Shape Metrics¶
This page explains the quantitative shape metrics computed by CGAspects.
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 Ų.
Volume¶
The volume enclosed by the convex hull surface, in ų.
SA:Vol Ratio¶
The surface area to volume ratio, in Å⁻¹:
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¶
When Crystallographic Direction Analysis (CDA) is enabled in Aspect Ratio Analysis, CGAspects additionally computes the extent of the crystal along each selected crystallographic direction.
For a direction vector d (unit vector), the extent is:
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 | Ų |
| Volume | Convex hull | ų |
| SA:Vol ratio | Convex hull | Å⁻¹ |
| Shape class | Zingg threshold | — |
| Direction extent | Dot product projection | Å |