Hodograph

Mathematics

The hodograph of the movement of a specific mobile M compared to a point fixes O is the place H points P such as:

$\ overrightarrow {COp} = \ overrightarrow {V}$

where $\ overrightarrow {V}$ is the vector Speed of M at the moment T .

Physics

The hodograph in physics is built at the same time with the physical plan and the plan of representation of the constraints to determine the field of the lines of slip in a material which becomes deformed plastically or the forces which are exerted in a system.

Thus one uses it in astronomy to represent the trajectory of planets. The vector connecting the Sun to planet being inversely proportional to the Speed and the tangent to the trajectory, the instanée Direction of displacement.

Meteorology

The hodograph is used in meteorology to point the data of Vent S of a Radiosondage. It takes the shape of a polar diagram where the direction is indicated by the angle to the center of the figure and the size by the distance with this one. In the figure on the right, there is in the part of bottom the size and the direction of the winds at various altitudes above the ground. Those are pointed like the vectors

\ vec V_0

with

\ vec V_4

. The hodograph is thus the line which joint the head of these vectors.

It has to notice that the angle from which the wind comes is pointed as in a mirror in this type of hodograph. Thus, the right upper part of the image shows how one must point the vectors (e.g. an angle of 180 degrees is traced to the top thus the winds coming from the south move towards north). The hodograph enables us to calculate various parameters in conjunction with the thermodynamic diagrams like the Téphigramme:

Shearing : the line which connects between them the ends of these vectors constitute the change, so much in direction than in size, of the wind with altitude. They represent the shearing winds in a layer of the atmosphere. This last is important to know for the forecast of the development of the Orage S and the behavior of the winds in the future.
Turbulence : according to the value of shearing in a layer, one can deduce from it the very important Turbulence for the Aviation,
Advection of temperature : to know the change of Température at an altitude, it is enough to look at the relation between the vector of the wind on this level and shearing between this level and that above. Indeed, the wind indicates the direction from which the wind comes and shearing in which direction one has a change of mass of air. In the northern hemisphere, the hot air is on the right shearing whereas it is the reverse in the southern hemisphere (see thermal Vent). For example, in the image the wind $\ vec V_3$ comes from south-west and shearing is towards the line. Thus there will be Advection hot air in this layer of the atmosphere: it will be heated.

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