SOME PROPERTIES OF THE HYPERSPHERE IN N-DIMENSIONAL SPACE

The study of the properties of surfaces contributes to the expansion of their use in solving various practical problems, especially if such properties can be generalized to manifolds of n-dimensional space. The most thoroughly studied are the properties of the simplest surfaces, including the properties of a sphere. That is why the simplest surfaces are most often used in practice. Each property not covered in the existing literature expands the indicated possibilities. Therefore, the purpose of this article is to identify the properties of the hypersphere unknown from the literature. Most of the properties of a circle and a sphere have been known since ancient times [1, 4, 5]. The generalized concept of a sphere into multidimensional spaces is based on the general principles of multidimensional geometry [3]. In [4], eleven basic properties of the sphere are listed and analyzed. In works [8, 10] it is shown that a circle can be considered as an isoline, and a sphere as an isosurface when modeling energy fields. In geometric modeling of energy fields with point energy sources, an essential role is played by the distances from the points of the field to the given energy sources [6, 7]. In [9], two schemes are given for determining the parameter t, taking into account the effect of the distance from the points of the field to the point sources of energy on the potentials of the points of the field. In a particular case, if this parameter is determined according to a simplified scheme with f(l)=al2, then the formula for calculating the potential of an arbitrary point of the energy field is a mathematical model of the energy field generated by the number n of point energy sources. The geometric model of the field will be a manifold that can be foliated into a one-parameter set of isospheres [8, 10]. Abstracting from the physical nature of the field, simplifying the equation for calculating the potential of an arbitrary point of the energy field and generalizing it to n-dimensional space, we can formulate the following properties: Property 1. A hypersphere can be considered as a locus of points, the sum of the squared distances from which to n given points is a constant value. Property 2. Arbitrary coefficients ki at distances li affect the parameters of the hypersphere without changing the type of surface.