this space by $C_p$. The disjoint union of all cross spaces of $M$ defines a fiber bundle that we refer to as the *cross bundle*. A section of the cross bundle, or a choice of one cross per cross space, is called a *cross field* on $M$. We make the natural identification between $T_p$ and the complex plane by the map $(a,b) \mapsto a + ib$. In this way, we can identify a cross, $[c_p]$, in $C_p$ as an unordered set of four orthogonal unit vectors in $T_p$, which we call the cross *component vectors*. This also allows us to define a representation map $R_p$ at each point with respect to the local coordinate basis, and a representation vector $u_p = R_p([c_p])$, which we identify with the corresponding unit tangent vector in $T_p$. For simplicity, we will use complex notation for equations throughout the paper.

In 2D cross field design, the goal of designing a smooth cross field is often formulated as designing a harmonic representation vector field $u$ [@viertel_approach_2019; @vaxman_directional_2016]. That is, to minimize the Dirichlet energy $$\label{eq:energy}
  E[u] \coloneqq \frac{1}{2}\int_M{|\nabla{u}|^2dA}$$

with the constraint that $|u| = 1$ at each point of the domain. We note that in general, 