Translation and Rotation of Axes

Learning Objectives

  • Understand the geometric meaning of translating and rotating coordinate axes.
  • Apply translation formulas to algebraically eliminate linear terms in conic equations.
  • Apply rotation formulas to algebraically eliminate the cross-product (xyxy) term.
  • Identify the invariant properties of conics under coordinate transformation.

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).

Translation of Axes

A rigid transformation where the standard origin (0,0)(0, 0) is physically shifted to a newly defined target point (h,k)(h, k) without altering the angular direction or parallel orientation of the original xx and yy axes. The new axes are denoted as x′x' and y′y'. Every point has absolute coordinates (x,y)(x, y) and relative coordinates (x′,y′)(x', y').

Transformation Simulator

Use the interactive simulation below to visualize how coordinates change under translation (shifting origin) and rotation (tilting axes) of the Cartesian grid.

Transformation of Axes

Translate and rotate coordinate axes, mapping point P(x,y) to P'(x',y')

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0.0
0°
Governing Equation
{x′=(x−h)cos⁡θ+(y−k)sin⁡θy′=−(x−h)sin⁡θ+(y−k)cos⁡θ\begin{cases} x' = (x - h)\cos\theta + (y - k)\sin\theta \\ y' = -(x - h)\sin\theta + (y - k)\cos\theta \end{cases}
Transformed Coordinates
P′(x′,y′)=P′(3.00,4.00)P'(x', y') = P'(3.00, 4.00)
Visual Guide:The original coordinate axes are drawn in light gray/slate, while the transformed (translated by h, k and rotated by θ) coordinate axes are shown in red. Point P(3, 4) is mapped into its new coordinate system values P'.

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

A rigid transformation where the origin (0,0)(0, 0) is kept strictly fixed in place while both the xx and yy axes are turned simultaneously through a specific angle θ\theta. The new, tilted coordinate axes are 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-

Matrix Explorer

Use the rotation of axes matrix explorer below to see how coordinates of a point project onto original and rotated axes, and how the linear transformation matrix transforms (x,y)(x, y) to (x′,y′)(x', y') dynamically.

Rotation of Axes Matrix Explorer

Rotation Control

Rotation Angle θ30°

Draggable point coordinates

Original Coordinates (x, y)
(5.0, 4.0)
Rotated Coordinates (x', y')
(6.33, 0.96)

Transformation Matrix solver

Rotation Transformation Matrix
(x′y′)=(cos⁡θsin⁡θ−sin⁡θcos⁡θ)(xy)\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}
Step 1: Trigonometric Values
θ=30∘  ⟹  cos⁡θ=0.866, sin⁡θ=0.500\theta = 30^{\circ} \implies \cos\theta = 0.866, \ \sin\theta = 0.500
Step 2: Linear Substitution
x′=5⋅(0.866)+4⋅(0.500)=6.33x' = 5 \cdot (0.866) + 4 \cdot (0.500) = 6.33
y′=−5⋅(0.500)+4⋅(0.866)=0.96y' = -5 \cdot (0.500) + 4 \cdot (0.866) = 0.96
xyx'y'Point P (5.0, 4.0)P(5.0, 4.0)

Drag point P or adjust angle θ to see original vs. rotated projection lines

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 a rotation angle for the stated coordinate transformation convention.

2θ=atan2⁡(B,A−C)2\theta = \operatorname{atan2}(B, A - C)

Variables

SymbolDescriptionUnit
θ\thetaRotation angle; atan2 uses (y, x) argument order-
A,B,CA, B, CCoefficients of the quadratic terms in the original equation-

Steps to Eliminate the xy-term

  1. Identify AA, BB, and CC in Ax2+Bxy+Cy2+Dx+Ey+F=0Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0.
  2. If B=0B=0, no rotation is needed to eliminate the cross-product term.
  3. Otherwise compute 2θ=atan2⁡(B,A−C)2\theta=\operatorname{atan2}(B,A-C), using the function's (y,x)(y,x) argument order, and take half of that angle.
  4. Evaluate sin⁡θ\sin\theta and cos⁡θ\cos\theta from the chosen angle. If using half-angle identities, select signs consistent with θ\theta; both principal positive roots are not generally correct.
  5. Substitute 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 into the original equation.
  6. Expand and collect the new cross-product coefficient B′=Bcos⁡(2θ)+(C−A)sin⁡(2θ)B'=B\cos(2\theta)+(C-A)\sin(2\theta). Verify that B′=0B'=0 before writing the transformed equation.
Rotate Axes to Eliminate the xy-Term

Choose and verify a rotation for a general conic using the lesson's stated coordinate transformation convention.

Rotate Axes to Eliminate the xy-TermChoose and verify a rotation for a general conic using the lesson's stated coordinate transformation convention.. Start with Ax² + Bxy + Cy² + Dx + Ey + F = 0 → Is B already zero?; Is B already zero? — Yes → No rotation is needed to remove the xy-term; Is B already zero? — No → Set 2θ = atan2(B, A − C); then take θ = (2θ)/2; Set 2θ = atan2(B, A − C); then take θ = (2θ)/2 → Evaluate sin θ and cos θ with signs consistent with θ; Evaluate sin θ and cos θ with signs consistent with θ → Substitute x = x′cosθ − y′sinθ; y = x′sinθ + y′cosθ; Substitute x = x′cosθ − y′sinθ; y = x′sinθ + y′cosθ → Expand and collect the coefficient B′ of x′y′; Expand and collect the coefficient B′ of x′y′ → Does the calculation give B′ = 0?; Does the calculation give B′ = 0? — Yes → Write the transformed conic in x′ and y′; Does the calculation give B′ = 0? — No → Check atan2 order, angle branch, signs, and substitution convention; Check atan2 order, angle branch, signs, and substitution convention → Set 2θ = atan2(B, A − C); then take θ = (2θ)/2

Start with Ax² + Bxy + Cy² + Dx + Ey + F = 0 → Is B already zero?; Is B already zero? — Yes → No rotation is needed to remove the xy-term; Is B already zero? — No → Set 2θ = atan2(B, A − C); then take θ = (2θ)/2; Set 2θ = atan2(B, A − C); then take θ = (2θ)/2 → Evaluate sin θ and cos θ with signs consistent with θ; Evaluate sin θ and cos θ with signs consistent with θ → Substitute x = x′cosθ − y′sinθ; y = x′sinθ + y′cosθ; Substitute x = x′cosθ − y′sinθ; y = x′sinθ + y′cosθ → Expand and collect the coefficient B′ of x′y′; Expand and collect the coefficient B′ of x′y′ → Does the calculation give B′ = 0?; Does the calculation give B′ = 0? — Yes → Write the transformed conic in x′ and y′; Does the calculation give B′ = 0? — No → Check atan2 order, angle branch, signs, and substitution convention; Check atan2 order, angle branch, signs, and substitution convention → Set 2θ = atan2(B, A − C); then take θ = (2θ)/2

  • Start with Ax² + Bxy + Cy² + Dx + Ey + F = 0: terminator
  • Is B already zero?: decision
  • No rotation is needed to remove the xy-term: terminator
  • Set 2θ = atan2(B, A − C); then take θ = (2θ)/2: process
  • Evaluate sin θ and cos θ with signs consistent with θ: process
  • Substitute x = x′cosθ − y′sinθ; y = x′sinθ + y′cosθ: process
  • Expand and collect the coefficient B′ of x′y′: process
  • Does the calculation give B′ = 0?: decision
  • Write the transformed conic in x′ and y′: terminator
  • Check atan2 order, angle branch, signs, and substitution convention: process

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.

Equation Transformation Explorer

Use the transformation explorer below to see how translating and rotating axes transforms the algebraic equation of a conic section and eliminates the xyxy term in real-time.

2.0
-1.0

Standard Ellipse (h=0, k=0):

x²/9 + y²/4 = 1

Translating axes shifts standard coordinates to: (x - h) and (y - k).

Shifted ellipse centered at (2, -1)Grid lineGrid backgroundOriginal Horizontal AxisOriginal Vertical AxisTranslated Horizontal AxisTranslated Vertical AxisShifted Ellipse CurveCenter Point at (2, -1)
(x−2.0)29+(y−(−1.0))24=1\frac{(x - 2.0)^2}{9} + \frac{(y - (-1.0))^2}{4} = 1
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 the xyxy-term: For the stated rotation convention, use 2θ=atan2⁡(B,A−C)2\theta=\operatorname{atan2}(B,A-C) when B≠0B\ne0, then verify that the transformed cross-product coefficient is zero.