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| A '''hodograph''' is a diagram that gives a [[Vector (geometric)|vectorial]] visual representation of the movement of a body or a fluid. It is the [[locus (mathematics)|locus]] of one end of a variable vector, with the other end fixed.<ref>{{Cite web
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| | title = AMS Glossary of Meteorology : Hodograph
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| | accessdate = 2007-05-30
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| | url = http://amsglossary.allenpress.com/glossary/search?id=hodograph1
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| }}</ref> The position of any plotted data on such a diagram is proportional to the [[velocity]] of the moving particle. It is also called a '''velocity diagram'''. It appears to have been used by [[James Bradley]], but its practical development is mainly from Sir [[William Rowan Hamilton]], who published an account of it in the ''[[Proceedings of the Royal Irish Academy]]'' in 1846.
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| ==Applications==
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| It is used in [[physics]], [[astronomy]] and [[fluid mechanics]] to plot deformation of material, motion of planet or any other data that involves the velocities of different part of a body.
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| See [[Swinging Atwood's machine]]
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| ===Meteorology===
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| [[Image:Hodographe NOAA.PNG|thumb|right|250px|Hodograph plot of upper air winds from [[radiosonde]] (Source: [[NOAA]])]]
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| In [[meteorology]], hodographs are used to plot [[wind]]s from sounding of the [[Earth's atmosphere]]. It is a polar diagram where wind direction is indicated by the angle from the center axis and its strength by the distance from the center. In the figure to the right, at the bottom one finds values of wind at 4 heights above ground. They are plotted by the [[Vector (geometric)|vector]]s <math>\vec V_0</math> to <math>\vec V_4</math>. One has to notice that direction are plotted as mentioned in the upper right corner.
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| With the hodograph and [[thermodynamic diagram]]s like the [[tephigram]], meteorologists can calculate:
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| * '''Wind shear''': The lines uniting the extremities of successive vectors represent the variation in direction and value of the wind in a layer of the atmosphere. [[Wind shear]] is important information in the development of [[thunderstorm]]s and future evolution of wind at these levels.
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| * '''Turbulence''': wind shear indicate the possible [[turbulence]] that would cause a [[hazard]] to [[aviation]].
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| * '''Temperature advection''': change of [[temperature]] in a layer of air can be calculated by the direction of the wind at that level and the direction of the wind shear with the next level. In the northern hemisphere, warm air is to the right of a wind shear between levels in the atmosphere. The opposite is true in the southern one (see [[thermal wind]]). So in the example hodograph, the wind <math>\vec V_3</math> from southwest meet the right side of the wind shear which means a warm [[advection]] and thus warming of the air at that level.
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| ==Hodograph transformation==
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| Hodograph transformation is a technique used to transform nonlinear [[partial differential equation]]s into linear version. It consists of interchanging the dependent and independent variables in the equation to achieve linearity.<ref>{{cite book|last1=Courant|first1=R.|authorlink1=Richard Courant|last2=Friedrichs|first2=K. O.|authorlink2=Kurt O. Friedrichs|title=Supersonic Flow and Shock Waves|url=http://books.google.co.in/books?id=Qsxec0QfYw8C&pg=PA248&dq=hodograph+transformation#v=onepage&q=hodograph%20transformation&f=false|year=1948|publisher=Springer}}</ref>
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| ==Further reading==
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| In his book "Matter and Motion, Maxwell writes:
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| "The study of the Hodograph, as a method of investigating the motion of a body, was introduced by Sir W. R. Hamilton. The hodograph may be defined as the path traced out by the extremity of a vector which continually represents, in direction and magnitude, the velocity of a moving body. In applying the method of the hodograph to a planet, the orbit of which is in one plane, we shall find it convenient to suppose the hodograph turned round its origin through a right angle, so that the vector of the hodograph is perpendicular instead of parallel to the velocity it represents."
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| and he applies these techniques to analyse KEPLER'S FIRST and SECOND LAWS.
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| Free "Matter and Motion" e-books are available on the Internet.
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| * [[Feynman's Lost Lecture: The Motion of Planets Around the Sun]] by David L. Goodstein & Judith R. Goodstein (ISBN 0-393-03918-8, W.W.Norton & Company: New York, 1996). In this book the hodograph is used to geometrically derive elliptical (Keplerian) orbits from Newton's laws of motion and gravitation.
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| ==See also==
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| * [[Visual calculus]], a related approach useful in solving a variety of integral calculus problems.
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| ==References==
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| {{reflist}}
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| ==External links==
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| * [http://www.du.edu/~jcalvert/phys/hodo.htm ''The Hodograph''] - Dr. James B. Calvert, University of Denver
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| [[Category:Classical mechanics]]
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| [[Category:Basic meteorological concepts and phenomena]]
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| [[Category:Wind]]
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Hello and welcome. My title is Irwin and I completely dig that title. For a whilst I've been in South Dakota and my mothers and fathers live close by. Doing ceramics is what her family and her appreciate. He utilized to be unemployed but now he is a meter reader.
Here is my blog post - hftirc.tk