3D Coordinate Transform Calculator

Apply rotation (Euler angles: X, Y, Z) and translation to any 3D point. Visualize original and transformed points in real‑time 3D space.

Original Point P
Translation T (dX,dY,dZ)
Scale before rotation/translation (S).
Examples:
Privacy-first: All computations local. 3D scene rendered in your browser – no data stored.
Transformed Point P'
( — , — , — )
Order: Scale → Rotate (XYZ) → Translate
4x4 Homogeneous Matrix (Row-major)
[ 1.00 0.00 0.00 0.00 ]
[ 0.00 1.00 0.00 0.00 ]
[ 0.00 0.00 1.00 0.00 ]
[ 0.00 0.00 0.00 1.00 ]
Original Point (P) Transformed Point (P') Origin (0,0,0) X (red) Y (green) Z (blue)

3D Coordinate Transformations: Mathematical Foundation & Applications

A 3D coordinate transformation changes the position and orientation of a point in Euclidean space. This calculator applies extrinsic Euler angles (XYZ order) followed by translation and optional uniform scaling. The combined transformation is represented as a single 4×4 homogeneous matrix, essential for computer graphics and robotics.

Transformation Pipeline

P' = T · Rz(γ) · Ry(β) · Rx(α) · S · P

Where Rx, Ry, Rz are rotation matrices, T translation, S uniform scale. Order matters: scaling first, then rotations (X→Y→Z), then translation.

Rotation X (θ)
[[1,0,0],[0,cosθ,-sinθ],[0,sinθ,cosθ]]
Rotation Y (φ)
[[cosφ,0,sinφ],[0,1,0],[-sinφ,0,cosφ]]

Homogeneous Coordinates: Using 4x4 matrices allows translation to be expressed as matrix multiplication, enabling concatenation of infinite transforms.

Real‑World Applications

  • Robotics: Forward kinematics for manipulator arms.
  • Computer Graphics: Object and camera transformations in 3D engines.
  • Aerospace: Aircraft attitude (roll, pitch, yaw) and navigation.
  • Medical Imaging: Rigid registration of CT/MRI scans.

Gimbal lock occurs when pitch (Y rotation) is ±90°, causing X and Z rotations to align. You can test it by setting Y=90° and noticing X and Z produce similar effects. Quaternions avoid this issue.

Double-precision arithmetic, accurate to 12+ decimal places. Validated against MATLAB and SciPy rotation libraries.
Last update: May 2026. Fully interactive WebGL-based 3D visualization.