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The Scaling Behavior of Viewing Transformations
May/June 1993 (vol. 13 no. 3)
pp. 48-54

Using formal analysis of parallel and perspective viewing transformation behavior, closed-form form expressions are obtained for immediate evaluation of maximum, minimum, and average scales at a given point in space. While these are constant for parallel transformations, they vary from point to point for perspective transformations. The average scale of a transformation is expressed in ways useful for heuristic computations. The expressions indicate how transformations perform in general without providing guarantees. It is shown that these results apply to the dynamic tessellation of curved surfaces. The maximum scale of a transformation across a bounded region can guarantee that one will meet postviewing approximation thresholds specified in display coordinates, based on derivative bounds precomputed in modeling coordinates.

1. J.M. Lane, L.C. Carpenter, T. Whitted, and J.F. Blinn, "Scanline Methods for Displaying Parametrically Defined Surfaces,"CACM, Vol. 23, No. 1, 1980, pp. 23-34.
2. S. Lien, M. Shantz, and V. Pratt, "Adaptive Forward Differencing for Rendering Curves and Surfaces,"Computer Graphics, Vol. 21, No. 4, July 1987, pp. 111-117.
3. S. Abi-Ezzi,The Graphical Processing of B-splines in a Highly Dynamic Environment, doctoral dissertation RDRC-TR-89001, Rensselaer Polytechnic Institute, Troy, NY, May 1989.
4. S. Abi-Ezzi and M. Wozny, "Factoring a Homogeneous Transformation for a More Efficient Graphics Pipeline,"Proc. Eurographics 90, Sept. 1990, pp. 245-255.
5. W. Press, B. Flannery, S. Teukolsky, and W. Vetterling,Numeric Recipes in C-The Art of Scientific Computing.Cambridge, UR: Cambridge University Press, 1988.
6. S. Abi-Ezzi and L. Shirman, "Tessellation of Curved Surfaces under Highly Varying Transformations,"Proc. Eurographics 91, Sept. 1991, pp. 385-397.

Citation:
Salim A. Abi-Ezzi, Leon A. Shirman, "The Scaling Behavior of Viewing Transformations," IEEE Computer Graphics and Applications, vol. 13, no. 3, pp. 48-54, May-June 1993, doi:10.1109/38.210491
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