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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 : $A$ : $\arg(z-i)=\pi/4$, $B$ : $\Real(z)=1$, $C$ : $\vert z + 1\vert = 1$, $D$ : $\vert z+i\vert = \vert z - 1\vert$