This paper presents a new method for studying transverse flow effects in relativistic nuclear collisions using Fourier analysis of azimuthal particle distributions. The method involves analyzing the Fourier coefficients of particle distributions in narrow rapidity windows to determine the magnitude and type of flow. Directivity and two-dimensional sphericity tensor emerge naturally in this approach, as they correspond to the first and second harmonic coefficients, respectively. The method accounts for finite particle fluctuations and correlations. The paper discusses two main approaches to flow analysis: one based on local thermal equilibrium and hydrodynamical expansion, and another that analyzes azimuthal event shapes directly from experimental data. The proposed method combines both approaches and provides a clear physical interpretation of the results. It is particularly useful for detecting anisotropic collective flow, which can arise from various sources such as hydrodynamical flow, shadowing, or other collective effects. The Fourier expansion method is applied to the azimuthal distribution of particles, which can be constructed from quantities such as transverse momentum, multiplicity, or transverse energy. The Fourier coefficients of the distribution provide information about the anisotropy of the flow. The first harmonic coefficient corresponds to directed flow, while the second harmonic coefficient describes the eccentricity of an elliptical distribution. Higher harmonics can describe more complex flow patterns. The paper also addresses the issue of finite multiplicity fluctuations, which can affect the accuracy of flow analysis. It shows that the distribution of Fourier coefficients is sensitive to anisotropic flow and can be used to detect it. The method is also applicable to the study of radial flow and can be used to analyze the three-dimensional event shape by correlating Fourier coefficients from different longitudinal windows. The paper discusses the relationship between the proposed method and other existing methods, such as directivity and sphericity analysis. It shows that the method can be used to study correlations between different types of flow and between different rapidity windows. The method is also applicable to the study of azimuthal two-particle correlations and can be used to analyze the effects of transverse flow on particle distributions. The paper concludes that the Fourier expansion method is a powerful tool for studying transverse anisotropic collective flow in relativistic nuclear collisions. It provides a clear physical interpretation of the results and is particularly useful for detecting anisotropic flow. The method is also applicable to the study of three-dimensional event shapes and can be used to analyze the effects of transverse flow on particle distributions.This paper presents a new method for studying transverse flow effects in relativistic nuclear collisions using Fourier analysis of azimuthal particle distributions. The method involves analyzing the Fourier coefficients of particle distributions in narrow rapidity windows to determine the magnitude and type of flow. Directivity and two-dimensional sphericity tensor emerge naturally in this approach, as they correspond to the first and second harmonic coefficients, respectively. The method accounts for finite particle fluctuations and correlations. The paper discusses two main approaches to flow analysis: one based on local thermal equilibrium and hydrodynamical expansion, and another that analyzes azimuthal event shapes directly from experimental data. The proposed method combines both approaches and provides a clear physical interpretation of the results. It is particularly useful for detecting anisotropic collective flow, which can arise from various sources such as hydrodynamical flow, shadowing, or other collective effects. The Fourier expansion method is applied to the azimuthal distribution of particles, which can be constructed from quantities such as transverse momentum, multiplicity, or transverse energy. The Fourier coefficients of the distribution provide information about the anisotropy of the flow. The first harmonic coefficient corresponds to directed flow, while the second harmonic coefficient describes the eccentricity of an elliptical distribution. Higher harmonics can describe more complex flow patterns. The paper also addresses the issue of finite multiplicity fluctuations, which can affect the accuracy of flow analysis. It shows that the distribution of Fourier coefficients is sensitive to anisotropic flow and can be used to detect it. The method is also applicable to the study of radial flow and can be used to analyze the three-dimensional event shape by correlating Fourier coefficients from different longitudinal windows. The paper discusses the relationship between the proposed method and other existing methods, such as directivity and sphericity analysis. It shows that the method can be used to study correlations between different types of flow and between different rapidity windows. The method is also applicable to the study of azimuthal two-particle correlations and can be used to analyze the effects of transverse flow on particle distributions. The paper concludes that the Fourier expansion method is a powerful tool for studying transverse anisotropic collective flow in relativistic nuclear collisions. It provides a clear physical interpretation of the results and is particularly useful for detecting anisotropic flow. The method is also applicable to the study of three-dimensional event shapes and can be used to analyze the effects of transverse flow on particle distributions.