Singular Curve
For a surface parameterization x:U->R^3, a singular curve is a curve in U along which the differential of x has rank less than 2. Equivalently, the coordinate tangent vectors x_u and x_v are linearly dependent. The image of a singular curve consists of singular points of the parametrization. On any region disjoint from these curves where the rank is 2, x is a regular parameterization (Gray 1997, pp. 281-286).
See also
Regular Parameterization, Regular SurfaceExplore with Wolfram|Alpha
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References
Gray, A. "The Definition of a Regular Surface in R^n." §12.4 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 281-286, 1997.Cite this as:
Weisstein, Eric W. "Singular Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SingularCurve.html