Rotating A Line Around A N Axis Calculator

Rotating a line around an arbitrary axis in 3D space is a fundamental operation in computer graphics, physics simulations, and engineering design. This calculator provides an accurate method to determine the new position of a line after rotation around any given axis.

Introduction

When working with 3D geometry, rotating a line around an arbitrary axis requires careful application of rotation matrices. The process involves defining the axis of rotation, determining the angle of rotation, and applying the transformation to the line's endpoints.

This calculator assumes the axis of rotation passes through the origin (0,0,0) for simplicity. For axes not centered at the origin, you would need to translate the line and axis to the origin first, perform the rotation, and then translate back.

How to Use the Calculator

To use the calculator:

Enter the coordinates of the line's starting point (x1, y1, z1)
Enter the coordinates of the line's ending point (x2, y2, z2)
Specify the axis of rotation (nx, ny, nz)
Enter the rotation angle in degrees
Click "Calculate" to see the rotated line coordinates

The calculator will display the new coordinates of both endpoints after rotation and provide a visualization of the rotation.

Formulas and Assumptions

The rotation of a point around an arbitrary axis is calculated using the following steps:

Rotation Matrix Construction

The rotation matrix R for rotating a point around an axis defined by unit vector (nx, ny, nz) by angle θ (in radians) is:

R = [nx²(1-c) + c, nx*ny(1-c) - nz*s, nx*nz(1-c) + ny*s]
[ny*nx(1-c) + nz*s, ny²(1-c) + c, ny*nz(1-c) - nx*s]
[nz*nx(1-c) - ny*s, nz*ny(1-c) + nx*s, nz²(1-c) + c]

where c = cos(θ) and s = sin(θ)

Point Rotation

To rotate a point (x, y, z) around the axis, multiply it by the rotation matrix R:

x' = R[0][0]*x + R[0][1]*y + R[0][2]*z
y' = R[1][0]*x + R[1][1]*y + R[1][2]*z
z' = R[2][0]*x + R[2][1]*y + R[2][2]*z

This calculator assumes the axis vector is normalized to a unit vector before applying the rotation.

Worked Example

Let's rotate the line from (1, 0, 0) to (1, 1, 0) around the axis (0, 0, 1) by 90 degrees.

The axis (0, 0, 1) is the z-axis. Rotating around this axis by 90 degrees should transform the line to (0, 1, 0) to (0, 1, 0).

Using the calculator with these values will show the rotated coordinates and visualize the transformation.

FAQ

Can I rotate a line around an axis that doesn't pass through the origin?: No, this calculator assumes the axis passes through the origin. For other cases, you would need to translate the line and axis to the origin first.
What if my axis vector isn't normalized?: The calculator will normalize the axis vector before applying the rotation.
How accurate are the calculations?: The calculations use standard rotation matrix mathematics and should be accurate to within floating-point precision limits.
Can I rotate multiple lines with this calculator?: This calculator is designed for single lines. For multiple lines, you would need to apply the rotation to each line individually.