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In this video, we show the example data sets included in the paper, animated over time.

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These are four example data sets of three trajectories rotating around a central point on a circle in different manners.

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We show the motion of the particles in a
fixed global reference system.

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The trajectories are the result of a super-
position of two rotational movements:

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a rotation of the local reference
system with the angular speed p

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and a rotation of the particles in this local reference system with the angular speed q.

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When observed in this optimal reference system, their objective rotations become clear: we can see their varying direction and speed.

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In another, non-optimal reference system though (here taken from the top left example) their behavior becomes more obscure.

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We now show our new measure, TRV, on a number of data sets.

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These trajectories show drifter buoys advected by ocean currents in the North Atlantic Ocean.

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They are sparsely distributed, their nearest neighbors several hundred kilometers away.

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Still, our new method, TRV, is able to pick up objective rotational behavior.

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High rotation can be observed on a few trajectories in the North Atlantic Gyre,
around Iceland and Greenland.

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As a second example, we applied TRV on insect trajectories.

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These 3D trajectories were reconstructed from videos of midges in a large translucent box.

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While their behavior seems erratic, TRV highlights some insects of especially high and low rotational behavior

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These TRV values can change rapidly when the set of neighbors changes.

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This happens as midges disappear from the tracked set, or move too far away.

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With this data set, the movement of socially distancing pedestrian crowds has been studied.

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Here, 32 study participants move quickly within an enclosed space, while attempting to keep a social distance of 1.5m from each other.

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The group at large tends to move in a counterclockwise rotation, but only six individuals show high objective rotation.

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Finally, we show the changing objective rotation over time in a simulation of bird-like agents (boids).

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In an initial chaotic phase, the boids all show strong rotational behavior.

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To simulate emergent behavior found in bird flocks or fish swarms, each individual agent follows a simple set of rules.

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Swarms form as the boids align with their neighbors, showing a decrease in their rate of rotation by TRV.

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At this point, all boids find themselves as part of established swarms, and seize to rotate.

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Only intermittent rotational behavior can be observed in single individuals.