There are two non parallel line segments $U$ and $V$ in $\mathbb{R}^3$, each given by their two end points $u_0, u_1$ and $v_0, v_1$, such that if they are projected into $\mathbb{R}^2$ by their $x$ and $y$ coordinates, they intersect. We want to know whether the line $U$ is above $V$. More exactly, suppose their projections on $\mathbb{R}^2$ intersects at point $(x, y)$. We want to know whether the $z$ coordinates of line segment $U$ at point $(x, y)$ is larger than the $z$ coordinates of line segment $V$ at point $(x, y)$.
Assuming that $u_0, u_1, v_0, v_1$ are all lattice points, is there a way to do this that doesn't explicitly calculates the $x$, $y$ and $z$ coordinates? In order words, how to check this without floating point arithmetic?
Motivation
This comes up when I am trying to compute the 3D convex hull of the projection of a triangulation of polygon from $\mathbb{R}^2$ into $\mathbb{R}^3$ during the computation of the weighted delaunay triangulation of the set of points.