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Frames & rigid motion

Rigid body motion in 2D and 3D

Special orthogonal groups, Euler vs axis–angle, and homogeneous coordinates for compact composition.

~60 min read + exercises

Rigid body motion in 2D and 3D

You will unify rotation representations and see why homogeneous coordinates are the standard bookkeeping trick in robotics, graphics, and vision calibration.

Figure

Planar rigid motion: rotate θ, then translate t

Planar rigid motion: rotate θ, then translate (x, y)Three degrees of freedom describe any body pose in the plane.xᵂyᵂworldt = (x, y)xᵇyᵇbodyθpᵂ = R(θ) pᵇ + tSE(2): 3 DoF (x, y, θ)
The body frame's axes are R(θ) applied to the world axes, anchored at the translated origin. Three numbers (x, y, θ) describe any pose.

Learning objectives

  • Parameterize planar rotation with one angle; describe the 2D rotation matrix.
  • Name three 3D rotation parameterizations and one trade-off for each.
  • Build a 4×4 homogeneous transform that combines rotation and translation.

Prerequisites

  • Coordinate frames lesson.
  • Comfort multiplying 3×3 matrices.

Step 1 — Planar rigid motion (SE(2))

A rigid body in the plane has three degrees of freedom: translation (x,y)(x, y) and rotation θ\theta.

Rotation:

R(θ)=[cos⁡θ−sin⁡θsin⁡θcos⁡θ]R(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}

Transform a point p\mathbf{p} in body frame to world:

pw=R(θ) pb+t\mathbf{p}^w = R(\theta)\, \mathbf{p}^b + \mathbf{t}

Checkpoint: How many degrees of freedom in SE(2)?

Step 2 — SO(3): the space of 3D rotations

The set of valid 3D rotation matrices is SO(3) (“special orthogonal”).

  • Matrix storage: 9 numbers with 6 constraints — redundant but easy to compose.
  • Euler angles: 3 numbers but suffer gimbal lock in some conventions and can be ambiguous.
  • Axis–angle / rotation vector: compact, useful for optimization and IMU integration.
  • Unit quaternions: 4 numbers with 1 constraint, smooth interpolation (SLERP), no gimbal lock.

Figure

Four ways to store a 3D rotation

Three popular ways to store a 3D rotationEach is convenient for some task — almost no team uses just one.Rotation matrixR ∈ SO(3)+ easy to compose− 9 numbers + 6 constraintsEuler angles(φ, θ, ψ)+ human-readable− gimbal lockϑûAxis–angle(û, ϑ)+ compact, IMU-friendly− discontinuous near πS³Unit quaternionq ∈ S³+ smooth, no gimbal lock− double cover (±q)
Almost no real codebase uses just one — matrices for composing, quaternions for integration, axis–angle for IMUs, Euler only for human-readable logs.

Exercise: When would you avoid Euler angles in estimation code?

Step 3 — Homogeneous transforms in SE(3)

Pack rotation and translation into one 4×44 \times 4 matrix:

T=[Rt0⊤1]T = \begin{bmatrix} R & \mathbf{t} \\ \mathbf{0}^\top & 1 \end{bmatrix}

(0\mathbf{0} is the 1×31 \times 3 zero row under RR, then a 11 in the bottom-right.)

A homogeneous point [x,y,z,1]⊤[x, y, z, 1]^\top in frame bb maps to frame ww by multiplying by the transform from bb to ww (read direction per your software’s convention — always verify against a known pose).

Checkpoint: Why is the bottom row (0, 0, 0, 1) useful rather than storing R and t separately?

Step 4 — Inverting rigid transforms

The inverse of a rigid transform is:

T−1=[R⊤−R⊤t0⊤1]T^{-1} = \begin{bmatrix} R^\top & -R^\top \mathbf{t} \\ \mathbf{0}^\top & 1 \end{bmatrix}

Intuition: “undo translation in the rotated coordinates, then undo rotation.”

Exercise: Derive −R⊤t-R^\top \mathbf{t} in words: why not just −t-\mathbf{t}?

Step 5 — Velocities preview (twists)

Rigid motion over time leads to angular velocity ω\boldsymbol{\omega} and linear velocity v\mathbf{v}. In advanced courses, these are packaged as twists and mapped through Jacobians — the kinematics lessons next.

Check your understanding

  1. Why are unit quaternions preferred over Euler angles for integrating IMU orientation?
  2. What does det⁡(R)=−1\det(R) = -1 imply physically?
  3. Compute R(90∘)R(90^\circ) about the zz axis and verify R⊤R=IR^\top R = I.

Lab-style stretch goal (optional)

Implement axis–angle → rotation matrix (Rodrigues) and test on random vectors:

∥Rv∥≈∥v∥\|R\mathbf{v}\| \approx \|\mathbf{v}\|