Zawirowany świat ekonomii - próba pomiaru wirów równowagi - Juzwiszyn, Jacek (Wydawnictwo Akademii Ekonomicznej im. Oskara Langego we Wrocławiu, 2007)
While observing the development of market indexes in the three-dimensional space in the system (price - value, volume - time), a rotational spiral curve is encountered. This curve traces cones in R₊³. Orthogonal projections of this curve onto the planes going through the cone axis are two-dimensional stock market zigzags in systems (price - time, volume - time). The shape of the orthogonal projection of the trajectory onto the plane perpendicular to the cone's axis resembles logarithmic spiral or, known from economics, so called arachnoic model formed in the system (volume - price). The model parametric equation of rotational spiral market trajectory is of the following form: x(φ) = a cos φ exp(λφ), y(φ) = a sin φ ехр(λφ), z(φ) = sl⁻¹exp(λφ). Where a, λ∈ R₊, λ = ctg ψ = const , l, s - the values of, respectively, the radius and the altitude of the cone. When z(φ) = 0 , we obtain a logarithmic spiral equation in R² (in polar coordinates) i.e. r=аехр(λφ). Determining the parameters of market trajectories in R₊³ allows market to be represented by cones whose axes form the trajectory of market equilibrium. (original abstract)
Prace Naukowe Akademii Ekonomicznej we Wrocławiu; 2007; nr 1183, s. 133-146