  For quadrotor trajectory planning, describing a polynomial trajectory through
coefficients and end-derivatives both enjoy their own convenience in energy
minimization. We name them double descriptions of polynomial trajectories. The
transformation between them, causing most of the inefficiency and instability,
is formally analyzed in this paper. Leveraging its analytic structure, we
design a linear-complexity scheme for both jerk/snap minimization and parameter
gradient evaluation, which possesses efficiency, stability, flexibility, and
scalability. With the help of our scheme, generating an energy optimal (minimum
snap) trajectory only costs 1 $\mu s$ per piece at the scale up to 1,000,000
pieces. Moreover, generating large-scale energy-time optimal trajectories is
also accelerated by an order of magnitude against conventional methods.
