Shape Metrics¶

This page explains the quantitative shape metrics computed by CGAspects.

Length units are scaled by the unit-cell a length

The coordinates in a CrystoGen .XYZ file are in Ångströms, but they are scaled by the unit-cell a lattice parameter — i.e. they are expressed in units of a, not raw Å. CGAspects performs no rescaling by default: every length-valued metric below (L/M/S dimensions, surface area, volume, SA:Vol) is reported in whatever units the input file uses. The "Å"-based units in the table are therefore only literally correct when the a length is 1 Å; otherwise multiply by a (or a², a³, 1/a as appropriate) to recover physical Ångström values. Dimensionless quantities (aspect ratios, shape class) are unaffected.

Principal Component Analysis (PCA)¶

CGAspects uses Principal Component Analysis (PCA) to find the three principal dimensions of a crystal point cloud. PCA is performed via Singular Value Decomposition (SVD) of the centered coordinate matrix.

The three principal components define orthogonal directions: - PC1 (first principal component) — the direction of greatest extent (long axis, L) - PC2 — the direction of second greatest extent (middle axis, M) - PC3 — the direction of smallest extent (short axis, S)

The magnitudes of PC1, PC2, PC3 give the dimensions L, M, S used for Zingg classification.

Why PCA Instead of Bounding Box?¶

A simple axis-aligned bounding box would give the X, Y, Z extents, which depend on crystal orientation. PCA is orientation-independent: it finds the intrinsic dimensions of the crystal regardless of how it is oriented in space.

Convex Hull¶

The convex hull is the smallest convex shape that contains all points in the crystal. CGAspects uses the convex hull (via scipy.spatial.ConvexHull) to compute:

Surface Area¶

The sum of the areas of all triangular faces on the convex hull, in Å².

Volume¶

The volume enclosed by the convex hull surface, in Å³.

SA:Vol Ratio¶

The surface area to volume ratio, in Å⁻¹:

SA:Vol = Surface Area / Volume

This ratio is important in dissolution: a higher SA:Vol means more surface exposed per unit of crystal mass, leading to faster dissolution. Needles and plates have higher SA:Vol ratios than blocks of the same volume.

Aspect Ratios¶

From the PCA dimensions:

Ratio	Formula	Name
Primary (S:M)	S / M	Short / Middle
Secondary (M:L)	M / L	Middle / Long

Both ratios range from 0 to 1. See Zingg Classification for how these ratios map to crystal shape classes.

Crystal Extent Along Directions¶

Crystallographic Direction Analysis (CDA) describes the extent of the crystal along selected crystallographic directions.

CDA is currently read from simulation output, not computed

At present CGAspects does not measure CDA itself. The per-direction values used in Aspect Ratio Analysis are read from the CrystoGen simulation_parameters.txt output. The in-code projection described below is planned but not yet implemented.

Once computed in-code, the extent along a direction vector d (unit vector) will be:

extent = max(points · d) - min(points · d)

where the dot product projects each point onto the direction. This gives the crystal's "width" in that specific crystallographic direction, regardless of orientation.

Shape Classification¶

CGAspects assigns a morphological class to each crystal frame using the Zingg threshold (2/3):

Class	S:M	M:L	Description
Block	≥ 2/3	≥ 2/3	Equant, roughly cubic
Plate	≥ 2/3	< 2/3	Disk-like, one thin dimension
Needle	< 2/3	≥ 2/3	Rod-like, one elongated dimension
Lath	< 2/3	< 2/3	Elongated and flat

Summary of Computed Metrics¶

Metric	Method	Unit
S, M, L dimensions	PCA / SVD	Å*
S:M aspect ratio	PCA	dimensionless
M:L aspect ratio	PCA	dimensionless
Surface area	Convex hull	Å²*
Volume	Convex hull	Å³*
SA:Vol ratio	Convex hull	Å⁻¹*
Shape class	Zingg threshold	—
Direction extent	Read from `simulation_parameters.txt` (in-code projection planned)	Å*

* Scaled by the unit-cell a length — see the warning at the top of this page.