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Translation and Rotation of Axes - Theory & Concepts - Analytic Geometry Transformation

Transformation of coordinates through translation and rotation of the axes, and eliminating the xy-term in conic sections.

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Transformation of Axes

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

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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'.

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