We assume a global input such that the particles start in an arc along the right side of the disk. The left-hand image shows the particle distribution in green as they propagate in the

direction, as is shown in blue. The

direction is indicated by two red arrows, each starting from either the left or right endpoint of the arc. The particle dispersion

is shown with transparent circle sectors, and the path from their current position to an ending arc is shown with a transparent blue sector, which ends along an orange arc whose extent is highlighted by green and purple points. If "show paths" is checked, additional starting points along the current arc are shown.

The arc length may be initialized from 0 to

radians.

Any initial distribution can be steered to a minimum arc length of

and a maximum arc length of

. Once the arc length exceeds the minimum, it can never be reduced to less than

. Similarly, if the arc length is ever less than the maximum, it can never be increased to more than

.

Every propagation step can add no more than

to the arc length or decrease the arc length by

.

This Demonstration was inspired by a similar problem in polygonal workspaces by Lewis and O'Kane [1].

[1] J. S. Lewis and J. M. O'Kane, "Planning for Provably Reliable Navigation Using an Unreliable, Nearly Sensorless Robot,"

*The International Journal of Robotics Research*,

**32**(11), 2013 pp. 1339–1354.

doi:10.1177/0278364913488428.