We provide some remarks on the very early developments of the visualization techniques conducted during the European Renaissance. It is shown that the basic principle of ray tracing was already presented by Albrecht Diirer (1471-1528) in 1525. This article is intended to be of common interest; it is not a scientific report.

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The notion of a linear Coxeter system introduced by Vinberg generalizes the geometric representation of a Coxeter group. Our main theorem asserts that if v is an element of the Tits cone of a linear Coxeter system and W is the corresponding Coxeter group, then Wv ⊆ v − Cv, where Cv is the convex cone generated by the coroots α, for which α(v) > 0. This implies that the convex hull of Wv is completely determined by the image of v under the reflections in W. We also apply an analogous result for convex hulls of W-orbits in the dual space, although this action need not correspond to a linear Coxeter system. Motivated by the applications in representation theory, we further extend these results to Weyl group orbits of locally finite and locally affine root systems. In the locally affine case, we also derive some applications on minimizing linear functionals on Weyl group orbits.