Rotation of Axes

Definition

Rotation of axes is a coordinate transformation in which the $x$ and $y$ axes are rotated about the origin by an angle $\theta$ to create a new set of axes $x'$ and $y'$. This technique is primarily used to simplify the general second-degree equation of a conic by eliminating the $xy$ term (the cross-product term), which represents a rotation in the plane.

• How to read: “Rotate the x and y axes about the origin by angle theta to obtain x-prime and y-prime axes.”
• Meaning: Change the observer’s frame so the conic’s natural axes align with the coordinate system.

Why It Matters

Rotation of axes is the key to simplifying ‘unfriendly’ geometry; by realigning the coordinate system, we can turn complex, slanted equations into standard forms that are much easier to calculate and visualize.

Core Concepts

• General Equation of a Conic: $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.

• How to read: “The A x squared plus B x y plus C y squared plus D x plus E y plus F equals zero.”

  • Meaning: The master form for any conic; the $Bxy$ term indicates the conic is tilted.

• Rotation formulas:

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

• How to read: “The X equals x-prime cosine theta minus y-prime sine theta; the y equals x-prime sine theta plus y-prime cosine theta.”

  • Meaning / when to use: Convert coordinates between the old and new (rotated) systems—substitute into the conic equation to eliminate $xy$.

• Eliminating the $xy$ Term: To transform the general equation into one without a $B'x'y'$ term, the angle of rotation $\theta$ must satisfy: $\cot(2\theta) = \frac{A - C}{B}, \quad 0 < 2\theta < \pi$

• How to read: “The cotangent of two theta equals A minus C divided by B.”

  • Meaning / when to use: Solve for $\theta$ to zero out the cross term. When $B = 0$, no rotation needed; when $A = C$, rotate $45^\circ$.

• Discriminant Invariance: The quantity $B^2 - 4AC$ is invariant under rotation. It determines the conic type regardless of the orientation:

  • $B^2 - 4AC < 0$: Ellipse (or circle).
  • $B^2 - 4AC = 0$: Parabola.
  • $B^2 - 4AC > 0$: Hyperbola.

• How to read: “The B squared minus four A C” compared to zero.

  • Meaning: Classifies the conic before and after rotation—tilting doesn’t change the genus.

Connected Concepts

• Area of a Triangle
• Average Rate of Change and Secant Lines
• Circle of Curvature
• Radius of Curvature
• Circle of Competence
• Comparison Properties of Integrals
• Fundamental Theorem of calculus
• Symmetry