Overview
Procrustes analysis centers corresponding point sets, optionally normalizes scale, and solves for the rotation that best aligns them in a least-squares sense.
CoreSource checked
Procrustes Analysis
Aligns two point sets by choosing the translation, scale, and rotation that minimize squared error.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Geometry
- Difficulty
- Intermediate
- Family
- Shape Alignment
- Tier
- Core
- Status
- Verified
- Review
- Source Checked
- Sources
- 2
- Formulas
- 0
- Relationships
- 3
- Code example
- Ready
- Visualization
- Ready
Core Concepts
shape alignmentgeometric primitivessequence alignmentgeometry
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(n)
Average
O(n + m)
Worst
O(n + m)
Space
O(n + m)
When to Use
- Use when point correspondences are known and rigid or similarity alignment is appropriate.
- Use when squared alignment error is a suitable objective.
When Not to Use
- Avoid when correspondences are unknown without a matching step.
- Avoid when nonrigid deformation or outliers dominate the shape difference.
What It Does
- Centers corresponding point sets
- Optionally normalizes scale
- Finds the best-fitting rotation or similarity transform
Use Cases
- Shape alignment
- Comparing landmark configurations
- Preprocessing point clouds before similarity measurement
Inputs
- Points, segments, polygons, or geometric primitives
- Query or construction parameters
Outputs
- Geometric decision, measurement, or constructed structure
Pseudocode
- 1.Check that point correspondences and dimensions match
- 2.Center both point sets
- 3.Optionally normalize scale
- 4.Compute the best rotation from the cross-covariance matrix
- 5.Apply transform and report residual error
Canonical Example
- Input
- triangle points P and the same triangle Q shifted by (2, 1)
- Output
- translation aligning Q back to P with near-zero residual
- Explanation
- Because the shapes match exactly, centering removes the shift and no deformation is needed.
Implementation Notes
- Center both point sets before solving for rotation.
- State whether reflection and scaling are allowed.
- Validate with a point set transformed by a known rotation and translation.
Common Pitfalls
- Feeding unmatched point orders produces meaningless transforms.
- Forgetting whether scale normalization is enabled changes the interpretation.
- Nearly collinear or duplicate landmarks can make the fitted transform unstable.
Interactive Mechanism Walkthrough
Spatial Primitive & Sweep-Line CanvasShow translating, scaling, and rotating point sets to minimize squared error on a small shape alignment example.
Step 1 of 5: Check that point correspondences and dimensions match
geometry-mechanismNotes
Procrustes analysis measures alignment under a chosen transform family; it does not solve correspondence by itself.
Sources (2)
What each reference supports for this entry.
Computational Geometry: Algorithms and Applications
BookTextbook support for Procrustes Analysis's computational geometry problem family, predicates, structures, or geometric construction.
Open sourceThe Algorithm Design Manual
BookAlgorithm catalog support for Procrustes Analysis in the computational geometry benchmark area.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Alternative toFrechet Distance
Geometry · Core
Procrustes Analysis and Frechet Distance solve related shape alignment problems with different assumptions and tradeoffs.
Compare with thisRelationships
View full relationship pageAlternative toVoronoi-Based Medial Axis
Procrustes Analysis and Voronoi-Based Medial Axis solve related shape alignment problems with different assumptions and tradeoffs.
Alternative toHausdorff Distance
Procrustes Analysis and Hausdorff Distance solve related shape alignment problems with different assumptions and tradeoffs.
Alternative toFrechet Distance
Procrustes Analysis and Frechet Distance solve related shape alignment problems with different assumptions and tradeoffs.
Related Algorithms
Tags
GeometryComputational GeometryStrings