Kepler's three laws are key achievements in the history of modern astronomy, which laid the foundation for the discovery of Newton's laws of motion. The mathematical basis of Kepler's three laws mainly includes the concepts of ellipse, focus, constant and semi-major axis.

First of all, an ellipse is the trajectory of a moving point P, and the sum of the distances between the moving point P and the fixed points f 1 and f2 on the plane is equal to a constant (greater than the focal length), and f 1 and f2 are called the two focuses of the ellipse. The properties of an ellipse are as follows: for any point p, the sum of the distances to the two focal points F 1 and F2 is equal, and this equal distance is called the long axis, and the length of the long axis is 2a; The difference between the distances to the two focal points F 1 and F2 is equal. This equal distance is called the minor axis, and the length of the minor axis is 2b.

Secondly, the focal point is the point on the ellipse where the sum of the distances between two focal points is equal to a constant. For any point p on the ellipse, the sum of the distances to the two focal points F 1 and F2 is equal to the constant 2a.

A constant is a fixed number, that is, the sum of f 1P and f2P is fixed. In an ellipse, the constant 2a is the length of the major axis, which is a fixed value. Finally, the semi-major axis is half of the major axis of the ellipse, that is, A. The semi-major axis is an important parameter of the ellipse, which determines the size and shape of the ellipse.

Kepler's first law shows that the orbits of all planets around the sun are elliptical, and the sun is at a focus of the elliptical orbit. The mathematical basis of this law is the nature of ellipse and the concept of focus. Kepler's second law shows that for any planet, its line with the sun sweeps the same area at the same time. The mathematical basis of this law is the concept of area.

To sum up, the mathematical basis of Kepler's three laws mainly includes the concepts of ellipse, focus, constant and semi-major axis. These mathematical concepts played a key role in Kepler's research and laid the foundation for the discovery of Newton's laws of motion. For any point p, the sum of the distances to the two focal points F 1 and F2 is equal. This isometric is called the major axis, this isometric is called the minor axis, and the length of the minor axis is 2b.