Abstract. The condition of poloidal closure of the contours of the second adiabatic invariant for all reflected particles is studied for stellarators with poloidally closed contours of the magnetic field strength B on the magnetic surfaces through computational stellarator optimization. It is shown that this is possible in a major fraction of the plasma volume and results in excellent alpha-particle confinement. Most importantly the magnetic axis curvature vanishes in all cross-sections with an extremum of B on the magnetic axis. Toroidal mirror traps are investigated with respect to isometry, i.e. the lengths of field lines on a magnetic surface between any two contours of B are equal and the contours of the second adiabatic invariant coincide with the magnetic surfaces. Analytical methods are used to design a vacuum field toroidal mirror trap satisfying this isometry condition. The boundary surface of such a configuration is a well-defined finite-sized toroidal surface. This provides the interface to the VMEC, JMC and MCT codes which allow analyses of equilibrium and neoclassical properties.