So you have a voltage source between your two biconical parts. So the E field is established between those two parts and is parallel to the biconical parts. Its like the Dipole antenna, the E field is parallel to the Dipole parts since the voltage delta is between the two parts of the dipoles. The E field is always coplanar withe the antenna, it means parallel to the parts and to the voltage direction between the two parts. The H field is perpendicular to the E field or parallel to the currents in the fed lines.