Shape Rotation 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:

Rotated Point:

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1. What is Shape Rotation?

Shape rotation is a transformation that turns a shape around a fixed point (in 2D) or axis (in 3D) by a certain angle. This calculator handles 3D rotations around the X, Y, or Z axes.

2. How Does the Calculator Work?

The calculator uses rotation matrices:

Where:

\( \theta \) — Rotation angle in radians
\( R_x \) — Rotation matrix around X-axis
\( R_y \) — Rotation matrix around Y-axis
\( R_z \) — Rotation matrix around Z-axis

Explanation: The calculator multiplies the point coordinates by the appropriate rotation matrix to get the new coordinates.

3. Importance of Rotation Calculation

Details: Rotation calculations are essential in computer graphics, robotics, physics simulations, and engineering applications where objects need to be transformed in 3D space.

4. Using the Calculator

Tips: Enter the original coordinates (x,y,z), the rotation angle in radians, and select the rotation axis. The calculator will output the new coordinates after rotation.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between degrees and radians?
A: Radians are the standard unit for angular measurement in mathematics. 1 radian ≈ 57.3 degrees. To convert degrees to radians, multiply by π/180.

Q2: Can I rotate around an arbitrary axis?
A: This calculator handles only principal axes (X,Y,Z). For arbitrary axes, a more complex rotation matrix is needed.

Q3: What's the right-hand rule in rotation?
A: The right-hand rule determines positive rotation direction: point your thumb along the axis, fingers curl in the positive rotation direction.

Q4: How are multiple rotations handled?
A: For multiple rotations, you would multiply the rotation matrices in sequence (order matters) before applying to the point.

Q5: What's the difference between active and passive rotation?
A: Active rotation moves the object in a fixed coordinate system. Passive rotation rotates the coordinate system around the object. This calculator performs active rotation.