I’m trying to calculate an axis-aligned bounding box of a circle that is transformed through a 2D affine matrix that can contain any operations (translation, rotation, scale, skew).

I think that good approach would be to transform the circle with the matrix and obtain a rotated ellipse, but I don’t know how to do it properly. What I’ve so far is a code that can calculate a bounding box of a rotated ellipse (not attached) and a code that partially can transform circle to an ellipse (center and radiuses, but ellipse missing rotation):

transformCircle(in matrix, in circle, out ellipse) {
    ellipse.center = matrix * circle.center

    // Calculate conjugated half-diameters
    f1 = matrix * vec2(circle.radius, 0)
    f2 = matrix * vec2(0, circle.radius)
    m = f1.x^2 + f1.y^2 + f2.x^2 + f2.y^2
    n = cross(b, c)
    p = sqrt(m + 2*n)
    q = sqrt(m - 2*n)

    ellipse.radiusX = 0.5 * (p + q);
    ellipse.radiusY = 0.5 * (p - q);

    ellipse.rotation = ???;  // Need help here
}

My questions would be:

• How to calculate ellipse rotation from conjugated half-diameters?
• Alternatively how to calculate AABB of circle transformed by a 2D matrix without ellipses, if possible?