3D Rotation of a Triangle about a Diagonal Line

When dealing with 3D transformations, particularly rotations, it's crucial to clearly understand the configuration of the points and the axis of rotation. In this article, we will walk through the process of rotating a triangle with vertices A(0, 0, 0), B(2, 2, 0), and C(4, 2, 0) about a line passing through points (0, 0, 0) and (-2, -2, 0) by an angle of 45°. This involves several steps, including finding the equation of the rotation axis, constructing the rotation matrix, and applying it to each vertex of the triangle.

1. Finding the Equation of the Rotation Axis

The first step is to find the equation of the line passing through the given points (0, 0, 0) and (-2, -2, 0). This is a straightforward process using the point-slope form of a linear equation in 3D. The slope of the line is:

m (-2 - 0) / (-2 - 0) 1

Using the point (0, 0, 0), the equation of the line in point-slope form becomes:

y x

Since there are no z-coordinates specified, we assume z 0 for simplicity, and the line is confined to the xy-plane.

2. Constructing the Rotation Matrix

The next step is to construct the rotation matrix for a 45° counterclockwise rotation about the line y x. The rotation matrix for a counterclockwise rotation of θ degrees about the origin is:

( cos(θ)-sin(θ) sin(θ)cos(θ) )

Substituting θ 45° (or π/4 radians), we get:

( 2/2-2/2 2/22/2 )

3. Applying the Rotation Matrix

Finally, we apply this rotation matrix to each vertex of the triangle to find their new positions:

Vertex A (0, 0, 0)

For vertex A at (0, 0, 0), the new coordinates are:

x22?0-22?0, y22?0 22?0

The result is (0, 0), as A remains stationary.

Vertex B (2, 2, 0)

For vertex B at (2, 2, 0), the new coordinates are:

x22?2-22?20, y22?2 22?222/122

The result is (0, 2√2).

Vertex C (4, 2, 0)

For vertex C at (4, 2, 0), the new coordinates are:

x22?4-22?222, y22?4 22?232/132

The result is (2√2, 3√2).

New Position of the Triangle

After the 45° counterclockwise rotation about the (0, 0) and (-2, -2) line, the new position of the triangle ABC is:

Conclusion

By following the steps of finding the line of rotation, constructing the rotation matrix, and applying it to each vertex of the triangle, we can accurately determine the new positions of the vertices after the rotation. This process is fundamental in various fields, including computer graphics, robotics, and engineering.