Device for assisting an air or maritime traffic controller in avoiding a potential conflict, detected within a predetermined path-prediction horizon, between a first path of a first aircraft and a second path of a second aircraft or between a first path of a first vessel and a second path of a second vessel, each path ( PN1, PN2) comprising a plurality of segments formed between a plurality of navigation points, the device comprising: - a determination unit for determining at least one peripheral lateral envelope of the first path, the peripheral lateral envelope defining a lateral navigation surface attainable by the first aircraft or by the first vessel from a current position of the first aircraft or of the first vessel, wherein the determination unit determines the peripheral lateral envelope from a maximum authorized delay related to the last navigation point of the navigation plan of the first path, - a division unit for dividing the peripheral lateral envelope into a longitudinal plurality of adjacent sections (TR0, TR1, TR2,..., TR) next to one another and delineated by transition lines (LT0, LT1, LT2,..., LT(N-1)) marking the change of section, each transition line intersecting a segment of the first path at a first point of intersection, and an edge of the peripheral lateral envelope at a second point of intersection, - a discretization unit for discretizing each transition line into a plurality of transition points ( I0, I1, I2,..., I), and - a calculation unit for determining for each transition point of each transition line, using a conflict detection algorithm, a potential conflict between an avoidance path and the second path, the avoidance path comprising the current position of the first aircraft or of the first vessel, the position of the current transition point, and a predetermined meeting point, a potential conflict being detected when a horizontal separation distance between the avoidance path and the second path is less than a predetermined horizontal separation distance; and for calculating at least one outline of a conflict avoidance surface ( SEC, SEC1, SEC2), from a plurality of transition point positions for which the conflict detection algorithm has not determined a potential conflict between the respective avoidance path and the second path. - a determination unit for determining at least one peripheral lateral envelope of the first path, the peripheral lateral envelope defining a lateral navigation surface attainable by the first aircraft or by the first vessel from a current position of the first aircraft or of the first vessel, wherein the determination unit determines the peripheral lateral envelope from a maximum authorized delay related to the last navigation point of the navigation plan of the first path, - a division unit for dividing the peripheral lateral envelope into a longitudinal plurality of adjacent sections (TR0, TR1, TR2,..., TR) next to one another and delineated by transition lines (LT0, LT1, LT2,..., LT(N-1)) marking the change of section, each transition line intersecting a segment of the first path at a first point of intersection, and an edge of the peripheral lateral envelope at a second point of intersection, - a discretization unit for discretizing each transition line into a plurality of transition points ( I0, I1, I2,..., I), and - a calculation unit for determining for each transition point of each transition line, using a conflict detection algorithm, a potential conflict between an avoidance path and the second path, the avoidance path comprising the current position of the first aircraft or of the first vessel, the position of the current transition point, and a predetermined meeting point, a potential conflict being detected when a horizontal separation distance between the avoidance path and the second path is less than a predetermined horizontal separation distance; and for calculating at least one outline of a conflict avoidance surface ( SEC, SEC1, SEC2), from a plurality of transition point positions for which the conflict detection algorithm has not determined a potential conflict between the respective avoidance path and the second path.