Translation, Rotation, Scaling

Complex Analysis · Axiom Academy

EXAMPLE Translation, Rotation, Scaling Reading a single map as a scaling, a rotation, and a translation — and mapping a whole square with it. Find the image of the unit square with vertices under the linear map What kind of region is it, and how is it scaled and rotated relative to the original? Picture: the square and its image Left: the unit square (vertices 0, 1, 1+i, i). Right: its image under f — a square turned 45° clockwise, enlarged by 2, and shifted so its center sits at 3. Nice work — you mapped an entire region by mapping its corners. Every map of the form f(z)=az+b is built from the same three moves, read straight off a and b . Scale by |a| : here , so the square grows by a factor of (its area by |a|^2=2 ). Rotate by : , so the square turns clockwise about the origin. Translate by b : adding 2 slides the rotated, scaled square so its center lands at 3 . Map the corners, get the region: a linear map sends straight edges to straight edges, so the four images fully determine the image square. Translation ( f(z)=z+b ), rotation ( ), and scaling ( f(z)=rz ) are just the special cases — every az+b does all three at once.

This is the written version of the interactive lesson above. See the full Complex Analysis course.