On the algebraic and geometric foundations of computer graphics

ACM Transactions on Graphics

Volumn 21, Issue 1, 2002, Pages 52-86

  Goldman, Ron ^a  

^a Rice University   (United States)

Author keywords

Algorithms; Computational Geometry and Object Modeling Hierarchy and geometric transformations; Grassmann space; Homogeneous coordinates; I.3.5 Computer Graphics ; Mass points; Projective space; Theory

Indexed keywords

EID: 0003371079     PISSN: 07300301     EISSN: None     Source Type: Journal    
DOI: 10.1145/504789.504792     Document Type: Article

References (25)

1. BLINN, J. 1977. A homogeneous formulation of a line in 3-space. In Proceedings of the ACM Conference on Computer Graphics (SIGGRAPH '77). ACM, New York, pp. 237-241.
2. BLINN, J. 1996. Hyperbolic interpolation. In Jim Blinn's Corner: A Trip Down the Graphics Pipeline. Morgan-Kaufmann, San Francisco, Calif., pp. 167-178.
3. BLINN, J. AND NEWELL, M. 1978. Clipping using homogeneous coordinates. In Proceedings of the ACM Conference on Computer Graphics (SIGGRAPH '78). ACM, New York, pp. 245-251.
4. COX, D., LITTLE, J., AND O'SHEA, D. 1998. Using Algebraic Geometry. Springer-Verlag, New York.
5. DEROSE, T. D. 1989. A coordinate-free approach to geometric programming. In Theory and Practice of Geometric Modeling, W. Strasser and H. P. Seidel, Eds. Springer-Verlag, Berlin, Germany, pp. 291-305.
6. FARIN, G. 1997. Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide, fourth edition. Academic Press, San Diego, Calif.
7. FIOROT, J. AND JEANNIN, P. 1989. Courbes et Surfaces Rationnelles. Applications à la CAO. RMA12, Masson, Paris, France. (English version: Rational Curves and Surfaces. Applications to CAD. Wiley and Sons, Chichester, England, 1992.)
8. FIOROT, J. AND JEANNIN, P. 1989. Courbes et Surfaces Rationnelles. Applications à la CAO. RMA12, Masson, Paris, France. (English version: Rational Curves and Surfaces. Applications to CAD. Wiley and Sons, Chichester, England, 1992.)
9. FOLEY, J. D., VAN DAM, A., FEINER, S. K., AND HUGHES, J. F. 1990. Computer Graphics: Principles and Practice, second edition. Addison-Wesley, New York.
10. GOLDMAN, R. N. 1985. Illicit expressions in vector algebra. ACM Trans. Graph. 4, 3, 223-243.
11. GOLDMAN, R. N. 1991. More matrices and transformations: Shear and pseudoperspective. In Graphics Gems II, Jim Arvo, Ed. Academic Press, Orlando, Fla., pp. 338-341.
12. GOLDMAN, R. N. 2000. The ambient spaces of computer graphics and geometric modeling. IEEE CG&A 20, 76-84.
13. GOLDMAN, R. N. 2001. Baseball arithmetic and the laws of pseudoperspective. IEEE Comput. Graph. Appl. 21, 70-78.
14. GOLDMAN, R. N. 2002. Applications of the mass distributions and vector spaces associated with rational Bezier curves and surfaces. In preparation.
15. GRASSMANN, H. G. 1894-1911. Gesammelte Mathematische und Physikalische Werke. B. G. Teubner, Leipzig.
16. HECKBERT, P. AND MORETON, H. 1991. Interpolation for Polygon Texture Mapping and Shading. In State of the Art in Computer Graphics: Visualization and Modeling, D. Rogers and R. Earnshaw, Ed. Springer-Verlag, New York.
17. KOGAN, B. Y. 1974. The Application of Mechanics to Geometry. The University of Chicago Press, Chicago, Ill.
18. MURRAY, R., LI, Z., AND SASTRY, S. 1994. A Mathematical Introduction to Robotic Manipulation. CRC Press, Boca Raton, Fla.
19. NEWMAN, W. AND SPROULL, R. 1973. Principles of Interactive Computer Graphics. McGraw-Hill, New York.
20. PATTERSON, R. AND PENNA, M. 1986. Projective Geometry and Its Applications in Computer Graphics. Prentice-Hall, Englewood Califfs, N.J.
21. RAMSHAW, L. 2001. On Multiplying Points: The Paired Algebras of Forms and Sites. SRC Research Report #169, COMPAQ Corporation, Palo Alto, Calif.
22. RIESENFELD, R. 1981. Homogeneous coordinates and the projective plane in computer graphics. IEEE CG&A 1, 50-55.
23. ROBERTS, L. G. 1965. Homogeneous matrix representation and manipulation of N-dimensional constructs. The Computer Display Review, C. W. Adams Associates, Inc., Cambridge, Mass., pp. 1-16.
24. SWIMMER, A. 2002. Ceva, Menelaus and the associative law. Unpublished manuscript.
25. USPENSKII, V. A. 1961. Some applications of mechanics to mathematics. Blaisdell Publishing Company, New York.