Triangle Rotation Calculator

Rotation Matrix:

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

Vertex 1 (x,y,z):

Vertex 2 (x,y,z):

Vertex 3 (x,y,z):

Rotation Angle (θ):

radians

Rotation Axis:

Rotated Vertex 1:

Rotated Vertex 2:

Rotated Vertex 3:

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1. What is Triangle Rotation?

Triangle rotation is a transformation that rotates a triangle's vertices around a specified axis (X, Y, or Z) by a given angle. This is fundamental in computer graphics, physics simulations, and geometric calculations.

2. How Does the Calculator Work?

The calculator uses rotation matrices:

Where:

\( \theta \) — Rotation angle in radians
\( \cos\theta \) — Cosine of the angle
\( \sin\theta \) — Sine of the angle

Explanation: Each vertex is multiplied by the rotation matrix to get its new position.

3. Importance of Rotation Matrices

Details: Rotation matrices preserve distances and angles, making them essential for rigid transformations in 3D space.

4. Using the Calculator

Tips: Enter triangle vertices as (x,y,z) coordinates, rotation angle in radians, and select rotation axis. Default values create a triangle in the XY plane.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between degrees and radians?
A: Radians are the natural unit for angular measurement (2π radians = 360°). Most mathematical functions use radians.

Q2: How do I convert degrees to radians?
A: Multiply degrees by π/180 (e.g., 45° = 45 × π/180 ≈ 0.7854 radians).

Q3: What's the right-hand rule?
A: Point your thumb along the positive axis; fingers curl in the positive rotation direction.

Q4: Can I rotate around an arbitrary axis?
A: This calculator handles principal axes only. Arbitrary axis rotation requires a more complex matrix.

Q5: Why are my z-coordinates changing when rotating around x-axis?
A: X-axis rotation affects YZ plane coordinates, changing both y and z values.