Rotation of Axes

Definition The rotation of the plane about the origin through the angle $\theta$, counterclockwise when $\theta > 0$, sends each point $(x, y)$ to

To see why, write the point as $(r\cos\varphi, r\sin\varphi)$. The rotation adds $\theta$ to the angle, so the image is $(r\cos(\varphi + \theta), r\sin(\varphi + \theta))$, and the sum formulas expand these into the expression above. A rotation keeps all distances and angles. If a curve is the solution set of $G(x, y) = 0$, the curve rotated through $\theta$ is the solution set of

because a point lies on the rotated curve exactly when rotating it back through $-\theta$ lands on the original curve.

Definition Turn the $x$-axis and the $y$-axis counterclockwise about the origin through the angle $\theta$, and call the turned axes the $X$-axis and the $Y$-axis. Every point stays where it is but gets new coordinates $(X, Y)$, measured along the new axes. Old and new coordinates are related by

and, solved the other way,

The first pair is the rotation through $\theta$ applied to the point $(X, Y)$, because the new axes are the old axes rotated through $\theta$. The second pair is the rotation through $-\theta$.

Definition In a second degree equation

with $A$, $B$ and $C$ not all zero, the term $Bxy$ is called the mixed term or cross term. Rotating the axes through $\theta$ turns the equation into one of the same kind in $X$ and $Y$, with the coefficients

For a parabola, a hyperbola or an ellipse that is not a circle, the mixed term is nonzero exactly when the axes of the conic are not parallel to the coordinate axes.

Definition The rotation angle $\theta$ is measured from the positive $x$-axis to the positive $X$-axis, counterclockwise when positive, and it may be given in radians or degrees. The mixed term disappears when $B' = 0$, that is when

When $B = 0$ there is nothing to remove and no rotation is needed. When $B \ne 0$ the standard choice is the unique angle with $0 < \theta < \frac{\pi}{2}$ and

Since $0 < 2\theta < \pi$, the value $\sin 2\theta$ is positive and $\cos 2\theta$ has the sign of $\frac{A - C}{B}$. So $\theta < \frac{\pi}{4}$ when that fraction is positive, $\theta = \frac{\pi}{4}$ when $A = C$, and $\theta > \frac{\pi}{4}$ when it is negative.

Definition To remove the mixed term from an equation with $B \ne 0$, first find the angle $\theta$ of the convention. When it is not a familiar angle, compute

which holds because $\sin 2\theta > 0$. The half-angle formulas, with positive roots because $0 < \theta < \frac{\pi}{2}$, then give

Finally substitute $x = X\cos\theta - Y\sin\theta$ and $y = X\sin\theta + Y\cos\theta$, or use the coefficient formulas. The result has no $XY$ term, and completing the square finishes the job.

Definition The discriminant of the equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ is the number $B^2 - 4AC$. It does not change under a rotation of axes,

After a rotation that removes the mixed term it equals $-4A'C'$, so the test for shifted conics gives the following.

• If $B^2 - 4AC < 0$, the graph is an ellipse (possibly a circle), a single point, or empty.
• If $B^2 - 4AC = 0$, the graph is a parabola, two parallel lines, one line, or empty.
• If $B^2 - 4AC > 0$, the graph is a hyperbola or two crossing lines.

So when the graph is known to be a nondegenerate conic, the sign names its type. The sign alone cannot tell a true conic from a degenerate one. For that, rotate and complete the square, or factor.