Mapping of curves from the physical to the hodograph plane, when the Jacobian vanishes along a curve is discussed in detail here. A certain quadratic expression occuring in the equations has been identified as the important factor in the transformation. The nature of the transformation depends on whether the roots are complex or real distinct and so on. A geometrical representation of the referred quadratic expression is also presented, resulting in a better understanding of the transformation technique.

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Journal Journal of the Franklin Institute
Rajan, S, & Damodaran, K.A. (K. A.). (1979). Hodograph transformation. Journal of the Franklin Institute, 307(6), 379–394.