For direct volume rendering of post-classified data, we propose an algorithm that analytically splits a ray through a cubical cell at the control points of a piecewise-polynomial transfer function. This splitting generates segments over which the variation of the optical properties is described by piecewise cubic functions. This allows using numerical quadrature rules with controlled precision to obtain an approximation with prescribed error bounds. The proposed splitting scheme can be used to

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... ind all piecewise linear or monotonic segments along a ray, and it can thus be used to improve the accuracy of direct volume rendering, scale-invariant volume rendering, and multi-isosurface rendering.