This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Deriving Linear Transformations in Three Dimensions
May/June 2003 (vol. 23 no. 3)
pp. 66-71
Ron Goldman, Rice University

Elementary problems in 3D geometry can often be reduced to solving two simple simultaneous vector equations-one involving only the dot product and the other only the cross product-in one vector unknown. This article presents a general technique for solving such equations and then applies this method to derive explicit formulas for many of the linear transformations used in the standard 3D graphics pipeline.

1. A.E. Hirst, Vectors in 2 or 3 Dimensions, Arnold, 1995.
2. R.N. Goldman, "Matrices and Transformations," Graphics Gems, A. Glassner, ed., Academic Press, 1990, pp. 472-475.
3. R.N. Goldman, "More Matrices and Transformations: Shear and Pseudo-Perspective," Graphics Gems II, J. Arvo, ed., Academic Press, 1991, pp. 338-341.
4. P-G. Maillot, "Using Quaternions to Code 3D Transformations," Graphics Gems, A. Glassner, ed., Academic Press, 1990, pp. 498-515.
5. K Shoemake, "Quaternions and Matrices," Graphics Gems II, J. Arvo, ed., Academic Press, 1991, pp. 351-354.
6. L. Dorst and S. Mann,"Geometric Algebra: A Computation Framework for Geometrical Application, Part 1," IEEE Computer Graphics and Applications, vol. 22, no. 3, May/June 2002, pp. 24-31.
7. S. Mann and L. Dorst,"Geometric Algebra: A Computation Framework for Geometrical Application, Part 2," IEEE Computer Graphics and Applications, vol. 22, no. 4, July/Aug. 2002, pp. 58-67.

Index Terms:
rendering, illumination effects, radiosity, texture mapping, illumination model, lightmaps, ambient light, form factor, obscurance, production rendering
Citation:
Ron Goldman, "Deriving Linear Transformations in Three Dimensions," IEEE Computer Graphics and Applications, vol. 23, no. 3, pp. 66-71, May-June 2003, doi:10.1109/MCG.2003.1198264
Usage of this product signifies your acceptance of the Terms of Use.