For a parametrized surface 
, a singular curve is a curve
in 
along which the differential
of 
has rank less
than 2. Equivalently, the coordinate tangent vectors 
and 
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, 
is a regular parameterization
(Gray 1997, pp. 281-286).
Singular Curve
See also
Regular Parameterization, Regular SurfaceExplore with Wolfram|Alpha
References
Gray, A. "The Definition of a Regular Surface in
." §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