Check if a matrix is axis aligned

Question

How can I check if a matrix is axis aligned? Therefor I mean any rotation a multiple of 90 degrees (including 0).

Right now the best I came up with is creating 2 points that are axis aligned, pass them through the matrix and see if the points are still axis aligned.


// return 0, 90, 180 or 270 if axis aligned, else returns -1
static public int test_matrix_axis_aligned(Mat3 m) {

    int rot = -1;

    Vec2 a = create_vec2(0, 0);
    Vec2 b = create_vec2(1000, 0);

    a = mult(m, a);
    b = mult(m, b);

    // TODO rounding with a certain threshold?
    a.x = round(a.x);
    a.y = round(a.y);
    b.x = round(b.x);
    b.y = round(b.y);

    boolean axis_aligned = a.x == b.x || a.y == b.y;

    if (axis_aligned) {

        //float angle = atan2(b.y - a.y, b.x - a.x);
        //println("a: "+angle);
        //if (angle < 0) angle += PI; // wrong for 270
        //rot = (int) round(degrees(angle));

        float dx = a.x - b.x;
        float dy = a.y - b.y;

        if (dx < 0 && dy == 0) {
            rot = 0;
        } else if (dx == 0 && dy < 0) {
            rot = 90;
        } else if (dx > 0 && dy == 0) {
            rot = 180;
        } else if (dx == 0 && dy > 0) {
            rot = 270;
        }

    }

    return rot;
}

But I wonder if it can be done more efficient.

`Vec2` means you're working in 2D? Because that greatly simplifies things. If you exclude scaling and mirroring, there are only 4 matrices which perform a rotation. You can just test if your matrix is exactly one of those four. — MSalters, Jun 18 '19 at 08:58
Being axis-aligned does not imply that the transformation is a rotation. As said in a previous comment, nonuniform scaling and reflections can have the same effect. I mention it because it makes your question ambiguous. The title says "axis-aligned" but the description says "rotation of 90 degrees". It's a subtle but important difference to get a correct solution for your problem. — Gilles-Philippe Paillé, Jun 18 '19 at 14:46

score 0 · Answer 1 · answered Jun 18 '19 at 09:34

If you have components of affine matrix that might be composed from translation, scale and rotation, then you can extract rotation angle as

 fi = atan2(-m12, m11)

where m11 and m12 are the first and the secons components of the first matrix row (note than matrices might be left-handed and right handed, so you perhaps would need to use column rather then row)

Addition: Something similar

score 0 · Answer 2 · answered Jun 18 '19 at 09:34

If you want a routine that will work in any number of dimensions, you could do two steps.

Check that each row (or column) contains all zeros except for one number which is either 1 or -1. If you want to allow scaling in your "rotations" this number could be any non-zero value.
Check that the determinant of the matrix is 1. If you want to allow reflections in your "rotations" you would also allow allow a determinant of -1. If you want to allow scaling in your "rotations" you allow any non-zero determinant.

As a comment said, your job is easier if you know your matrix has only two dimensions.

Check if a matrix is axis aligned

2 Answers2