Solve 2D and 3D frame transformations precisely. Enter rotations, offsets, and points for chained analysis. See matrices, coordinates, graphs, and exports in one place.
Use the grid below. Large screens show three columns, smaller screens show two, and phones show one.
| Case | Dimension | Input Point | Translation | Rotation | Output Point |
|---|---|---|---|---|---|
| Example A | 2D | (2, 1) | (4, 3) | 90° about Z | (3, 5) |
| Example B | 3D | (1, 2, 3) | (5, 1, 2) | 90° about Z | (3, 2, 5) |
| Example C | 3D Chained | (1, 0, 2) | T1 then T2 | Order ZYX | Use the form to inspect intermediate values |
T = [ [cosθ, -sinθ, tx], [sinθ, cosθ, ty], [0, 0, 1] ]
P′ = T × P, where P = [x, y, 1]T.
T = [ R t; 0 0 0 1 ]
P′ = T × P, where P = [x, y, z, 1]T.
The rotation block is composed by the selected order, such as R = RZ × RY × RX for ZYX.
Ttotal = T2 × T1
This means the first transform acts first on the point, and the second transform acts on that intermediate result.
T-1 = [ RT , -RTt; 0 1 ] for 2D, and the 4×4 equivalent for 3D.
It is a matrix method that combines rotation and translation in one operation. It lets you transform points, vectors, and coordinate frames using one consistent multiplication rule.
They let translation fit into matrix multiplication. Without homogeneous coordinates, translation needs a separate addition step, which makes chained spatial operations less convenient.
3D rotations are not commutative. Changing the order changes the final orientation. That is why XYZ and ZYX can produce different transformed coordinates from the same angle values.
The inverse transform maps transformed coordinates back to the original frame. It is useful for robot kinematics, camera pose recovery, and checking whether your transform behaves as expected.
Yes. Enable the chained option and enter Transform 2. The calculator computes the composite matrix and also shows the intermediate point after Transform 1.
A proper rigid rotation matrix usually has determinant 1. If the value drifts far from 1, your transform may include an invalid rotation or strong numerical distortion.
No. This page focuses on rigid transformations made from rotation and translation. That keeps the matrices practical for robotics, graphics, navigation, and analytic geometry tasks.
The graph plots the original frame, the transformed frame, the original point, the intermediate point, and the final transformed point. It helps you verify orientation and displacement visually.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.