Plane analytic geometry is a mathematical subject that studies geometric problems by algebraic method on the basis of coordinate system. The main problem studied is the equation of (1) plane curve. (2) Through the equation, study the properties of plane curve and make the graph of curve.

The meaning of "analytic" in plane "analytic" geometry is to explain and analyze plane geometry by algebraic method.

The generation of analytic geometry (that is, where did you ask? )

/kloc-After the 6th century, due to the development of production and science and technology, astronomy, mechanics and navigation put forward new demands for geometry. For example, the German astronomer Kepler found that the planet runs around the sun along an ellipse, and the sun is at a focus of this ellipse; Italian scientist Galileo discovered that throwing objects tested parabolic motion. These findings all involve conic curves. In order to study these complex curves, the original set of methods is obviously not applicable, which leads to the emergence of analytic geometry.

1637, Descartes, a French philosopher and mathematician, published Methodology. There are three appendices at the back of this book, one is called refractive optics, the other is called meteorology, and the other is called geometry. At that time, this "geometry" actually referred to mathematics, just like "arithmetic" and "mathematics" in ancient China.

Cartesian geometry is divided into three volumes. The first volume discusses the rule drawing method. The second volume is the nature of the curve; The third volume is the drawing method of three-dimensional and "super-three-dimensional", which is actually an algebraic problem and discusses the properties of the roots of equations. Mathematicians and historians of mathematics in later generations regard Descartes' geometry as the starting point of analytic geometry.

It can be seen from Descartes' Geometry that Descartes' central idea is to establish a "universal" mathematics and unify arithmetic, algebra and geometry. He imagined that transforming any mathematical problem into an algebraic problem is to simplify any algebraic problem into solving an equation.

In order to realize the above hypothesis, Descartes pointed out the corresponding relationship between points on the plane and real number pairs (x, y) from the latitude and longitude system of astronomical geography. Different values of x and y can determine many different points on the plane, so we can study the properties of curves by algebraic method. This is the basic idea of analytic geometry.

Specifically, the basic idea of plane analytic geometry has two main points: one is to establish a coordinate system on the plane, and the coordinates of a point correspond to a set of ordered real number pairs; Secondly, after the coordinate system is established on the plane, a curve on the plane can be expressed by a binary algebraic equation. It can be seen that the application of coordinate method can not only solve geometric problems through algebraic methods, but also closely relate important concepts such as variables, functions, numbers and shapes.

The appearance of analytic geometry is not accidental. Before Descartes wrote geometry, many scholars used two intersecting straight lines as coordinate systems to study it. While studying astronomical geography, some people put forward that a position can be determined by two "coordinates" (longitude and latitude). All these have a great influence on the establishment of analytic geometry.

In the history of mathematics, it is generally believed that Fermat, a contemporary French amateur mathematician with Descartes, is also one of the founders of analytic geometry and should share the honor of the establishment of this discipline.

Fermat is an amateur scholar engaged in mathematical research and has made important contributions to number theory, analytic geometry and probability theory. He is modest and quiet, and has no intention of publishing his book. But from his correspondence, we know that Descartes had written a short article about analytic geometry long before he published Geometry, and he already had the idea of analytic geometry. It was not until 1679 that Fermat's thoughts and works were published in Letters to Friends.

Descartes' Geometry, as a work of analytic geometry, is incomplete, but it is important to bring forth the old and bring forth the new and make contributions to opening up a new garden of mathematics.

The Basic Content of Analytic Geometry

In analytic geometry, coordinate system is first established. As shown above, two mutually perpendicular straight lines with a certain direction and measurement unit on the plane are called rectangular coordinate system oxy. Using the coordinate system, a one-to-one relationship can be established between a point on a plane and a pair of real numbers (x, y). In addition to rectangular coordinate system, there are oblique coordinate system, polar coordinate system, spatial rectangular coordinate system and so on. There are also spherical coordinates and cylindrical coordinates in the spatial coordinate system.

The coordinate system establishes the close relationship between geometric objects and numbers, geometric relations and functions, which simplifies the study of spatial morphology into a relatively mature and easy-to-control quantitative relationship. Learning geometry in this way is usually called analytic method. This analysis method is not only important for analytic geometry, but also for studying various branches of geometry.

The establishment of analytic geometry introduced a series of new mathematical concepts, especially the introduction of variables into mathematics, which made mathematics enter a new development period, which is the variable mathematics period. Analytic geometry promotes the development of mathematics. Engels once commented: "The turning point in mathematics is Descartes' variable. With the change of books, sports entered mathematics; With variables, dialectics enters mathematics; With variables, differentiation and integration will become necessary immediately, ... "

Application of analytic geometry

Analytic geometry is divided into plane analytic geometry and space analytic geometry.

In plane analytic geometry, besides the properties of straight lines, the properties of conic curves (circle, ellipse, parabola and hyperbola) are mainly studied.

In spatial analytic geometry, besides the properties of plane and straight line, cylinders, cones and rotating surfaces are mainly studied.

Some properties of ellipse, hyperbola and parabola are widely used in production or life. For example, the reflector of the spotlight bulb of a movie projector is oval, the filament is in one focus and the movie door is in another focus; Searchlights, spotlights, solar cookers, radar antennas, satellite antennas and radio telescopes are all made by using the principle of parabola.

Generally speaking, analytic geometry can solve two basic problems by using coordinate method: one is to satisfy the trajectory of a given point and establish its equation through coordinate system; The other is to study the curve properties expressed by the equation through the discussion of the equation.

The steps of solving problems by coordinate method are as follows: firstly, establish a coordinate system on the plane and "translate" the geometric conditions of known point trajectories into algebraic equations; Then use algebraic tools to study the equation; Finally, the properties of algebraic equations are described in geometric language, and the answers to the original geometric problems are obtained.

The idea of coordinate method urges people to use various algebraic methods to solve geometric problems. It used to be regarded as a difficult problem in geometry, but once algebraic methods are used, it becomes bland. The coordinate method also provides a powerful tool for the mechanization proof of modern mathematics.