slop.boutique

A lab for computation and intelligence as material and medium. Run by Manuel and Ramon.

026 · 23 October 2026 · math · robotics · education

SO(3) Playground

An interactive guide to the rotation group: turn an airplane from a ball of rotation vectors, compare geodesics with straight lines, watch composition depart from addition, and check Jacobians against the exact perturbation.

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Perturbation lab in the dark theme: the exact image of a sphere of perturbations pushed through the exp map, drawn in teal, tangled with the violet linear prediction from the Jacobian

In early access for subscribers until 22 November 2026. Subscribe to run it now.

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A hands-on guide to the rotation group SO(3), the maths behind robot arms, cameras and state estimators. Every point in a ball of radius π is a rotation: the direction is the axis and the length is the angle. Move a point and watch an airplane turn with it; then perturb a rotation and check the Jacobian against what actually happens.

  • The rotation-vector ball: drag the point or the airplane, spin about a fixed world axis, and watch the point reach the surface at 180° and reappear on the opposite side, with the angle, axis and rotation matrix read out live.
  • Truncated series: draw the airplane with only N terms of the exponential series and see it shear, with its distance from SO(3) measured.
  • Geodesic vs straight line: interpolate between two rotations along the geodesic (slerp) and along a straight line in the ball, with a plot of how fast each one turns and their total lengths.
  • Composing is not adding: drag two rotation vectors and watch the Baker–Campbell–Hausdorff series walk from a + b toward the true composition, order by order, with the error of each order and of the reversed product.
  • Perturbation lab: push a small sphere of perturbations through rotating a point, a rotation residual, the exp map itself or the inverse, with right or left perturbations, and compare the exact image with the linear prediction from the Jacobian.
  • Catch the common mistake: switch to a typical wrong Jacobian and the log-log error plot shows it, with the measured order dropping from 2 to 1. It is the numerical check to run on any Jacobian you derive by hand.

Every panel shows the formulas it uses, and the page follows your light or dark theme. Its sibling, the Lie Group IK Playground, uses the same maths to solve inverse kinematics for a robot arm.

Full interface on the Rotation ball tab: the rotation-vector ball, the airplane it rotates with its dashed rotation axis, and the side panel with angle, axis, rotation matrix and the Exp formula
Full interface on the Rotation ball tab: the rotation-vector ball, the airplane it rotates with its dashed rotation axis, and the side panel with angle, axis, rotation matrix and the Exp formula
Close-up without the interface, dark theme: the rotation ball with vectors a and b, a + b and the two compositions, next to the airplane turned by Exp(a)Exp(b) with a dashed outline of Exp(a + b)
Close-up without the interface, dark theme: the rotation ball with vectors a and b, a + b and the two compositions, next to the airplane turned by Exp(a)Exp(b) with a dashed outline of Exp(a + b)
Composing is not adding: the BCH chain of dots from a + b toward Exp(a)Exp(b), with a table of errors for orders 1 to 4 and for the reversed order, and the BCH series written out
Composing is not adding: the BCH chain of dots from a + b toward Exp(a)Exp(b), with a table of errors for orders 1 to 4 and for the reversed order, and the BCH series written out
Close-up of the shaded airplane in the dark theme, turned by a rotation vector, with its dashed rotation axis
Close-up of the shaded airplane in the dark theme, turned by a rotation vector, with its dashed rotation axis
Jacobian check for rotating a point with a common mistake, the left-perturbation Jacobian used with a right perturbation: the error plot's slope and the measured order drop to 1
Jacobian check for rotating a point with a common mistake, the left-perturbation Jacobian used with a right perturbation: the error plot's slope and the measured order drop to 1
Close-up of the perturbation lab for rotating a point: the exact image of the epsilon-sphere curving around the sphere of radius |p|, against the flat disk the linear prediction gives
Close-up of the perturbation lab for rotating a point: the exact image of the epsilon-sphere curving around the sphere of radius |p|, against the flat disk the linear prediction gives
Geodesic vs straight line: the curved geodesic from A to B and the dashed straight line through the ball, the solid and outlined airplanes, and a plot of angular speed along both paths with their total lengths
Geodesic vs straight line: the curved geodesic from A to B and the dashed straight line through the ball, the solid and outlined airplanes, and a plot of angular speed along both paths with their total lengths
Close-up of the perturbation lab for a rotation residual Log(Rm transpose R): the exact and predicted images of the epsilon-sphere almost coincide
Close-up of the perturbation lab for a rotation residual Log(Rm transpose R): the exact and predicted images of the epsilon-sphere almost coincide
The airplane drawn with a two-term truncated exponential series, visibly sheared, with its distance from SO(3) highlighted in the panel
The airplane drawn with a two-term truncated exponential series, visibly sheared, with its distance from SO(3) highlighted in the panel
Close-up of the airplane halfway along the geodesic, with the dashed outline of the straight-line interpolation turned differently
Close-up of the airplane halfway along the geodesic, with the dashed outline of the straight-line interpolation turned differently
The perturbation lab's side panel: function and perturbation choices, the Jacobian matrix used, and the log-log plot of error against epsilon with the measured orders 2.00 and 1.00
The perturbation lab's side panel: function and perturbation choices, the Jacobian matrix used, and the log-log plot of error against epsilon with the measured orders 2.00 and 1.00
Spinning about a fixed axis near 180 degrees: the ball marks where the point wraps to on the opposite side, while the airplane turns smoothly
Spinning about a fixed axis near 180 degrees: the ball marks where the point wraps to on the opposite side, while the airplane turns smoothly