The realization process of butterfly theorem of conic curve can be described by parameter equation. Suppose we mark the coordinates of focus P as (f, 0) and the coordinates of directrix L as (x, y), where (x, y) is not equal to (f, 0). Then the parametric equation of the generated conic curve is as follows: x=A* sin(t), y=B*sin(c*t).

Where a and b are amplitude parameters, which determine the shape and size of the curve. C is a frequency parameter, which affects the number and period of curve fluctuation. Parameter t is a variable value, usually an integer, which is used to represent the change of the curve in a period.

By adjusting the values of a, b and c, we can control the shape of the generated conic curve. For example, when A= 1, B=2, and c=3, the generated curve is a butterfly-like pattern. When the parameter value changes, the shape will change accordingly.

First of all, the butterfly theorem:

Butterfly theorem is one of the most wonderful results in ancient Euclidean plane geometry. This proposition first appeared in 18 15 and was proved by W.G. Horner. The name "Butterfly Theorem" first appeared in the February issue of American Mathematical Monthly (1944), with the title like a butterfly. There are countless proofs of this theorem, math lovers are still studying it, and there are various deformations in the exam.

Second, the conical butterfly theorem:

The application of Cone Butterfly Theorem is diverse. In physics, Vercas curve can be used to study vibration, fluctuation and * * * vibration. In the field of music, it can be used to visualize the relationship between tone and harmony. In electronic engineering, Vercas curves can be used to generate complex signal patterns, such as image display and modulation.

In a word, the Cone Butterfly Theorem is an interesting and widely used mathematical phenomenon, which shows the beauty and diversity of mathematics and also brings many opportunities for practical application.