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    Tessellation of Curved Surfaces under Highly Varying Transformations
    (Eurographics Association, 1991) Abi-Ezzi, Salim S.; Shirman, Leon A.
    We pursue the problem of step size determination for tessellating arbitrary degree polynomial and rational Bezier patches, under highly varying modeling and viewing transformations, to within post-viewing size and/or deviation thresholds specified in display coordinates. The technique involves the computation of derivative bounds of surfaces in modeling coordinates, and the mapping of these bounds into world coordinates (or lighting coordinates), where tessellation takes place by using norms of modeling transformations. A key result of this work is a closed form expression for the maximum scale a perspective transformation is capable of at an arbitrary point in space. This result allows the mapping of thresholds from DC into WC (LC). In practice, while the step size determination needs to take place during every traversal, the costly operations of finding derivative bounds, computing norms of modeling transformations, and factoring viewing transformations take place at creation time.
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    Factoring a Homogeneous Transformation for a more Efficient Graphics Pipeline
    (Eurographics Association, 1990) Abi-Ezzi, Salim S.; Wozny, Michael J.
    We identify an intermediate coordinate system situated between world coordinates and display coordinates, which exhibits unique features for lighting calculations and for clipping in homogeneous coordinates. Our key contribution is an algorithm for extracting such a coordinate system from a homogeneous viewing transformation that relates WC to DC. The algorithm is based on factoring the transformation into a product of a Euclidean factor and a sparse (computationally cheap) but non- Euclidean factor. A particularly strong application of the proposed technique is the graphical processing of curved surface primitives, such as what is needed in the PHIGS PLUS viewing pipeline. Furthermore, in PHIGS PLUS the graphical data is retained by the graphics system, therefore, it is possible to perform the factoring of the viewing transformation at creation time, and to take advantage of this factored form at traversal time.