We present a novel metric to analyze the similarity between the physical environment and the virtual environment for natural walking in virtual reality. Our approach is general and can be applied to any pair of physical and virtual environments. We use geometric techniques based on conforming constrained Delaunay triangulations and visibility polygons to compute the Environment Navigation Incompatibility (ENI) metric that can be used to measure the complexity of performing simultaneous navigation. We demonstrate applications of ENI for highlighting regions of incompatibility for a pair of environments, guiding the design of the virtual environments to make them more compatible with a fixed physical environment, and evaluating the performance of different redirected walking controllers. We validate the ENI metric using simulations and two user studies. Results of our simulations and user studies show that in the environment pair that our metric identified as more navigable, users were able to walk for longer before colliding with objects in the physical environment. Overall, ENI is the first general metric that can automatically identify regions of high and low compatibility in physical and virtual environments. Our project website is available at https://gamma.umd.edu/eni/.
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