want to show algebraically that ''Joukowski transformation'' $$w = z + {1 \over z}$$ maps every circle passing through both -1 and +1 onto a curved segment or arc. if a circle only passes through z=-1, it is mapped onto an airfoil that is sharp on one side only. Now, if the circle passes through both points, its image becomes an arc
In the pictures below, the circle is shown in blue and its image is shown in red
