The secant circle method starts with a known circle inscribed with a regular hexagon, and then a regular polygon is constructed by multiplying the number of sides. The more sides this regular polygon has, the closer its shape is to a circle. When the number of sides approaches infinity, a regular polygon approaches a complete circle.
Liu Hui found that when the number of sides of a regular polygon is 12, its area is very close to that of a circle. So he can get an approximate pi value by calculating the area of this regular polygon. Specifically, he found that the area of 12 polygon is about 3. 14, while that of 96 polygon is about 3. 14 16, and that of 192 polygon is about 3.14/kloc-.
By this method, Liu Hui got a very high-precision approximate value of pi. His calculation results are very close to those used in modern mathematics, hundreds of years earlier than those of western mathematicians.
Although the principle of secant circle method is simple, it shows the superb attainments of ancient Chinese mathematicians in mathematical theory. It not only created a new method to find pi, but also provided a way for later mathematicians to solve similar problems.
The use of cyclotomy:
1, calculate the area and perimeter of a circle: the most basic application of secant is to calculate the area and perimeter of a circle. By increasing the number of divisions of cyclotomy, the circle can be divided into more and more small pieces with the same area, thus obtaining more and more accurate area and circumference of the circle. This method can be used in various practical applications, such as calculating land area and designing graphics.
2. Study circles and curves: Secant can be used not only to calculate the area and perimeter of a circle, but also to study the basic properties of circles and curves. By increasing the number of division, the circle can be divided into smaller and smaller arcs, thus obtaining more and more accurate standard equations and polar coordinates equations of the circle. This method can be used to study the basic properties and geometric structure of curves, such as curvature and torsion of curves.
3. Numerical approximation and computer graphics: Secant is also a numerical approximation method, which can be used to find approximate solutions of some mathematical problems. For example, we can use secant technology to solve the approximate area and perimeter of a circle, thus obtaining approximate solutions to some mathematical problems. In addition, cyclotomy is also widely used in computer graphics. For example, when drawing a circle or curve, you can use the circle cutting method to generate a pixel image that is closer to the real shape.