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Geometry 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
Elementary transformations 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 : $$A$$ : $$\arg(z-i)=\pi/4$$, $$B$$ : $$\Real(z)=1$$, $$C$$ : $$\vert z + 1\vert = 1$$, $$D$$ : $$\vert z+i\vert = \vert z - 1\vert$$
Geometrical figures in the complex plane : $$A$$ : $$\arg(z-i)=\pi/4$$, $$B$$ : $$\Real(z)=1$$, $$C$$ : $$\vert z + 1\vert = 1$$, $$D$$ : $$\vert z+i\vert = \vert z - 1\vert$$