Math 2.0

Geophysics Geodynamics and heat flow Theorem

Euler pole of plate motion

Builds on Plate tectonicsEuler's rotation theoremSpecial orthogonal groupAngular velocity

Statement

Let S⊂ℝ3S \subset \R^3 be the sphere of radius RR centred at the origin. Let a plate be a region of SS that moves rigidly on SS.

  1. Finite motion. The displacement of the plate between two times is given by some rotation A∈SO(3)A \in SO(3). If A≠IA \neq I, then AA fixes exactly two points ±𝐞\pm \mathbf{e} of SS, and these are antipodal. The map AA is a rotation by an angle θ∈(0,π]\theta \in (0, \pi] about the axis through ±𝐞\pm \mathbf{e}, and 𝐞\mathbf{e} is a finite rotation pole.
  2. Instantaneous motion. If the motion is differentiable in time, there is an angular velocity 𝝎(t)∈ℝ3\boldsymbol{\omega}(t) \in \R^3. Each point 𝐫\mathbf{r} of the plate has velocity 𝐯=𝝎×𝐫\mathbf{v} = \boldsymbol{\omega} \times \mathbf{r}. If 𝝎≠0\boldsymbol{\omega} \neq 0, the point R 𝝎/∣𝝎∣R \, \boldsymbol{\omega}/\abs{\boldsymbol{\omega}} is the Euler pole. A point at angular distance δ\delta from it has speed ∣𝐯∣=∣𝝎∣Rsin⁡δ\abs{\mathbf{v}} = \abs{\boldsymbol{\omega}} R \sin \delta.
  3. Relative motion. Let two plates PP and QQ have angular velocities 𝝎P\boldsymbol{\omega}_P and 𝝎Q\boldsymbol{\omega}_Q. The velocity of PP seen from QQ is (𝝎P−𝝎Q)×𝐫(\boldsymbol{\omega}_P - \boldsymbol{\omega}_Q) \times \mathbf{r}.

Remarks

Part 1 is Euler's rotation theorem. A rigid motion of SS that fixes the centre is an orthogonal map, and it has determinant 11 because it is connected to the identity. Every element of the special orthogonal group SO(3)SO(3) other than II has eigenvalue 11 with a one-dimensional eigenspace. Part 2 follows by differentiating 𝐫(t)=A(t)𝐫0\mathbf{r}(t) = A(t) \mathbf{r}_0: the matrix A˙A−1\dot A A^{-1} is skew-symmetric, and every skew-symmetric 3×33 \times 3 matrix acts as 𝐫↦𝝎×𝐫\mathbf{r} \mapsto \boldsymbol{\omega} \times \mathbf{r}. Part 3 holds because angular velocities add, while finite rotations in general do not commute. Example: with ∣𝝎∣=1∘\abs{\boldsymbol{\omega}} = 1^\circ per million years and R=6371R = 6371 km, a point at δ=90∘\delta = 90^\circ moves 2πR/360≈1112\pi R/360 \approx 111 km per million years, about 1111 cm per year. Points near the Euler pole move slowly, and transform faults between two plates lie along small circles centred on their relative Euler pole.

Going deeper

For a student

This result makes plate tectonics quantitative. Spreading rates at a ridge vary with the sine of the distance from the pole, and transform faults trace small circles about it. Both facts are used to locate the pole from data. Angular velocities of plates around a closed circuit of boundaries must sum to zero, which checks the consistency of global plate motion models. Finite rotations are composed by matrix multiplication and are the basis of plate reconstructions.

Reading path

To understand this page from the ground up, read these 27 pages in order. The foundations come first.

Show the path as a graph
Cross product, concept, no evidence levelCross productconceptDerivative, concept, no evidence levelDerivativeconceptDeterminant, concept, no evidence levelDeterminantconceptEigenvalue, concept, no evidence levelEigenvalueconceptEigenvector, concept, no evidence levelEigenvectorconceptEuclidean isometry, concept, no evidence levelEuclidean isometryconceptEuclidean plane, object, no evidence levelEuclidean planeobjectEuclidean vector, concept, no evidence levelEuclidean vectorconceptIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionPlate tectonics, concept, no evidence levelPlate tectonicsconceptTrace of a matrix, concept, no evidence levelTrace of a matrixconceptVector space, definition, no evidence levelVector spacedefinitionVelocity, concept, no evidence levelVelocityconceptRotation, definition, no evidence levelRotationdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionOrthogonal group, object, no evidence levelOrthogonal groupobjectDeterminant, definition, no evidence levelDeterminantdefinitionSpecial orthogonal group, object, no evidence levelSpecial orthogonal gr…objectAngular velocity, concept, no evidence levelAngular velocityconceptEuler's rotation theorem, statement, no evidence levelEuler's rotation theo…statementEuler pole of plate motion, statement, no evidence levelEuler pole of plate m…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cross product concept
  2. Derivative concept
  3. Determinant concept
  4. Eigenvalue concept
  5. Eigenvector concept
  6. Euclidean isometry concept
  7. Euclidean plane object
  8. Euclidean vector concept
  9. Identity matrix object
  10. Matrix definition
  11. Plate tectonics concept
  12. Trace of a matrix concept
  13. Vector space definition
  14. Velocity concept
  15. Rotation definition
  16. Square matrix definition
  17. Transpose definition
  18. Linear map definition
  19. Multilinear map definition
  20. Orthogonal matrix definition
  21. Alternating multilinear map definition
  22. Orthogonal group object
  23. Determinant definition
  24. Special orthogonal group object
  25. Angular velocity concept
  26. Euler's rotation theorem statement
  27. Euler pole of plate motion statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Plate_reconstruction
About this page
Tag6SGQDKV15M
URIm2://6SGQDKV15M
Kindstatement, theorem
WritingW0 stub
MSC86A60, 70E15
StatusThis copy of the site has no build data. Run m2 status 6SGQDKV15M.
AboutPlate tectonics
UsesPlate tectonics, Euler's rotation theorem, Special orthogonal group, Angular velocity
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by