Calculate Position of Point That Rotates Around All Axis

This calculator helps you determine the new position of a point after it has been rotated around all three axes (X, Y, and Z) in 3D space. The calculation uses rotation matrices to apply the rotations sequentially.

Introduction

When a point in 3D space is rotated around multiple axes, its final position depends on the order of rotations. This calculator applies rotations around the X, Y, and Z axes in sequence, which is a common approach in computer graphics and physics simulations.

The rotation matrices used are:

Rotation around X-axis: [1, 0, 0] [0, cos(θx), -sin(θx)] [0, sin(θx), cos(θx)] Rotation around Y-axis: [cos(θy), 0, sin(θy)] [0, 1, 0] [-sin(θy), 0, cos(θy)] Rotation around Z-axis: [cos(θz), -sin(θz), 0] [sin(θz), cos(θz), 0] [0, 0, 1]

The final position is calculated by multiplying the original point coordinates by these matrices in the order X, Y, then Z.

How to Use This Calculator

Enter the original coordinates of the point (X, Y, Z).
Enter the rotation angles in degrees for each axis (X, Y, Z).
Click "Calculate" to see the new position after rotation.
Use the "Reset" button to clear all inputs.

The calculator will display the new coordinates and show a visualization of the rotation path.

Formula

The final position (x', y', z') of a point after rotation is calculated by applying the rotation matrices in sequence:

x' = x * cos(θy) * cos(θz) + y * (sin(θx) * sin(θy) * cos(θz) - cos(θx) * sin(θz)) + z * (cos(θx) * sin(θy) * cos(θz) + sin(θx) * sin(θz)) y' = x * cos(θy) * sin(θz) + y * (sin(θx) * sin(θy) * sin(θz) + cos(θx) * cos(θz)) + z * (cos(θx) * sin(θy) * sin(θz) - sin(θx) * cos(θz)) z' = -x * sin(θy) + y * sin(θx) * cos(θy) + z * cos(θx) * cos(θy)

Where θx, θy, and θz are the rotation angles in radians for the X, Y, and Z axes respectively.

Worked Example

Let's calculate the new position of a point at (1, 0, 0) after rotating 90° around the X-axis, 90° around the Y-axis, and 0° around the Z-axis.

Convert angles to radians: 90° = π/2 radians.
Apply the rotation formulas:
x' = 1 * cos(π/2) * cos(0) + 0 * (sin(π/2) * sin(π/2) * cos(0) - cos(π/2) * sin(0)) + 0 * (cos(π/2) * sin(π/2) * cos(0) + sin(π/2) * sin(0)) = 0 * 1 * 1 + 0 + 0 = 0 y' = 1 * cos(π/2) * sin(0) + 0 * (sin(π/2) * sin(π/2) * sin(0) + cos(π/2) * cos(0)) + 0 * (cos(π/2) * sin(π/2) * sin(0) - sin(π/2) * cos(0)) = 0 * 0 + 0 + 0 = 0 z' = -1 * sin(π/2) + 0 * sin(π/2) * cos(π/2) + 0 * cos(π/2) * cos(π/2) = -1 * 1 + 0 + 0 = -1
The new position is (0, 0, -1).

This shows that rotating a point on the X-axis by 90° around the Y-axis moves it to the negative Z-axis.

FAQ

What is the order of rotations?: The calculator applies rotations in the order X, Y, then Z. Changing the order can produce different results.
Can I rotate around all three axes simultaneously?: No, this calculator applies rotations sequentially. Simultaneous rotation would require a different approach using quaternions.
What units should I use for the angles?: Enter the angles in degrees. The calculator will convert them to radians for the calculations.
Is there a limit to the rotation angles?: No, you can use any angle value. Large angles may produce unexpected results due to floating-point precision limits.