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We analyze two recently-introduced flow measured that are based on a single trajectory only: trajectory stretching exponent (TSE) to detect hyperbolic (stretching) behavior, and trajectory angular velocity (TRA) to detect elliptic (rotation) behavior. Haller et al. [2021] and Haller et al. [2022] introduced TSE, TRA as well as the concept of quasi-objectivity, and formulated theorems about the objectivity and quasi-objectivity of TSE and TRA. In this paper, we present two counter-examples showing that all theorems in Haller et al. [2021] and Haller et al. [2022] are incorrect.

Reference frame optimization is a generic framework to calculate a spatially-varying observer field that views an unsteady fluid flow in a reference frame that is as-steady-as-possible. In this paper, we show that the optimized vector field is objective, i.e., it is independent of the initial Euclidean transformation of the observer. To check objectivity, the optimized velocity vectors and the coordinates in which they are defined must both be connected by an Euclidean transformation. In this paper we show that a recent publication [1] applied this definition incorrectly, falsely concluding that reference frame optimizations are not objective. Further, we prove the objectivity of the variational formulation of the reference frame optimization proposed in [1], and discuss how the variational formulation relates to recent local and global optimization approaches to unsteadiness minimization.

We introduce a representation of a 2D steady vector field \\( v\\) by two scalar fields \\(a\\), \\(b\\), such that the isolines of \\(a\\) correspond to stream lines of \\( v\\), and \\(b\\) increases with constant speed under integration of \\( v\\). This way, we get a direct encoding of stream lines, i.e., a numerical integration of \\( v\\) can be replaced by a local isoline extraction of \\(a\\). To guarantee a solution in every case, gradient-preserving cuts are introduced such that the scalar fields are allowed to be discontinuous in the values but continuous in the gradient. Along with a piecewise linear discretization and a proper placement of the cuts, the fields \\(a\\) and \\(b\\) can be computed. We show several evaluations on non-trivial vector fields.

Sparse trajectory data consist of a low number of trajectories such that the reconstruction of an underlying velocity field is not possible. Recently, approaches have been introduced to analyze flow behavior based on a single trajectory only: trajectory stretching exponent (TSE) to detect hyperbolic (stretching) behavior, and trajectory angular velocity (TRA) to detect elliptic (rotation) behavior. In this paper, we analyze these approaches and in particular show that they are -- contrary to what is claimed in the literature -- not objective in the extended phase space. Furthermore, we introduce the first objective measure of rotation behavior that is based on only few trajectories: at least 3 in 2D, and at least 4 in 3D. For this measure -- called trajectory vorticity (TRV) -- we show that it is objective and that it can be introduced in two independent ways: by approaches for unsteadiness minimization and by considering the relative spin tensor. We apply TRV to a number of constructed and real trajectory data sets, including drifting buoys in the Atlantic, midge swarm tracking data, and a simulated vortex street.