rotation of axes

Tilting the x- and y-axes themselves so a slanted conic lines up straight.

The rotation of axes technique replaces $x$ and $y$ with a new pair of coordinates $x'$ , $y'$ measured along axes tilted by an angle $\theta$ , chosen so a conic's $xy$ -cross term vanishes and the curve's standard form becomes recognizable again.

Examples

Rotating by $45°$ turns $xy=1$ (a tilted hyperbola) into the familiar $\frac{x'^2}{2}-\frac{y'^2}{2}=1$ .