There are different opinions about the earliest discovery of conic curve. Some people say that when the ancient Greek mathematicians solved the "cubic product" problem, they found the conic curve: let x and y be the median term in the ratio of a and 2a, that is, then, then, then, and then, again. It is also said that ancient Greek mathematicians found the same result as the "cubic product" problem when studying the cross section between a plane and a cone. It is also believed that ancient astronomers discovered conic curves when making sundials. A sundial is an inclined disk with a rod perpendicular to the disk in the center. When the sun shines on the sundial, the motion of the rod shadow can be timed. At different latitudes, the top of the pole is drawn into different conic curves. However, the invention of sundial was lost in ancient times.

The most outstanding achievement in the early systematic study of conic curves can be said to be the ancient Greek mathematician Apollo (262~ 190). He and Euclid were contemporaries, and his magnum opus "Conic Curve" and Euclid's "Elements of Geometry" were considered as the pinnacles of ancient Greek geometry.

In "Conic Curve", Apolloni summed up the previous work (Menehomos of Plato School discovered the conic curve to solve the problem of multiple cubes), especially Euclid's work, carried out the work of removing the rough and keeping the fine, summarized and refined the previous achievements and made them systematic. On this basis, he put forward many original ideas of his own. There are 8 books and 487 propositions in the book, which completely cover the properties of conic curves, so that there is almost no room for future scholars to set foot in it for more than 1000 years.

As we all know, when cutting a biconical surface with a plane, circles, ellipses, parabolas, hyperbolas and their degenerated forms will be obtained: two intersecting straight lines, a straight line and a point, as shown in figure 1.

Here, only Apolloni's definition of conic curve is introduced. As shown in Figure 2, given a circle BC and a point A out of its plane, a straight line passing through A and moving along the circle generates a bipyramid.

This circle is called the base of the cone, and the straight line from A to the center is called the axis of the cone (not shown), which may not be perpendicular to the base.

Let a section of the cone intersect with the base line DE, and take a diameter BC of the base circle perpendicular to DE, so △ABC containing the axis of the cone is called the axis triangle. The axis triangle intersects the conic at p, P', PP' is not necessarily the axis of the conic, PP'm is the straight line determined by the intersection of the axis triangle and the section, and PM is not necessarily perpendicular to DE. Let QQ' be the chord of the conic parallel to DE, and QQ' be divided equally by PP', that is, VQ=QQ'.

Now it's af∨pm, passing through BM to F, and then PL⊥PM on the section. As shown in Figure 3, PL⊥PP'

For ellipses and hyperbolas, let l satisfy, for parabolas, let l satisfy. For ellipse and hyperbola, there is qv = PV VR, and for parabola, there is qv = PV PL, which are two conclusions that can be proved.

In these two conclusions, QV is called the ordinate line of conic curve, so the conclusion shows that the square of the ordinate line is equal to the area of a rectangle on PL. For ellipse, rectangle PSRV does not fill rectangle PLJV；; The hyperbolic case is VR & gtPL, and the rectangular PSRV exceeds the rectangular PLJV；; And parabola, short PLJV just fills up. Therefore, the original names of ellipse, hyperbola and parabola are called "deficient curve", "hypercurve" and "homogeneous curve" respectively. This is the definition of conic curve introduced by Apolloni.

The two conclusions given by Apolloni can also be easily expressed by modern mathematical symbols:

When approaching infinity, LS=0, that is, parabola, that is, the limit form of ellipse or hyperbola.

After the publication of Apolloni's "Conic Curve", there has been no new progress in the research of conic curve in the whole mathematical field in the13rd century. 1 1 century, Arabic mathematicians used conic curves to solve cubic algebraic equations. /kloc-Since the 20th century, conic curves have been introduced to Europe through Arabia, but there was still no breakthrough in the study of conic curves at that time. Until16th century, two things prompted people to make further research on conic curves. First of all, the German astronomer Kepler (157 1 ~ 1630) inherited Copernicus' Heliocentrism and revealed the fact that planets orbit the sun in elliptical orbits. Second, Italian physicist Galileo (1564 ~ 1642) concluded that the trajectory of oblique throwing motion of an object is a parabola. It is found that conic curve is not only a static curve attached to a conical surface, but also a common form of natural object motion. As a result, the machining method of conical curve began to have some small changes. For example, 1579 (Guido Baldo del Monte, 1545 ~ 1607) ellipse is defined as the trajectory of a moving point, and the sum of the distances from this point to two focal points is a fixed length. Thus changing the definition of conic curve in the past. However, this has not greatly promoted the research on the properties of conical curves, nor has it put forward more new theorems or new proof methods.

/kloc-At the beginning of the 7th century, under the influence of the new idea that mathematical objects can be continuously changed from one shape to another, Kepler made a new exposition on the properties of conic curves. He discovered the focus and eccentricity of conic curve, and pointed out that parabola also has a focus at infinity, and a straight line is a circle with its center at infinity. Therefore, he was the first to grasp the fact that an ellipse, a parabola, a hyperbola, a circle and a degenerate conic curve composed of two straight lines can be changed from one to another continuously, only considering the various moving modes of the focus. For example, an ellipse has two focal points F 1 and F2, as shown in Figure 4. If the left focus F 1 is fixed, considering the movement of F2, when F2 moves to the left, the ellipse gradually tends to be round, and when F 1 coincides with F2, it becomes a circle; When F2 moves to the right, the ellipse gradually tends to parabola, and when F2 reaches infinity, it becomes parabola. F2 is a hyperbola when it returns from the left to the axis of the conic from infinity. When F2 continues to move to the right and coincides with F 1, it is two intersecting straight lines, that is, a degenerate conic curve. This provides a logical and intuitive basis for the modern unified definition of conical curve.

With the establishment of projective geometry, the projection and silhouette methods, which were originally helpful to painters, may be used to study conical curves because of their natural connection with conical surfaces. In this respect, there are three French mathematicians, namely, Gillard Girard Desargues (Debate1591-161), Pascal (Pascal, 1623- 1662) and Rachel (. When two other French mathematicians Descartes and Fermat founded analytic geometry, people's understanding of conic sections entered a new stage. The research method of conic is different from Apollo, projection and silhouette, but develops to analytical method, that is, the equation of conic is obtained by establishing coordinate system, and then the conic is studied by using the equation, so as to get rid of geometric intuition and achieve abstract purpose, and at the same time, the research of conic is highly summarized and unified.

In the18th century, people discussed analytic geometry extensively. In addition to rectangular coordinate system, polar coordinate system was established, and these two coordinate systems can be converted to each other. In this case, the quadratic equation representing the conic curve is also transformed into several standard forms, or the parametric equation of the curve is introduced. 1745, Euler published Introduction to Analysis, which is an important work in the history of analytic geometry and a classic work in the study of conic curves. In this book, Euler systematically expounded the modern form of conic curve. Starting from the general quadratic equation, various situations of conic curves can always be transformed into one of the following standard forms through appropriate coordinate transformation:

After Euler, three-dimensional analytic geometry has also developed vigorously, and many important surfaces have been derived from conic curves, such as cylindrical surface, ellipsoid surface, hyperboloid, various paraboloids and so on.

In a word, conic plays an important role in mathematics and other fields of science and technology, as well as in our real life. People's research on it continues to deepen, and its research results have been widely used. This just reflects the purpose and law of people's understanding of things.