If the curve is, say, $\displaystyle \gamma(u), u\in [a,b]$ then a parametrization may be given by $\displaystyle X(u,v)=v\gamma(u), u\in [a,b],v\geq 0$.
What needs to be removed is the origin (where $\displaystyle v=0$) and the lines $\displaystyle v\gamma(a),v\gamma(b), v>0$.