# Intersection of 3D curves parameterised by piecewise defined functions

I need to calculate the intersection of two 3D parametric curves $\vec{C_1}$ and $\vec{C_2}$.

Those curves are parameterised by piecewise functions.

$\vec{C_1}= \begin{pmatrix}x_1(t)\\y_1(t)\\z_1(t)\end{pmatrix}\\ \vec{C_2}= \begin{pmatrix}x_2(t)\\y_2(t)\\z_2(t)\end{pmatrix}$

For example, $x_1(t)$ could look like this: