Combined Translation and Rotation

Combined Translation and Rotation

In general, simplifying the equation of a conic whose axes are tilted with respect to the coordinate axes and whose centre (or vertex) is away from the origin requires both operations together: first a translation of the origin to remove the linear terms, and then a rotation of the (already translated) axes to remove the $xy$ term.

Suppose the origin is first shifted to a point $(h,k)$, giving intermediate coordinates $(x_1,y_1)$ related to the original coordinates $(x,y)$ by

Next, the translated axes are rotated through an angle $\theta$ about the new origin $(h,k)$, giving final coordinates $(X,Y)$ related to $(x_1,y_1)$ by

Combining the two steps, the original coordinates relate to the final coordinates by

The order matters: translation is carried out first (about the original origin), and rotation is carried out second (about the new, translated origin). Doing the two operations in the opposite order — rotating first, then translating — gives a different, generally more complicated relationship, because the direction in which the origin is shifted would then be measured along the rotated axes rather than the original ones. …