Rotation of axes
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A rotation of axes is a form of Euclidean transformation in which the entire xy-coordinate system is rotated in the counterclockwise direction with respect to the origin (0, 0) through a scalar quantity denoted by θ.
Rotation of loci
If a locus is defined on the xy-coordinate system as , then it is denoted as on the rotated x'y'-coordinate system. Likewise, if a locus is defined on the x'y'-coordinate system as , then it is denoted as on the "un-rotated" xy-coordinate system.
Elimination of the xy term by the rotation formula
For a general, non-degenerate second-degree equation , the term can be removed by rotating the xy-coordinate system by an angle , where
If then .
Derivation of the rotation formula
Now, the equation is rotated by a quantity , hence
Expanding, the equation becomes
Collecting like terms,
In order to eliminate the x'y'-term, the coefficient of the x'y'-term must be set equal to 0.
Identifying rotated conic sections
A non-degenerate conic section with the equation can be identified by evaluating the value of :