The dark matter density is formally infinite at the location of caustic surfaces, where dark matter sheet folds in phase space. The caustics are boundaries between the regions with different number of streams n_str(x) in Eulerian space. Alternatively they can be defined as boundaries between the regions with different number of flip-flops n_ff(q) in Lagrangian space. The number of flip-flops equals the amount of turns inside out experienced by a fluid element of a collision-less medium. Physically both definitions are equivalent but discreteness of numerical models may result in some distinctions. After n_str(x) or n_ff(x) field is numerically evaluated the identification of caustics becomes a purely geometrical procedure which is independent on any numerical parameters. Both approaches are used in identifying a compact closed caustic surface around potential halos in an idealized N-body simulation. The set of all caustics should be the same in both cases, but comparing n_str(x) and n_ff(q) is not straightforward because there is no simple relation between the number of streams and number of flip-flops. The halo boundary in this simulation is found to be neither spherical nor ellipsoidal nor oval but remarkably asymmetrical. However, a convex hull is a good approximation to the halo boundaries. The analysis of the kinetic and potential energies of individual particles and the halo as a whole concludes that it is gravitationally bound. In addition, the examination of the two-dimensional phase space confirms the above conclusion. The recent finding that common shells in a sample of halos obtained from the suite of large simulations are non-ellipsoidal ovals is quite encouraging for carrying out a more detailed analysis of this approach on higher resolution simulations.

暗物质的密度在苛性表面的位置是形式无限的，在该位置，暗物质在相空间中折叠。苛性碱是流数不同的区域之间的边界n _str（x）在欧拉空间。或者，可以将它们定义为具有不同数量触发器的区域之间的边界n _ff（q）在拉格朗日空间中。触发器的数量等于无碰撞介质的流体元件经历的由内而外的匝数。在物理上，这两个定义是等效的，但数值模型的离散性可能导致某些区别。后n _str（x） 或者 n _ff（x）对场进行数值评估后，苛性碱的识别就变成了一种纯粹的几何程序，它与任何数值参数无关。在理想的N体模拟中，两种方法都用于识别潜在光晕周围的致密闭合苛性碱表面。在两种情况下，所有焦散的集合应该相同，但是比较n _str（x） 和 n _ff（q）这不是简单明了的，因为在流的数量和触发器的数量之间没有简单的关系。发现该模拟中的光晕边界既不是球形也不是椭圆形也不是椭圆形，而是明显不对称。但是，凸包非常适合光晕边界。对单个粒子以及整个光环的动能和势能的分析得出结论，它是受重力约束的。另外，对二维相空间的检查证实了以上结论。最近的发现表明，从大型仿真套件中获得的光环样本中的常见壳体为非椭圆形椭圆，这对于在更高分辨率的仿真上对该方法进行更详细的分析非常令人鼓舞。