The visual display transformation for virtual reality (VR) systems is typically much more complex than the standard viewing transformation discussed in the literature for conventional computer graphics. The process can be represented as a series of transformations, some of which contain parameters that must match the physical configuration of the system hardware and the user's body. Because of the number and complexity of the transformations, a systematic approach and a thorough understanding of the mathematical models involved are essential. This paper presents a complete model for the visual display transformation for a VR system; that is, the series of transformations used to map points from object coordinates to screen coordinates. Virtual objects are typically defined in an object-centered coordinate system (CS), but must be displayed using the screen-centered CSs of the two screens of a head-mounted display (HMD). This particular algorithm for the VR display computation allows multiple users to independently change position, orientation, and scale within the virtual world, allows users to pick up and move virtual objects, uses the measurements from a head tracker to immerse the user in the virtual world, provides an adjustable eye separation for generating two stereoscopic images, uses the off-center perspective projection required by many HMDs, and compensates for the optical distortion introduced by the lenses in an HMD. The implementation of this framework as the core of the UNC VR software is described, and the values of the UNC display parameters are given. We also introduce the vector-quaternion-scalar (VQS) representation for transformations between 3D coordinate systems, which is specifically tailored to the needs of a VR system. The transformations and CSs presented comprise a complete framework for generating the computer-graphic imagery required in a typical VR system. The model presented here is deliberately abstract in order to be general purpose; thus, issues of system design and visual perception are not addressed. While the mathematical techniques involved are already well known, there are enough parameters and pitfalls that a detailed description of the entire process should be a useful tool for someone interested in implementing a VR system.

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