I have question about curves in space. What is the difference between regular curve and smooth curve?
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 $t_0 \in [a,b]$