p$, $\gamma_{pq}(t_q) = q$, and $t_p, t_q \in (0,1)$. Let $\phi_p$ be the signed angle from the velocity vector $\gamma'_{pq}(t_p)$ to $\frac{\partial }{\partial x^1}|_p$ and let $\phi_q$ be the signed angle from $\gamma'_{pq}(t_q)$ to $\frac{\partial }{\partial x^1}|_q$. Then if $\phi_{pq} = \phi_p - \phi_q$, $P_{pq}({v}) = e^{i\phi_{pq}}{v}$ gives the Levi-Civita parallel transport function, $P_{pq}$, in coordinates with respect to the basis $\{\frac{\partial }{\partial x^1}|_p,\frac{\partial }{\partial x^2}|_p\}$.

Then, $Q_{pq}$, the parallel transport function for *representation vectors* between $p$ and $q$, must satisfy $$\label{eq:parallel-transport}
  Q_{pq}(R_p([c_p])) = R_q([P_{pq}(c_p)]).$$

We can write $c_p = e^{i(\theta_p +2k\pi/4)}$ for some $k \in \{0,1,2,3\}$ where $\theta_p$ is the signed angle from $\frac{\partial }{\partial x^1}|_p$ to one of the component vectors of $[c_p]$. We can now write equation \[eq:parallel-transport\] as $$Q_{pq}(e^{4i\theta_p}) = e^{4i(\theta_p + \phi_{pq})}.$$

It follows that $Q_{pq}({v}) = e^{4i\phi_{pq}}{v}$. In addition, we can define parallel transport on the cross bundle from $C_p$ to $C_q$ by $R^{-1}_q \circ Q_{pq} \circ R_p$. We can now use $\nabla^Q$ in equation \[eq:energy\] to define a smooth cross field. In the following section, we define the discrete Laplace equation corresponding to this energy and describe the diffusion generated method for cross field design in detail.

Discrete Formulation {#sec:discrete-MBO}
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On each 