On a connected manifold, a smooth bracket-generating family of vector fields connects every pair of points by a piecewise smooth horizontal curve. For a free step-N nilpotent Lie group, this says that every group element is the path signature of a sufficiently regular path. In the planar step-two case, a closed rectangle prescribes the signed area and a following straight segment prescribes the displacement.
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The Chow–Rashevskii theorem is a fundamental result in differential geometry and the theory of control systems. It pertains to the accessibility of points in a control system defined by smooth vector fields.