I have question about curves in space. What is the difference between regular curve and smooth curve?

  • 1
    $\begingroup$ smooth curve means the parametrization is differentiable whereas regularity requires the derivative to be non-zero. $\endgroup$
    – Paladin
    Mar 20, 2016 at 11:21

1 Answer 1


Suppose the curve is defined by the parametric equations

$(x,y,z) = (f(t), g(t), h(t))$, where $t \in [a,b]$

The curve is a smooth curve if the derivatives $f'$, $g'$, $h'$ exist and are continuous in $[a,b]$

The curve is a regular curve or a regular smooth curve, if it's smooth and also the three derivatives $f'$, $g'$, $h'$ are not simultaneously zero for any $t_0 \in [a,b]$


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