Consider a unit square centred at origin. The coordinates of the square are translated by a factor \(\left(\frac{1}{2}, 1\right)\) and rotated by an angle of 90°. What shall be the coordinates of the new square ?
\(\left(\frac{-1}{2}, 0\right), \left(\frac{-1}{2}, 1\right), \left(\frac{-3}{2}, 1\right), \left(\frac{-3}{2}, 0\right)\)
Unit square centred at the origin: \(\left(\pm\frac{1}{2}, \pm\frac{1}{2}\right)\).
Translate by \(\left(\frac{1}{2}, 1\right)\): \(\left(0, \frac{1}{2}\right), \left(1, \frac{1}{2}\right), \left(1, \frac{3}{2}\right), \left(0, \frac{3}{2}\right)\).
Rotate by 90° anticlockwise, \((x, y) \to (-y, x)\): \(\left(-\frac{1}{2}, 0\right), \left(-\frac{1}{2}, 1\right), \left(-\frac{3}{2}, 1\right), \left(-\frac{3}{2}, 0\right)\).
Check: the four points still form a unit square (side 1); lists that mix positive and negative x do not.
Hence, \(\left(\frac{-1}{2}, 0\right), \left(\frac{-1}{2}, 1\right), \left(\frac{-3}{2}, 1\right), \left(\frac{-3}{2}, 0\right)\).
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