Tangential surface of a smooth algebraic space curve

Question

We know that for a smooth space curve parametrized $$(x_1(t),\; x_2(t),\; x_3(t))$$ the tangential surface (the union of all tangent lines to the curve) is parametrized $$(x_1(t)+sx_1'(t), \; x_2(t)+sx_2'(t), \; x_3(t)+sx_3'(t))$$ But what about tangential surface of algebraic smooth curves not defined by rational parametrization? For example $$\begin{cases} XZ+Y^2+W^2 &=0 \\ X^2+XY-Z^2-W^2 &=0 \end{cases}$$ is a smooth curve in $\mathbb{P}^3_\mathbb{C}$, not contained in every plane, and having genus 1. It's not possible to find a rational parametrization. So how to compute the tangential surface (find the implicit equation) in these cases?