Redirected walking (RDW) aims to reduce the collisions in the physical space for VR applications. However, most of the previous RDW methods do not consider future possibilities of collisions after imperceptibly redirecting users. In this paper, we combine the subtle RDW methods and reset strategy in our method design and propose a novel solution for RDW that can make better use of physical space and trigger fewer resets. The key idea of our method is to discretize the representation of possible user positions and orientations by a series of standard poses and rate them based on the possibilities of hitting obstacles of their reachable poses. A transfer path algorithm is proposed to measure the accessibility among standard poses and is used to support the calculation of the scores of standard poses. Using our method, the user can be redirected imperceptibly to the optimal pose with the best score among all the reachable poses from the user’s current pose during walking. Experiments demonstrate that our method outperforms state-of-the-art methods in various environment sizes and obstacle layouts.

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We establish the intrinsic Harnack inequality for non-negative solutions of a class of degenerate, quasilinear, parabolic equations, including equations of the$p$-Laplacian and porous medium type. It is shown that the classical Harnack estimate, while failing for degenerate parabolic equations, it continues to hold in a space-time geometry intrinsic to the degeneracy. The proof uses only measure-theoretical arguments, it reproduces the classical Moser theory, for non-degenerate equations, and it is novel even in that context. Hölder estimates are derived as a consequence of the Harnack inequality. The results solve a long standing problem in the theory of degenerate parabolic equations.

Let$F$be families of meromorphic functions in a domain$D$, and let$R$be a rational function whose degree is at least 3. If, for any$f∈$$F$, the composite function$R(f)$has no fixed-point in$D$, then$F$is normal in$D$. The number 3 is best possible. A new and much simplified proof of a result of Pang and Zalcman concerning normality and, shared values is also given.

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