Explicit expressions of the full Hamiltonian of tri-atomic system in the hyperspherical elliptic (HSE) coordinates are derived. The derivation is made from the expressions in the Delves coordinates. A numerical algorithm is also presented to evaluate the surface eigenfunctions including all the effects of Coriolis coupling terms. The whole formalism is numerically tested by using the Cl + DH and O(D-1) + HCl reaction systems. The HSE coordinate system, which is well-known to be powerful to elucidate reaction mechanisms especially for heavy-light-heavy systems, is now ready to be applied for clarifying full quantum dynamics of such systems.