The length cross-ratio $l^{\prime} r_{i j}$ is a geometric quantity associated with a common edge $i j$ shared by two adjacent triangles in a triangulated surface. It is defined as:

$$ \operatorname{lcr}{i j}=\frac{\ell{i l} \ell_{j k}}{\ell_{l j} \ell_{k i}} $$

where:

• $\ell_{a b}$ represents the length of the edge between vertices $a$ and $b$,
• The two triangles under consideration are $\triangle i j k$ and $\triangle j i l$,
• The orientation of the surface plays a role in ensuring a consistent definition.

Interpretation and Properties

1. Relation to Complex Cross-Ratio:

   If the triangulation is embedded in the complex plane, the length cross-ratio corresponds to the modulus of the complex cross-ratio of the four vertices. The complex cross-ratio is a conformal invariant and measures how the four points are arranged relative to each other.

2. Product Around a Vertex is 1 :

The given property states that for any interior vertex $i$, the product of the length cross-ratios for all edges $i j$ connected to $i$ satisfies:

$$ \prod_{i j \ni i} \operatorname{lcr}_{i j}=1 $$

This result arises from the way edge lengths cancel when forming the product. Essentially, each edge length appears in both the numerator and denominator across different terms, leading to a telescoping effect.

1. Significance in Discrete Geometry:
   • The length cross-ratio is important in discrete differential geometry and computational geometry, particularly in the study of discrete conformal mappings.
   • It plays a role in defining conformal structures on triangulated surfaces, where preserving the cross-ratio maintains a discrete notion of conformal equivalence.
2. Behavior on Non-Orientable Surfaces:
   • The definition assumes an orientation, which is necessary for defining cross-ratios consistently.
   • On non-orientable surfaces, the length cross-ratio is well-defined on the oriented double cover, where orientation is locally restored.

Applications

• Discrete Conformal Geometry: Length cross-ratios are used in algorithms for conformal parameterization and discrete Ricci flow.
• Computational Mesh Processing: The preservation of cross-ratios is useful in shape deformation and texture mapping.
• Mathematical Physics: The cross-ratio appears in discrete integrable systems and connections with statistical mechanics.

Understanding the Length Cross-Ratio with an Example

Consider two adjacent triangles $\triangle i j k$ and $\triangle j i l$, sharing the edge $i j$, as shown below:

$$ \triangle i j k \text { and } \triangle j i l $$

We assign edge lengths as follows:

• $\ell_{i j}$ : length of edge $i j$