Continuous Shape Measures

Given two shapes described as set of $N$ points $P$ and $Q$ in ${\mathbb{R}}^3$ (normally $n=3$), find the transformation $T$ that best maps $Q$ onto $P$, that is, the one that minimizes the sum of distances between paired points $p_i$ and $T{q}_i$.

The Continuous Shape Measure of a problem shape $Q$ relative to a reference shape $P$ is the squared distance from $TQ$ to $P$, normalized and expressed as a percentage ^1–3:

$$S_P(Q)=\underset{T,\pi }{\min }\left(\frac{{\sum }_i^N|T{q}_{\pi (i)}-p_i{|}^2}{{\sum }_i^N|p_i-p_0{|}^2}\right)\times 100$$

where $q_j$ and $p_i$ are the position vectors of the points of $Q$ and $P$, $p_0$ is the geometric centre of the reference structure $P$ and $\pi$ is the permutation that matches points of $Q$ onto $P$. For more information about how this minimization is performed see: Optimal transformation finding

From the definition it follows that $S_P(Q)=0$ if and only if $P$ and $Q$ have exactly the same shape. The maximum value of $S_P(Q)$ is 100, reached in the unrealistic case where all points of $Q$ collapse into a single point in space ³.

For coordination compounds, $S_P(Q)<1$ is generally read as a small distortion from the reference shape, while larger values indicate a progressively more distorted, or different, geometry. The measure is independent of size and orientation, and is therefore comparable across structures ³.

1.

Pinsky, M. & Avnir, D. Continuous Symmetry Measures. 5. The Classical Polyhedra. Inorganic Chemistry 37, 5575–5582 (1998).

2.

Casanova, D. et al. Minimal Distortion Pathways in Polyhedral Rearrangements. Journal of the American Chemical Society 126, 1755–1763 (2004).

3.

Alvarez, S. et al. Shape maps and polyhedral interconversion paths in transition metal chemistry. Coordination Chemistry Reviews 249, 1693–1708 (2005).