Translation and Rotation of Axes

In analytic geometry, the complex algebraic equations describing conic curves can often be significantly simplified by strategically choosing a completely different, customized coordinate system. This crucial mathematical process is formally called the transformation of coordinates. Rather than trying to solve a complicated equation in a standard grid, we mathematically pick up the entire grid and move it so it aligns perfectly with the shape we are studying. The two primary types of rigid transformation operations are translation (physically shifting the (0,0)(0,0) origin to a new location while strictly maintaining the original parallel orientation of the axes) and rotation (keeping the origin pinned but angularly turning the xx and yy axes around that central pivot).

Transformation of Axes

Translation of Axes

When we execute a mathematical translation of the axes, we physically shift the standard origin (0,0)(0,0) to a newly defined target point (h,k)(h, k). Importantly, during a pure translation, we do not alter the angular direction or parallel orientation of the original x and y axes. We define the newly shifted coordinate axes using "prime" notation as x′x' and y′y'. Every single point in the plane now has two valid sets of addresses: its old absolute coordinates (x,y)(x,y) and its new relative coordinates (x′,y′)(x', y').

Concept

If a point has absolute coordinates (x,y)(x, y) in the original system and relative coordinates (x′,y′)(x', y') in the translated system (where the new origin is situated at (h,k)(h, k)), the fundamental relationships between the coordinates are simply additive.

Translation Formulas (Original in terms of New)

Calculates the old coordinates based on the new grid.

x=x′+h,y=y′+kx = x' + h, \quad y = y' + k

Variables

SymbolDescriptionUnit
x,yx, yOriginal coordinates-
x′,y′x', y'New translated coordinates-
h,kh, kCoordinates of the new origin-

Translation Formulas (New in terms of Old)

Calculates the new coordinates based on the original grid.

x′=x−h,y′=y−kx' = x - h, \quad y' = y - k

Variables

SymbolDescriptionUnit
x′,y′x', y'New translated coordinates-
x,yx, yOriginal coordinates-
h,kh, kCoordinates of the new origin-

Algebraic Purpose of Translation

Translation is the geometric equivalent of the algebraic method of "completing the square." It is specifically utilized to systematically eliminate the linear terms (DxDx and EyEy) from the general equation of a conic section Ax2+Cy2+Dx+Ey+F=0Ax^2 + Cy^2 + Dx + Ey + F = 0, perfectly centering the conic at the origin of the new coordinate system to reveal its standard form.

Rotation of Axes

When we mathematically rotate the axes, we keep the origin (0,0)(0,0) strictly fixed in place but turn both the x and y axes simultaneously through a specific angle θ\theta. The new, tilted coordinate axes are again denoted as x′x' and y′y'.

Concept

If a point has standard coordinates (x,y)(x, y) in the original upright system and tilted coordinates (x′,y′)(x', y') in the rotated system (with an angle of counterclockwise rotation θ\theta), the geometric transformation equations are derived using trigonometry.

Rotation Formulas

Calculates original coordinates from rotated coordinates.

x=x′cos⁡θ−y′sin⁡θ,y=x′sin⁡θ+y′cos⁡θx = x' \cos \theta - y' \sin \theta, \quad y = x' \sin \theta + y' \cos \theta

Variables

SymbolDescriptionUnit
x,yx, yOriginal upright coordinates-
x′,y′x', y'New rotated coordinates-
θ\thetaAngle of counterclockwise rotation-

Eliminating the xy-Term

The most common and powerful application of the rotation of axes is to mathematically eliminate the cross-product term (BxyBxy) from the full general equation of the second degree: Ax2+Bxy+Cy2+Dx+Ey+F=0Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. The very presence of this BxyBxy term explicitly indicates that the conic section's axes of symmetry are tilted—they are neither perfectly horizontal nor perfectly vertical.

Concept

To successfully eliminate the xyxy-term, the axes must be rotated by a highly specific angle θ\theta such that the new B′B' coefficient evaluates to exactly zero.

Angle of Rotation Formula

Determines the exact angle needed to eliminate the Bxy term.

cot⁡(2θ)=A−CB\cot(2\theta) = \frac{A - C}{B}

Variables

SymbolDescriptionUnit
θ\thetaRequired angle of rotation-
A,B,CA, B, CCoefficients from the original general equation-

Steps to Eliminate the xy-term

  1. Identify the coefficients AA, BB, and CC from the original general equation Ax2+Bxy+Cy2+Dx+Ey+F=0Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0.
  2. Calculate the cotangent of the double angle using the formula: cot⁡(2θ)=A−CB\cot(2\theta) = \frac{A - C}{B}.
  3. Using the value of cot⁡(2θ)\cot(2\theta), construct a right triangle to visually determine the exact fractional values of cos⁡(2θ)\cos(2\theta) and sin⁡(2θ)\sin(2\theta). Ensure you pick the correct quadrant based on the sign of BB.
  4. Apply trigonometric half-angle formulas to precisely calculate the values of sin⁡θ\sin \theta and cos⁡θ\cos \theta: sin⁡θ=1−cos⁡(2θ)2\sin \theta = \sqrt{\frac{1 - \cos(2\theta)}{2}} and cos⁡θ=1+cos⁡(2θ)2\cos \theta = \sqrt{\frac{1 + \cos(2\theta)}{2}}.
  5. Substitute the algebraic expressions for xx and yy (from the Rotation Formulas) directly back into the original complex equation.
  6. Expand, group like terms, and simplify. The resulting equation in x′x' and y′y' will have no cross-product term.

Invariants Under Coordinate Transformation

When axes are translated or rotated, the geometric properties of a curve (like distance, area, and angles) remain unchanged. For a general second-degree equation Ax2+Bxy+Cy2+Dx+Ey+F=0Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, certain algebraic expressions involving the coefficients are invariant, meaning their values are exactly the same in both the original (x,y)(x, y) coordinate system and the transformed (x′,y′)(x', y') system.

Key Invariants

  • The Discriminant (I1I_1): B2−4ACB^2 - 4AC. This invariant fundamentally determines the type of conic section (e.g., if B2−4AC<0B^2 - 4AC \lt 0, it is an ellipse in any coordinate system).
  • The Trace (I2I_2): A+CA + C. The sum of the coefficients of the squared terms remains constant under rotation.
  • The Determinant (I3I_3): The determinant of the 3×33 \times 3 matrix formed by the coefficients. This helps determine if the conic is degenerate.
Key Takeaways
  • Translation: Physically shifts the grid origin to (h,k)(h,k) without tilting. Formulas: x=x′+hx = x' + h and y=y′+ky = y' + k. Primarily used to algebraically eliminate linear terms (DxDx and EyEy).
  • Rotation: Turns the grid axes by angle θ\theta around a fixed origin. Formulas: x=x′cos⁡θ−y′sin⁡θx = x'\cos\theta - y'\sin\theta and y=x′sin⁡θ+y′cos⁡θy = x'\sin\theta + y'\cos\theta.
  • Eliminating xy-term: Used to perfectly align a tilted conic section with the grid axes. The required angle is found using the trigonometric relation cot⁡(2θ)=(A−C)/B\cot(2\theta) = (A-C)/B.
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