the general representation of developable surface, and then gave the general representation of ruled surface, which is a special ruled surface.

In 1776, he studied how to move soil from one place to another most effectively. In fact, the focus of this article is not application, but geometric results. He started with dealing with the intersection of a family of straight lines or lines with two parameters, and then followed the work of Euler and Mesny, considering the normal family of surface S, and the surface normals of curvature lines constitute developable surfaces, which are called normal developable surfaces (these terms make me out of my mind ...). Similarly, the surface normal along the curvature line perpendicular to the first curvature line also constitutes a developable surface. Because there are two families of curvature lines on the surface, there are two families of developable surfaces that are orthogonal to each other. All the ridges of the developable surface family form a surface, which is called the central surface. The envelope of each family of developable surfaces is called focal plane.

Gaspard monge's research work on surface families satisfying nonlinear, linear first-order, second-order and third-order partial differential equations is of great significance to partial differential equations. He likes to clarify his ideas by discussing specific curves and surfaces. The popularization and application of his ideas were realized by mathematicians in the19th century. Gaspard monge faced the reality. In his paper 1795, he ended with how to apply theory to building construction.

Pierre Charles Fran? ois Pandit (1784- 1873), a student from gaspard monge, also contributed to the surface theory. Pandy is a shipbuilding engineer and also pays attention to application. One of his contributions is the Pandit index line, which clarifies the previous results of Euler and Mesny. Given the tangent plane of a surface at point m, draw a line segment in each direction from point m to the tangent plane, the length of which is equal to the square root of the radius of curvature of the normal of the surface in that direction, and the trajectory of the end points of these line segments is a quadratic curve, that is, the index line. Through the fact that the curves with the maximum and minimum curvatures of m points on the surface are curves tangent to m points with the axis of exponential line, Pandy also gives the theorem that three families of orthogonal surfaces intersect on the curvatures of each surface (curves with the maximum or minimum normal curvature).

Pandy popularized the results of online exchange in gaspard monge. If a line confluence is orthogonal to a family of surfaces, it is said to be orthogonal. Marius Marius, a French physicist (1775- 18 12), used gaspard monge's results to prove that the normal from a point is still normal after being reflected or refracted on a curved surface. 18 16 Pandy proved that this theorem still holds for any normal confluence after any number of reflections. Later, Kettler Lambert Adolphe Jacques Quetlet (1796-1874) proved that a normal line sink is still a normal line sink after repeated refraction. Line convergence and line bundle (a family of three-parameter curves introduced by Marius) are the research topics of many mathematicians in the19th century.