Under what condition is the transformation T(x,y)=(ax+by+e,cx+dy+f)T(x, y) = (ax + by + e, cx + dy + f)T(x,y)=(ax+by+e,cx+dy+f) an isometry?
The matrix (abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix}(acbd) is orthogonal
The determinant is 1
The trace is 0
The matrix is symmetric