Best rigid rotation

teaching-model

The Transparent Star-Map Table: matches known, pose unknown

This is a teaching analogy, not a story from the paper. Stars on two transparent maps are already paired by number. The task is not to discover matches, but to move one map so paired stars overlap as closely as possible.

1. Chapter 1: paired stars are already known

   The calibrator receives two maps with matching labels, so star 1 needs no match search. Only the pose of the whole map is unknown.

   Decoded paper language: The Kabsch problem assumes known correspondences and seeks an orthogonal rotation minimizing weighted squared error.

   Kabsch 1976, p. 922, eqs. (1)–(3).

2. Chapter 2: align the centers first

   The calibrator finds each map's average center and moves both to the table origin, temporarily removing overall translation.

   Decoded paper language: Subtracting each centroid separates translation from rotation.

   Kabsch 1976, p. 922.

3. Chapter 3: rotate the sheet without stretching it

   The transparent sheet may rotate as a whole but cannot stretch, shrink, or become a mirror image. The calibrator seeks the angle with minimum total error.

   Decoded paper language: The rotation matrix is orthogonal and preserves distances and angles. The paper derives the optimal rotation and treats rank-deficient cases.

   Kabsch 1976, pp. 922–923, eqs. (4)–(12).

4. Chapter 4: return the center to its proper place

   After rotation, the calibrator translates the sheet to the target map and checks the residual distance of every pair.

   Decoded paper language: Translation is recovered from the centroids after optimal rotation. RMSD summarizes residuals over matched points but does not validate the matches.

   Kabsch 1976, pp. 922–923.

A four-stop plain-language map of Kabsch

Follow one question only: with matches already known, how can one rigid motion fit two point sets as closely as possible?

1. 1. Treat correspondence as given

   Kabsch does not discover residue matches between proteins; it receives coordinates that are already paired.

   Keep only this: Match first, fit second; this paper solves the second half.

   Kabsch 1976, p. 922.

2. 2. Subtract each centroid

   Subtract each set's average position so both point sets are centered at the origin.

   Keep only this: Centering temporarily removes translation.

   Kabsch 1976, p. 922.

3. 3. Find the best orthogonal rotation

   Without stretching the geometry, find the rotation minimizing the total squared distance across all pairs.

   Keep only this: The optimum is a compromise across all points, not a perfect fit for each one.

   Kabsch 1976, pp. 922–923.

4. 4. Restore translation, then read RMSD

   After rotation, recover translation from the centroids and use RMSD to summarize the residual scale over matched points.

   Keep only this: Low RMSD only says whether the selected pairs fit closely.

   Kabsch 1976, pp. 922–923.

Check understanding

1. Does Kabsch 1976 discover residue matches by itself?

   • No; it assumes known correspondences
   • Yes, and it needs no coordinates
   • Only when RMSD is zero

   It solves optimal rigid rotation for known matches, not the full structural-search problem.

2. Why subtract centroids first?

   • To separate translation from rotation
   • To scale every point to the same length
   • To delete outliers

   Centering gives both sets a common origin so rotation can be solved separately.

3. Does low RMSD prove the residue matches are correct?

   • No; it only summarizes residuals for the current pairs
   • Yes; any low value proves homology
   • Yes, but only for three points

   Short or selectively chosen matches can look good in RMSD; aligned length, coverage, and correspondence plausibility still matter.

Completion task: Without looking above, use four sentences to state the input, why centering is used, what rotation may not do, and what RMSD does not answer.

paper-fact

Problem setup: matches known, pose unknown

Given matched vectors xₙ and yₙ with weights wₙ, the goal is an orthogonal matrix U that minimizes weighted squared deviations without stretching or scaling. Translation is removed first by centering each set.

This step does not discover residue correspondence. If the matches are wrong, the optimal rotation can only make a least-squares compromise over those wrong pairs.

E(U) = ½ Σₙ wₙ ‖U xₙ − yₙ‖², subject to UᵀU = I

E is the weighted sum of squared distances over known pairs; U must preserve lengths and angles.

Kabsch 1976, p. 922, eqs. (1)–(3).

paper-fact

The method in four steps

First subtract each point set's centroid to remove translation. Second form the relation matrix R and symmetric matrix S from matched points.

Third solve for the error-minimizing rotation U under the orthogonality constraint. Fourth recover translation from the target centroid and the rotated source centroid.

The paper also treats rank-deficient and planar cases, reminding us that nearly collinear or coplanar points leave some rotational directions weakly constrained.

Kabsch 1976, pp. 922–923, eqs. (4)–(12).

project-reading

How to read RMSD

RMSD is the root mean square residual after the best fit. It compresses overall error into one length-valued number, but it does not reveal which point carries the error or whether the correspondence is sensible.

A small number of outliers can pull the least-squares solution toward themselves. The lab's outlier preset is designed to expose this limitation.

project-reading

Its role in later methods

CE, FAST, and TM-align all need to superpose selected residue pairs at some stage; the Kabsch problem is the geometric subproblem inside that workflow. DALI first searches through internal distance matrices that are invariant to rigid rotation.

Worked example: a moved triangle

A contains (0,0,0), (2,0,0), and (0,1,0). B is the same triangle rotated by 90° and translated near (4,−2,1). All three matches are known.

1. Compute the centroid of A and B and subtract it from every point.
2. Build the relation matrix from centered points and solve for the length- and angle-preserving rotation.
3. Apply the rotation to B, then recover translation from the two centroids.

With no noise the three pairs coincide and fitted RMSD approaches zero. Noise or an outlier turns the result into a least-squares compromise across all points.

Glossary

Centroid／質心

The coordinate-wise mean position of a point set.

Orthogonal rotation／正交旋轉

A rigid rotation that preserves distances and angles without reflection.

RMSD

The square root of the mean squared distance between matched points.

Interactive lab

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Sources and limits

• DOI / IUCr

• It does not discover residue matches or search protein structures.
• Least squares is sensitive to outliers, and one rigid transform cannot represent flexibility or multi-domain motion.
• The point cloud is synthetic, not a biological conclusion from PDB data.