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["Brown, James 2" <>]

>   Vertex_handle move_point(const Vertex_handle& old_vertex,
>                            const Point_3& new_position,
>                            Outdated_cell_set& outdated_cells_set,
>                            Moving_vertices_set& moving_vertices) const;
> does exactly what you need.

Yes, this is the function that I am looking at.

>- collect the incident cells to v
>- remove v
>- triangulate the cavity --> triangulation T1
>- collect cells in conflict with the insertion of newpoint : T2
>- insert newpoint
>- collect incident cells to new vertex : T3
>the newly created cells after removal + insertion is T1-T2+T3.

Sorry if I am missing something, but this is not quite the process that is 
happening in move_point (though it must be equivalent?):
- In move_point the new (outdated) cells are just the conflict cells of the 
old position in the final triangulation. 
 - As well it is doing removal after insertion of the new point (so the 
insertion conflicts are calculated from the original triangulation before 
removal)
Is move_point taking a shortcut? It is not obvious to me that it yields the 
same results as your suggested process.