Investigating the near-field interaction of a dipole source such as a single quantum-dot with nearby dimers such as bowties is a challenging problem because of the lack of analytical models and the difficulty to perform the experiment. We tackle this problem here by using a conformal transformation. We show how, in the electrostatic approximation, the bowtie and point dipole source can be transformed into the analytically solvable configuration of a periodic metal-insulator-metal and point dipole source configuration. The analytical solutions are validated with full-wave simulations.