Model rigid rotations using stable Earth-fixed vector math. Compute site velocities, azimuth, and components instantly. Switch units, verify results, and share outputs easily today.
| Mode | Inputs | Expected output (illustrative) |
|---|---|---|
| Velocity at a site |
Pole: 60°, −90°; Rate: 0.5 deg/Ma; Site: 34°, 71°; Radius: 6371 km |
East/North in mm/yr; Speed and azimuth computed. |
| Pole to vector | Pole: 60°, −90°; Rate: 0.5 deg/Ma | ωx, ωy, ωz in rad/s with |ω|. |
| Vector to pole | ω = (1e−15, 0, 1e−15) rad/s | Pole near 45°N, 0°E with converted rate. |
Rigid plate motion can be expressed as a rotation about an Euler pole. The angular velocity vector is ω = Ω · p̂, where Ω is the angular speed and p̂ is the unit vector pointing to the pole.
This tool reports East, North, Up in mm/yr, plus horizontal speed and azimuth clockwise from North.
An Euler pole is the point where a rigid-body rotation axis intersects Earth’s surface. In plate kinematics, the relative motion of one plate with respect to another is represented as a rotation about this axis with an angular speed. On a spherical Earth, this single rotation reproduces the full horizontal velocity field of a rigid plate.
Euler pole solutions are often reported in degrees per million years (deg/Ma). A useful scale is that 1 deg/Ma gives about 111 mm/yr at a point 90 degrees from the pole for a 6371 km Earth. Many plate pairs are roughly 0.1 to 2 deg/Ma, while site speed still depends on distance to the pole.
The pole latitude and longitude define a unit vector p̂ in Earth-centered coordinates: p̂ = (cosφ cosλ, cosφ sinλ, sinφ). Multiplying by the angular speed Ω produces the angular velocity vector ω = Ω p̂. The sign follows a right-hand rule about p̂, so consistent longitude and sign conventions matter when comparing external models.
For a site at latitude θ and longitude L, the position vector is r = R (cosθ cosL, cosθ sinL, sinθ), with R typically 6371 km unless an ellipsoid is required. Rigid-body linear velocity is v = ω × r. The magnitude grows with perpendicular distance to the rotation axis and becomes small near the Euler pole itself.
Geodetic interpretation uses local East–North–Up components. After computing v in Earth-centered coordinates, the tool rotates it into the local tangent basis to report eastward and northward motion, plus an optional up component. Horizontal speed is sqrt(E² + N²) and azimuth is the clockwise angle from North, which is useful for map-ready vectors.
Validate outputs against typical plate motion. Fast oceanic plates can reach about 50 to 120 mm/yr, while many continental regions are around 1 to 20 mm/yr. Extreme values usually indicate a rate-unit mismatch, a sign convention issue, or motion relative to an unintended reference plate.
Small errors in pole location can shift predicted velocities noticeably, especially near plate boundaries or at sites close to the pole. Published Euler poles may also use different reference frames or opposite sign conventions. Document pole coordinates, rate unit, and the sign rule. If uncertainties are available, explore sensitivity by perturbing φ, λ, and Ω.
Keep inputs with outputs. CSV suits archiving and batch workflows, while PDF suits reports. Record Earth radius, rate units, and the computed ω vector so later comparisons with GPS or geological constraints remain straightforward.
An Euler pole is a kinematic rotation axis for a rigid plate pair. A plate boundary is the physical deformation zone. A boundary can be complex, while one Euler pole summarizes the best-fit rigid motion.
Speed depends on perpendicular distance to the rotation axis. Points far from the Euler pole move faster, while points near the pole move slowly. This follows directly from v = ω × r.
6371 km is a common spherical approximation and matches many tectonic applications. If you need higher accuracy, use a radius consistent with your reference model or switch to an ellipsoidal workflow externally.
Azimuth is the horizontal direction of motion measured clockwise from geographic North. It is computed from the East and North components using an arctangent and then wrapped to 0–360 degrees.
For ideal rigid rotation on a sphere, velocity is tangential, so Up should be near zero. Small nonzero values can appear from rounding, the chosen basis, or if inputs are near singular geometries.
It uses a right-hand rule about the pole unit vector p̂. If your reference lists the opposite sign, negate the rotation rate or flip ω to match the external convention before comparing values.
Yes. Enter the Euler pole and rate for the plate A relative to plate B in your chosen model. The computed velocities are then interpreted as the motion of sites fixed on plate A in that frame.