In astronomy, when considering the motion of cosmic bodies in orbits, the concept of "ellipse" is often used, since their trajectories are characterized by this particular curve. Consider the question in the article, what is the marked figure, and also give the formula for the length of the ellipse.
What is an ellipse?
According to the mathematical definition, an ellipse is a closed curve for which the sum of the distances from any of its points to two other defined points lying on the main axis and called foci is a constant. The following figure illustrates this definition.
In the figure, the sum of the distances PF 'and PF is 2 * a, that is, PF' + PF = 2 * a, where F 'and F are the foci of the ellipse, "a" is the length of its semimajor axis. The segment BB 'is called the minor axis, and the distance CB = CB' = b is the length of the minor axis. Here, point C defines the center of the figure.
The figure above also shows a simple method with a rope and two studs, which is widely used to image elliptical curves. Another way to get this figure is to perform a section of the cone at any angle to its axis, which is not equal to 90 o .
If the ellipse is rotated along one of its two axes, then it forms a three-dimensional figure, which is called a spheroid.
Ellipse Circle Formula
Although the figure in question is quite simple, its circumference can be precisely determined by calculating the so-called elliptic integrals of the second kind. However, the Hindu self-taught mathematician Ramanujan at the beginning of the 20th century proposed a rather simple ellipse length formula, which approaches the result of the noted integrals below. That is, the value of the considered value calculated from it will be slightly less than the actual length. This formula has the form: P ≈ pi * [3 * (a + b) - √ ((3 * a + b) * (a + 3 * b))], where pi = 3.14 is the number pi.
For example, let the lengths of the two semi-axes of the ellipse be equal to a = 10 cm and b = 8 cm, then its length P = 56.7 cm.
Everyone can verify that if a = b = R, that is, a regular circle is considered, then the Ramanujan formula reduces to the form P = 2 * pi * R.
Note that in school textbooks often given a different formula: P = pi * (a + b). It is simpler, but also less accurate. So, if it is applied to the considered case, then we get the value P = 56.5 cm.