10.4 Rotation of Axes

Conic Sections and Rotation of Axes

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General form of conic sections

Every conic section can be written in the general second-degree equation:

$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$

The coefficients $A$, $B$, and $C$ tell you which type of conic you're dealing with. The key factor is whether the $Bxy$ term is present, because that term is what indicates rotation.

When $B = 0$ (no rotation):

• $A = C$: circle
• $A = 0$ or $C = 0$ (but not both): parabola
• $A$ and $C$ have the same sign but $A \neq C$: ellipse
• $A$ and $C$ have opposite signs: hyperbola

When $B \neq 0$, the conic is rotated relative to the coordinate axes. You can still classify it using the discriminant $B^2 - 4AC$:

• $B^2 - 4AC < 0$: ellipse (or circle)
• $B^2 - 4AC = 0$: parabola
• $B^2 - 4AC > 0$: hyperbola

The discriminant is invariant under rotation, meaning it gives the same value no matter how the axes are oriented. That's what makes it so useful for classification.

Rotation of axes for conics

The whole point of rotating axes is to eliminate the $Bxy$ term. Once that term is gone, you're left with a standard-looking conic equation that's much easier to analyze.

Finding the rotation angle:

The angle $\theta$ that eliminates the $xy$ term satisfies:

$\cot(2\theta) = \frac{A - C}{B}$

This is equivalent to $\theta = \frac{1}{2}\arctan\!\left(\frac{B}{A - C}\right)$ when $A \neq C$. If $A = C$, then $\cot(2\theta) = 0$, which gives $\theta = 45°$.

Rotation formulas (original coordinates in terms of rotated coordinates):

$x = x'\cos\theta - y'\sin\theta$

$y = x'\sin\theta + y'\cos\theta$

These express the old $(x, y)$ coordinates in terms of the new rotated $(x', y')$ system.

Step-by-step process for eliminating the $xy$ term:

1. Start with the general equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$

2. Compute $\theta$ using $\cot(2\theta) = \frac{A - C}{B}$

3. Find $\cos\theta$ and $\sin\theta$ (use the half-angle identities if needed)

4. Substitute the rotation formulas for $x$ and $y$ into the original equation

5. Expand and simplify. The $x'y'$ terms will cancel, leaving:

$A'x'^2 + C'y'^2 + D'x' + E'y' + F' = 0$

The new coefficients $A'$, $C'$, $D'$, $E'$, and $F'$ will generally differ from the originals, but the $B'xy'$ term is now zero.

Standard form of rotated conics

After rotation eliminates the $xy$ term, you still may need to complete the square to reach true standard form.

Completing the square on the rotated equation:

Starting from $A'x'^2 + C'y'^2 + D'x' + E'y' + F' = 0$, group and complete the square:

$A'\!\left(x' + \frac{D'}{2A'}\right)^2 + C'\!\left(y' + \frac{E'}{2C'}\right)^2 = -F' + \frac{D'^2}{4A'} + \frac{E'^2}{4C'}$

(This form applies when both $A'$ and $C'$ are nonzero. For a parabola, one of them is zero, so you only complete the square on the surviving squared variable.)

You can then translate to center the conic at the origin by setting $x'' = x' + \frac{D'}{2A'}$ and $y'' = y' + \frac{E'}{2C'}$.

Standard forms by conic type:

• Ellipse: $\frac{x''^2}{a^2} + \frac{y''^2}{b^2} = 1$
• Hyperbola: $\frac{x''^2}{a^2} - \frac{y''^2}{b^2} = 1$
• Parabola: $y'' = ax''^2$ or $x'' = ay''^2$ (only one variable is squared)

Direct analysis of conic equations

Once you've rotated and translated to standard form, you can read off the key features directly.

Ellipse ($\frac{x''^2}{a^2} + \frac{y''^2}{b^2} = 1$, with $a > b$):

• Center at the origin of the translated/rotated system
• Vertices at $(\pm a, 0)$ along the major axis and $(0, \pm b)$ along the minor axis
• Foci at $(\pm c, 0)$, where $c^2 = a^2 - b^2$
• The defining property: the sum of distances from any point on the ellipse to the two foci is constant (equal to $2a$)

Hyperbola ($\frac{x''^2}{a^2} - \frac{y''^2}{b^2} = 1$):

• Center at the origin of the translated/rotated system
• Vertices at $(\pm a, 0)$ along the transverse axis
• Foci at $(\pm c, 0)$, where $c^2 = a^2 + b^2$
• The defining property: the absolute difference of distances from any point on the hyperbola to the two foci is constant (equal to $2a$)

Parabola (e.g., $x'' = \frac{1}{4p}y''^2$):

• Vertex at the origin of the translated/rotated system
• Focus at distance $p$ from the vertex along the axis of symmetry
• Directrix is a line perpendicular to the axis of symmetry, at distance $p$ on the opposite side of the vertex from the focus

To convert these features back to the original $(x, y)$ coordinate system, apply the rotation formulas in reverse and then undo the translation.

Advanced rotation concepts

• Matrix form: The rotation can be written as a matrix equation. The rotation matrix $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ maps $(x', y')$ coordinates to $(x, y)$ coordinates.
• Quadratic forms: The second-degree terms $Ax^2 + Bxy + Cy^2$ can be expressed as $\mathbf{x}^T M \mathbf{x}$, where $M = \begin{pmatrix} A & B/2 \\ B/2 & C \end{pmatrix}$. The eigenvalues of $M$ give the coefficients $A'$ and $C'$ in the rotated equation, and the eigenvectors point along the principal axes of the conic.
• Principal axes: These are the directions along which the conic has its maximum and minimum curvature. Finding them through eigenvalue analysis is equivalent to finding the rotation angle $\theta$ algebraically.