The Visual Display Transformation for Virtual RealityCyrus R. MoonComputer Integrated Surgery II (600.446)Critical SummaryThe Visual Display Transformation for Virtual RealityThe paper of interest is a technical report that discusses the potential problems that accompany the creation and use of any type of virtual reality system; it also presents solutions to some of these problems and justifies the practicality of these solutions.The report first outlines the basic concepts and components of a virtual reality system. Ingeneral, a VR system displays computer graphic stereoscopic images of a virtual environment to the user, usually through a head mounted display (HMD); the image that is displayed is dependent on the location of the user, and the direction in which he/she is facing. However, writing a program that produces accurate images on the display can be difficult; the objects in the virtual world, and the VR components in the real world must all be given correct relationships to one another in the software. As the paper states, “Making sure that the constants and algorithms in the display code both match the physical geometry of the HMD and produce correctly sized and oriented graphics is very difficult and slow work.”At the time this report was written, a significant amount of work had already been done on virtual reality systems, but few of the projects had created any organized documentation concerning the methods used to create the graphics software for them. Therefore there was needed a systematic and organized procedure for creating VR software. The first method discussed in the paper was use of the VQS representation in place of the4x4 matrix for some three-dimensional transformations. VQS, which stands for vector-quaternion-scalar, breaks up a frame transformation into three basic parts: a vector of three numbers representing displacement, a quaternion of four numbers for change in orientation of axes, and one number for change in scale. This representation has unique properties that make it good for use with VR software. First of all, VQS can only represent rotation, translation, and uniform scaling; it cannot be used for shear and non-uniform scaling. Although this limits what the use of VQS, those operations are usually unused and often unwanted in a frame transformation. Other advantages of the VQS representation are that the three transformation components are separated, making them more readable, intuitive to change, and easier to renormalize, and that quaternions, though unfamiliar to some, are a more robust, efficientrepresentation of three-dimensional rotation, and make interpolation between two rotations simpler than the Euler rotation matrix. The paper admits that the Euler matrix cannot be completely replaced, as some operations, such as perspective transforms cannot be represented byVQS, but also states that most operations requiring Euler transformations usually are not present in the application parts of the code. Equations for manipulation of VQS transforms, along with a list of quaternion operations are listed in the paper.The next concept introduced is the Coordinate System Graph, which is a diagram that contains organized information on every transformation and frame present within a given virtual reality system. The graph consists of nodes, which represent different coordinate systems, and lines between nodes, which represent independent transformations, independent meaning that they are unaffected by any changes in other transformations. Each independent transform can either be dynamic (meaning that the transform is updated every frame), or static (never changing throughout the running of the VR program). The transform from any given coordinate system to another can be determined by combining the transformations that are found on the path between them, in the order that the lines are traversed. Two properties of a Coordinate System Graph is that it is connected and acyclic; this means that for any two nodes, there is a path between them, but only one path. This ensures that every coordinate system can eventually be expressed in screen space, and that there is only one possible transform between nodes, which eliminates inconsistencies accompanying multiple possible transformations.For the virtual reality system, the primary transforms of interest are those that move fromobject coordinate systems to the screen coordinate system; these transforms essentially give the information needed to display objects on the screen in the correct position, orientation, and scale (if the object should be displayed at all). This transform is given in the report as a series of independent transforms:TS_O = TS_US * TUS_N * TN_E * TE_H * TH_HS * THS_TB * TTB_R * TR_W * TW_O,-with TB_A defined as the transformation from frame A to B, and:S = Screen HS = Head SensorUS = Undistorted Screen TB = Tracker BaseN = Normalized R = RoomE = Eye W = WorldH = Head O = ObjectThe majority of the transforms can be represented in VQS format, and are fairly intuitive.TW_O transforms an object from its local coordinate system to the coordinate system of the virtual world. TR_W represents the conversion from the virtual world to real world space; changing the displacement will place the room in a different part of the virtual world, rotation will ‘tilt’ the world, and scale will shrink/grow the world with respect to the user, or vice versa; the origin of the room coordinate system is chosen by the programmer, with the criteria of making computation easier. TR_TB is pre-calculated, and gives the origin of the room coordinate system asmeasured by the tracker. THS_R converts from tracker coordinates to the origin of the head sensor (located somewhere on the HMD); this would be found by solving for the inverse of the head sensor position in tracker coordinates. TH_HS moves from the position of the head sensor to the head coordinate system, whose origin is usually located midway between the eyes; this transform is pre-calculated. TE_H represents conversion from head space to eye space; there are two TE_H’s for each user (one for each eye), and though this transform would ideally be different for every user (since the distance between a user’s eyes would be different from another’s), a default transform is often used. If the HMD has angled displayed screens, there is a rotation element to his transform. TN_E represents the determination of what in the