Rotation Transformation Calculator

Rotation Matrix:

\[ R_x(\theta) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \\ \end{bmatrix} \] \[ R_y(\theta) = \begin{bmatrix} \cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos\theta \\ \end{bmatrix} \] \[ R_z(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} \]

X Coordinate:

Y Coordinate:

Z Coordinate:

Rotation Angle (θ):

radians

Rotation Axis:

Transformed Point:

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1. What is a Rotation Matrix?

A rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. It rotates points in the xy-Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system.

2. How Does the Calculator Work?

The calculator applies the standard rotation matrices:

\[ \text{X-axis: } \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \\ \end{bmatrix} \] \[ \text{Y-axis: } \begin{bmatrix} \cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos\theta \\ \end{bmatrix} \] \[ \text{Z-axis: } \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} \]

Where:

θ — Rotation angle in radians
(x, y, z) — Original point coordinates
(x', y', z') — Transformed point coordinates

Explanation: The matrix multiplication transforms the original coordinates to new coordinates after rotation.

3. Importance of Rotation Transformations

Details: Rotation matrices are fundamental in computer graphics, robotics, physics simulations, and any application involving 3D transformations.

4. Using the Calculator

Tips: Enter the original point coordinates, rotation angle in radians, and select the rotation axis. The calculator will output the new coordinates after rotation.

5. Frequently Asked Questions (FAQ)

Q1: Why use radians instead of degrees?
A: Radians are the natural unit for angular measurement in mathematics and most programming languages.

Q2: What's the difference between active and passive rotations?
A: Active rotations rotate the object itself, while passive rotations rotate the coordinate system. This calculator performs active rotations.

Q3: Can I chain multiple rotations?
A: Yes, you can multiply rotation matrices together to perform multiple rotations in sequence.

Q4: What about rotation around an arbitrary axis?
A: This calculator handles only principal axis rotations. Arbitrary axis rotation requires a more complex matrix.

Q5: How do I convert degrees to radians?
A: Multiply degrees by π/180 (approximately 0.0174533).