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Hodograph
Hodograph
Hodograph
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A hodograph is a diagram that gives a vectorial visual representation of the movement of a body or a fluid. It is the locus of one end of a variable vector, with the other end fixed.[1] The position of any plotted data on such a diagram is proportional to the velocity of the moving particle.[2] 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.[2]

Applications

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See also: Swinging Atwood's machine

It is used in physics, astronomy, solid and fluid mechanics to plot deformation of material, motion of planets or any other data that involves the velocities of different parts of a body.

Meteorology

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Hodograph plot of upper air winds from radiosonde

In meteorology, hodographs are used to plot winds from soundings 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, at the bottom one finds values of wind at 4 heights above ground. They are plotted by the vectors to . One has to notice that direction are plotted as mentioned in the upper right corner.

With the hodograph and thermodynamic diagrams like the tephigram, meteorologists can calculate:

  • 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 thunderstorms and future evolution of wind at these levels.
  • Turbulence: wind shear indicates possible turbulence that would cause a hazard to aviation.
  • 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 the wind shear vector between levels in the atmosphere. The opposite is true in the southern hemisphere (see thermal wind). Using the example hodograph, the wind from the southwest meets the right side of the wind shear vector which means warm advection is occurring and thus the air is warming at that level.

Distributed hodograph

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It is a method of presenting the velocity field of a point in planar motion. The velocity vector, drawn at scale, is shown perpendicular rather than tangent to the point path, usually oriented away from the center of curvature of the path. [3]

Hodograph transformation

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Hodograph transformation is a technique used to transform nonlinear partial differential equations into linear version. It consists of interchanging the dependent and independent variables in the equation to achieve linearity.[4]

See also

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References

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Further reading

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Hodograph

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from Grokipedia
A hodograph is the curve traced by the tip of the vector of a moving particle when all such vectors are translated to originate from a fixed point. The term derives from the Greek words hodos (way or path) and graphein (to write), reflecting its representation of a "path of ." Introduced by Irish mathematician in 1847, the hodograph provides a geometrical tool for analyzing motion by transforming second-order differential equations of position into equations in . In , the hodograph is particularly valuable for central force problems, where it simplifies the solution of trajectories under , such as in planetary motion. For Keplerian orbits—elliptical paths around a central body like a orbiting a star—the hodograph forms a circle in velocity space, with the circle's center offset from the origin by an amount related to the and the velocity at periapsis. This circular shape directly encodes and was historically used by Hamilton to deduce the of gravitation from observed elliptical orbits. Beyond , hodographs apply to in uniform gravitational fields, where they trace straight lines, aiding calculations of maximum range and optimal launch angles. The concept extends to other domains, including through hodograph transformations that interchange dependent and independent variables to solve nonlinear partial differential equations, such as those governing around objects. In , hodographs plot vertical profiles to predict storm development and rotation, with curved shapes indicating potential for formation. Additionally, in , Pythagorean-hodograph curves—a modern variant—enable rational parametrizations where the speed is a function, facilitating exact arc-length computation and applications in manufacturing. Despite its elegance, the hodograph method has declined in mainstream use with the rise of computational and analytical approaches, though it remains a insightful pedagogical tool for understanding dynamics.

Fundamentals

Definition

A hodograph is derived from the Greek words "hodos" (path or way) and "graphein" (to write or describe), a term coined by the mathematician in 1847 to denote a representing the path of . It serves as the locus of the tip of a , with the vector's tail fixed at a common origin, thereby tracing the evolution of the velocity's magnitude and direction for a moving body or fluid particle over time. The basic construction of a hodograph involves plotting the vector v(t)\mathbf{v}(t) in a velocity space, where the endpoints of successive vectors form the ; this contrasts with traditional position diagrams by focusing solely on components rather than spatial coordinates. Its primary purpose is to visualize and analyze motion through changes, providing insight into vector fields such as those encountered in trajectories or wind patterns, where understanding directional shifts is key. For instance, in uniform , where speed remains constant but direction varies continuously, the hodograph manifests as a centered at the origin with a equal to that constant speed. Unlike a position-time graph, which tracks displacement against time to infer average via , a hodograph directly plots magnitude and direction, enabling immediate assessment of as the to the curve.

Historical Development

The term "hodograph" was formally introduced in the 1840s by Irish mathematician , who developed it as a geometric tool for expressing the Newtonian law of attraction in symbolic language, particularly in his 1847 paper "The Hodograph, or a New Method of Expressing in Symbolic Language the Newtonian Law of Attraction." Hamilton expanded on this in his broader work on dynamics and optics, including applications to central force problems—such as planetary motion under inverse-square laws—and light propagation, as elaborated in his "Lectures on Quaternions" published in 1853. During the late 19th century, the hodograph benefited from advancements in vector analysis, independently formulated by American J. Willard Gibbs in his 1881–1884 notes "Elements of Vector Analysis" and by British engineer in his electromagnetic treatises, which provided a rigorous framework for handling vectorial representations of loci. In the , hodographs saw expanded use in during the 1920s, where they were applied to analyze fields in fluid flows around airfoils and propellers, building on early theoretical work in incompressible . Following , the concept gained prominence in for wind profiling, as upper-air observations from radiosondes enabled the plotting of vertical patterns to forecast convective storms, with key methodological refinements appearing in the late 1940s and 1950s. By the 1960s, hodographs were integrated into computational methods for , notably in applications for space trajectory optimization and analysis under gravitational influences. A significant milestone in the 1990s was the development of Pythagorean hodograph curves by Rida T. Farouki and Thomas N. Sakkalis, which introduced rational parametric curves with exact arc-length parametrization for and manufacturing.

Mathematical Description

Vector Formulation

In , the hodograph of a particle's is defined as the locus traced by the endpoint of its vector v(t)\mathbf{v}(t), with the vector's tail fixed at the origin of velocity space. This representation transforms the original position into a in the velocity domain, providing insights into the motion's . Consider a particle with position vector r(t)=(x(t),y(t))\mathbf{r}(t) = (x(t), y(t)) in the plane, parameterized by time tt. The vector is then v(t)=drdt=(vx(t),vy(t))\mathbf{v}(t) = \frac{d\mathbf{r}}{dt} = (v_x(t), v_y(t)), where vx(t)=dxdtv_x(t) = \frac{dx}{dt} and vy(t)=dydtv_y(t) = \frac{dy}{dt}. The hodograph is the parametric curve (vx(t),vy(t))(v_x(t), v_y(t)) plotted in the plane as tt varies, effectively mapping the time-dependent components of the . In Cartesian coordinates, the hodograph is directly given by these components (vx,vy)(v_x, v_y). Alternatively, in polar coordinates within the plane, the hodograph can be described by the speed v=∣v∣=vx2+vy2v = |\mathbf{v}| = \sqrt{v_x^2 + v_y^2}
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