Planar graphs are graphs which can be drawn on the plane without edges crossing.
Disk contact graphs are graphs obtained as follows. Place some disks in the plane without overlaps, allowing touching. Create a vertex per disk, and connect vertices by an edge if the corresponding disks touch.
As stated by graph classes, planar graphs and disk contact graphs are equivalent. What I understand from this equivalence is that the two sets contain the same graphs. It is easy to see that any disk contact graph is planar, however I am having trouble understanding how every planar graph is also a disk contact graph.
Is there an intuitive proof? I explicitly ask for an intuitive proof since the link cites a survey paper, which itself cites multiple sources for proofs, although these papers are either impossible to find due to ultimately not being published (or being too old for search engines?) or require some expertise in specific fields which I unfortunately do not have.