The Community for Technology Leaders
RSS Icon
Subscribe
Issue No.04 - July/August (1992 vol.12)
pp: 54, 58, 61-62
ABSTRACT
<p>A family of techniques for creating intuitively informative shaded images of 4-D mathematical objects is proposed. The rendering of an object in a 4-D world is described by considering step-by-step how objects might be rendered into images in simpler worlds. The mathematical principles needed to compute projected images of objects and their shadows in D dimensions are outlined. The issues involved in producing shaded images of objects in four dimensions, including extending rendering from 3-D to 4-D, smooth shading, and specularity, are discussed. Results of rendering a Steiner surface, torus, and knotted sphere in four dimensions are presented.</p>
CITATION
Andrew J. Hanson, Pheng A. Heng, "Illuminating the Fourth Dimension", IEEE Computer Graphics and Applications, vol.12, no. 4, pp. 54, 58, 61-62, July/August 1992, doi:10.1109/38.144827
REFERENCES
1. T.F. Banchoff,Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions, Scientific American Library, New York, 1990. This general book is accessible to people with a moderate level of mathematical interest: the graphics are excellent and the exposition very readable.
2. S. Feiner and C. Beshers, "Visualizingn-dimensional Virtual Worlds withn-Vision,"Computer Graphics(Proc. 1990 Symp. Interactive Graphics), Vol. 24, No. 2, March 1990, pp. 37-38.
3. C. Bajaj, "Rational Hypersurface Display,"Computer Graphics, Vol. 24, No. 2, Mar. 1990, pp. 117-128.
4. F. Apéry,Models of the Real Projective Plane, F. Vieweg und Sohn, Braunschweig/Wiesbaden, Germany, 1987.
5. T.F. Banchoff, "Visualizing Two-Dimensional Phenomena in Four-Dimensional Space: A Computer Graphics Approach," inStatistical Image Processing and Computer Graphics, E. Wegman and D. Priest, eds., Marcel Dekker, New York, 1986, pp. 187-202.
6. G.K. Francis,A Topological Picturebook, Springer-Verlag, New York, 1987.
7. J.R. Weeks,The Shape of Space, Marcel Dekker, New York, 1985.
8. D.W. Brisson, ed.,Hypergraphics: Visualizing Complex Relationship in Art, Science and Technology, AAAS Selected Symp., Vol. 24, Westview Press, Boulder, Colo., 1978.
9. K.V. Steiner and R.P. Burton, "Hidden Volumes: The 4th Dimension,"Computer Graphics World, Feb. 1987, pp. 71-74.
10. S.A. Carey, R.P. Burton, and D.M. Campbell, "Shades of a Higher Dimension,"Computer Graphics World, Oct. 1987, pp. 93-94.
11. A.J. Hanson and P.A. Heng, "Visualizing the Fourth Dimension Using Geometry and Light."Proc. Visualization 91, IEEE CS Press, Los Alamitos, Calif., 1991, pp. 321-328.
12. Foley, J.D., et al.,Computer Graphics: Principles and Practice, Addison-Wesley, Reading, Mass., 1990.
13. A.H. Barr, "Superquadrics and Angle-Preserving Transformations,"IEEE CG&A, Vol. 1, No. 1, Jan. 1981, pp. 11-23.
14. D. Hilbert and S. Cohn-Vossen,Geometry and the Imagination, Chelsea Publishing Co., New York, 1952.
15. B. K. P. Horn,Robot Vision. Cambridge, MA: M.I.T. Press, 1986.
22 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool