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Traditional vector field visualization has a close focus on velocity, and is typically constrained to the dynamics of massless particles. In this paper, we present a novel approach to the analysis of the force-induced dynamics of inertial particles. These forces can arise from acceleration fields such as gravitation, but also be dependent on the particle dynamics itself, as in the case of magnetism. Compared to massless particles, the velocity of an inertial particle is not determined solely by its position and time in a vector field. In contrast, its initial velocity can be arbitrary and impacts the dynamics over its entire lifetime. This leads to a four-dimensional problem for 2D setups, and a six-dimensional problem for the 3D case. Our approach avoids this increase in dimensionality and tackles the visualization by an integrated topological analysis approach. We demonstrate the utility of our approach using a synthetic time-dependent acceleration field, a system of magnetic dipoles, and N-body systems both in 2D and 3D.

The exploration and analysis of multidimensional data can be pretty complex tasks, requiring sophisticated tools able to transform large amounts of data bearing multiple parameters into helpful information. Multidimensional projection techniques figure as powerful tools for transforming multidimensional data into visual information according to similarity features. Integrating this class of methods into a framework devoted to data sciences can contribute to generating more expressive means of visual analytics. Although the Principal Component Analysis (PCA) is a well-known method in this context, it is not the only one, and, sometimes, its abilities and limitations are not adequately discussed or taken into consideration by users. Therefore, knowing in-depth multidimensional projection techniques, their strengths, and the possible distortions they can create is of significant importance for researchers developing knowledge-discovery systems. This research presents a comprehensive overview of current state-of-the-art multidimensional projection techniques and shows example codes in Python and R languages, all available on the internet. The survey segment discusses the different types of techniques applied to multidimensional projection tasks from their background, application processes, capabilities, and limitations, opening the internal processes of the methods and demystifying their concepts. We also illustrate two problems, from a genetic experiment (supervised) and text mining (non-supervised), presenting solutions through multidimensional projection application. Finally, we brought elements that reverberate the competitiveness of multidimensional projection techniques towards high-dimension data visualization, commonly needed in data sciences solutions.

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