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# Geometry in the complex plane

## Geometrical transformations in the complex plane

Simple operations on complex numbers correspond to elementary geometric transformations in the complex plane. We will use the convention that $z_0 = x_0+iy_0 = r_0e^{i\theta_0}$

• The translation of $(x_0,y_0)$ is the addition $z\mapsto z+z_0$.
• The scaling of factor $r_0$ (with centre $0$) is the multiplication $z\mapsto r_0z$.
• The rotation of angle $\theta_0$ (with centre $0$) is the multiplication $z\mapsto z e^{i\theta_0}$.
• The symmetry across the real axis is conjugation: $z\mapsto \overline{z}$.

Complex transformations can be obtained by the composition of elementary transformations. For instance a rotation-scaling of angle $\theta_0$ and ratio $r_0$ is the multiplication $z\mapsto z_0 z$.

Elementary transformations in the complex plane

## Geometrical figures in the complex plane

Geometrical figures can be described simply in the complex plane.

• The equation of the circle of centre $z_0$ and radius $r$ is $\vert z- z_0\vert =r$. This is because $\vert z- z_0\vert$ is the distance between $z$ and $z_0$.
• The equation of the disc is $\vert z- z_0\vert \le r$.
• The equation of the straight line parallel to the $y$-axis through $(a,0)$ has equation $\Real(z)=a$.
• The equation of the half-line starting at $a$ in the direction $\theta$ has equation $\arg(z-a)=\theta$.
• The equation of the perpendicular bisector of the line joining $a$ and $b$ (i.e. the line crossing perpendicularly in its midpoint the segment joining $a$ to $b$) has equation $\vert z- a\vert = \vert z-b\vert$. This is because the bisector is the set of points of the same distance to $a$ and $b$.
Geometrical figures in the complex plane : $A$ : $\arg(z-i)=\pi/4$, $B$ : $\Real(z)=1$, $C$ : $\vert z + 1\vert = 1$, $D$ : $\vert z+i\vert = \vert z - 1\vert$