  In 1981, Beta Pictoris showed strong and rapid photometric variations
possibly due to a transiting giant planet. Later, a planetary mass companion to
the star, Beta Pic b, was identified using imagery. Observations at different
epochs (2003 and 2009-2015) detected the planet at a projected distance of 6 to
9 AU from the star and showed that the planet is on an edge-on orbit. The
observed motion is consistent with an inferior conjunction in 1981, and Beta
Pic b can be the transiting planet proposed to explain the photometric event
observed at that time. Assuming that the 1981 event is related to the transit
or the inferior conjunction of Beta Pic b on an edge-on orbit, we search for
the planetary orbit in agreement with all the measurements of the planet
position published so far. We find two different orbits that are compatible
with all these constraints: (i) an orbit with a period of 17.97$\pm$0.08 years
along with an eccentricity of around 0.12 and (ii) an orbit with a period of
36.38$\pm$0.13 years and a larger eccentricity of about 0.32. In the near
future, new imaging observations should allow us to discriminate between these
two different orbits. We also estimate the possible dates for the next
transits, which could take place as early as 2017 or 2018, even for a
long-period orbit.
