Geometry of the SC method for a particular ray with direction n propagating from the upwind point u^(ijk) to a considered grid point p^(ijk). The downwind point d^(ijk) is required to set the slope of a Bézier curve representing the opacities and source functions along the ray. The middle and lower panel display all possible downwind and upwind intersection surfaces for a short characteristic at a grid point p^(ijk). For rays intersecting the xy-, xz-, and yz-planes, the 2D Bézier interpolation is obtained from given quantities at grid points located in the cyan, red, and magenta shaded surfaces, respectively. The coordinate system is indicated at the upper left, where α, β, γ determine the direction of the coordinate-axes and are defined in Sect. 3.3.