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Accurate Environment Mapped Reflections and Refractions by Adjusting for Object Distance

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This paper was brought up in another thread, but I don't understand it. They say that you can compute the intersection of the "environment sphere" by using

CorrectedVector = ReflecVector * Radius + (pos-center);

why do they then divide this by Radius to get

CorrectedVector / Radius = ReflecVector + ((pos-center)/Radius);

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