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''of a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047500/h0475001.png" /> along a curve''

The curve representing the ends of the variable vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047500/h0475002.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047500/h0475003.png" /> is a real variable, such as time) whose origin for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047500/h0475004.png" /> is a given fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047500/h0475005.png" />. Trajectory of a point Velocity hodograph Construction of a velocity hodograph.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/h047500a.gif" />

Figure: h047500a

The hodograph is a visual geometrical representation of the variation (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047500/h0475006.png" />) of the magnitude represented by the variable vector and of the rate of this change. Its direction is that of the tangent to the hodograph. For example, if the velocity of a moving point is represented by a variable vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047500/h0475007.png" />, then by drawing the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047500/h0475008.png" /> at different moments of time from the origin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047500/h0475009.png" />, one obtains the velocity hodograph. The magnitude describing the rate of the variation of the velocity at some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047500/h04750010.png" />, i.e. the acceleration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047500/h04750011.png" /> at that point, has at any point of time the direction of the tangent to the velocity hodograph at the respective point.