  A numerical integration method for guiding-center orbits of charged particles
in toroidal fusion devices with three-dimensional field geometry is described.
Here, high order interpolation of electromagnetic fields in space is replaced
by a special linear interpolation, leading to locally linear Hamiltonian
equations of motion with piecewise constant coefficients. This approach reduces
computational effort and noise sensitivity while the conservation of total
energy, magnetic moment and phase space volume is retained. The underlying
formulation treats motion in piecewise linear fields exactly and thus preserves
the non-canonical symplectic form. The algorithm itself is only quasi-geometric
due to a series expansion in the orbit parameter. For practical purposes an
expansion to the fourth order retains geometric properties down to computer
accuracy in typical examples. When applied to collisionless guiding-center
orbits in an axisymmetric tokamak and a realistic three-dimensional stellarator
configuration, the method demonstrates stable long-term orbit dynamics
conserving invariants. In Monte Carlo evaluation of transport coefficients, the
computational efficiency of quasi-geometric integration is an order of
magnitude higher than with a standard fourth order Runge-Kutta integrator.
