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Learning Resource

Rigid Body Transformations

Move and rotate rigid objects while learning how translation vectors and rotation matrices form a complete rigid body transform.

Workspace

Dimension Mode Object

Choose dimension first, then pick the mode for that dimension.

Quick Start

Pick an example.
Choose dimension and mode in Workspace.
Adjust values or enter your own matrix/basis.
Press Animate to watch identity transform to target transform.

Examples for current mode

Guided Controls

Yaw (rad) Pitch (rad) Roll (rad)

Tx Ty Tz

Euler order used here is ZYX (Yaw → Pitch → Roll).

Custom Input (Matrix / Basis)

Enter either a rotation matrix or basis vectors. Basis vectors become matrix columns: R = [u v w].

Rotation Matrix R (row-major)

Basis Vectors (u, v, w)

u = [ux, uy, uz] v = [vx, vy, vz] w = [wx, wy, wz]

Animation

Duration (seconds) Show trajectory trace Show local car axes

3D View

World frame is fixed. Car frame moves according to your transform.

Drag scene or drag the cube to orbit. Scroll to zoom. Click faces/corners to snap view.

World axes Local car axes Car origin trace

Live Summary (What You Are Seeing)

Trajectory Trace (purple path in the viewport)

Trace: On Samples: 0 Path Length: 0.0000 scene units

Live Graphs

The charts below update in real time. The x-axis is recent time in seconds, and all position values use scene units.

Position and Distance

tx ty tz |t| distance

Distance |t| tells how far the object is from the world origin. Positive and negative tx/ty/tz show direction in the world frame.

Rotation Quality

det(R) R^T R deviation det target = 1.0 deviation limit = 0.08

Good rotation matrices stay near det(R)=1 and keep R^T R deviation near 0 (below 0.08 in this tool).

Current Rotation Matrix R

Show cos/sin form

Translation Vector t

Homogeneous Transform T

Learning Notes

Rigid transform: keeps shape and size fixed.
Point transform: p' = R p + t.
Homogeneous form: [p'; 1] = T [p; 1].
Valid rotation matrix: R^T R = I and det(R) = 1.