 Second prolongation of the rotation generator for plane graphs

ID: second-prolongation-of-the-rotation-generator-for-plane-graphs

V = u \partial_{x} - x \partial_{u}

of rotations of the

(x, u)

plane has second prolongation

pr^{(2)} V = u \partial_{x} - x \partial_{u} - (1 + u_{x}^{2}) \partial_{u_{x}} - 3 u_{x} u_{xx} \partial_{u_{xx}} .

Consequently it preserves both

u_{xx} = 0

and

u_{xx} = (1 + u_{x}^{2})^{3/2}

, the equations for straight lines and consistently oriented unit-circle arcs.

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