The Double Pulsar, PSR J$0737$$-$$3039$A/B, is a unique system in which both neutron stars have been detected as radio pulsars. As shown in Ferdman et al., there is no evidence for pulse profile evolution of the A pulsar, and the geometry of the pulsar was fit well with a double-pole circular radio beam model. Assuming a more realistic polar cap model with a vacuum retarded dipole magnetosphere configuration including special relativistic effects, we create synthesized pulse profiles for A given the best-fit geometry from the simple circular beam model. By fitting synthesized pulse profiles to those observed from pulsar A, we constrain the geometry of the radio beam, namely the half-opening angle and the emission altitude, to be $30^circ$ and $10$ neutron star radii, respectively. Combining the observational constraints of PSR J$0737$$-$$3039$A/B, we are able to construct the full three-dimensional orbital geometry of the Double Pulsar. The relative angle between the spin axes of the two pulsars ($Delta_S$) is estimated to be ($138^circ pm 5^circ$) at the current epoch and will likely remain constant until tidal interactions become important in $sim$$85$ Myr, at merger.
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