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An Algebra of Geometric Shapes
September/October 1993 (vol. 13 no. 5)
pp. 50-59

A simple algebra of shapes with 2D planar regions is developed. The fact that a 2D region can be completely described by a one-dimensional, closed-boundary curve if it is homogeneous is used in the presented approach, which first converts the spatial description of the closed curve into an equivalent Fourier series description and then uses the Fourier-description to define binary composition operations that combine two planar shapes to form another planar shape. It is shown how the geometric system comprising the set of all planar shapes and the composition operations can be mapped onto the algebraic system of linear/vector space.

1. D. Pedoe,Geometry: A Comprehensive Course, Dover, New York, 1970.
2. P.K. Ghosh, "An Algebra of Polygons through the Notion of Negative Shapes,"Computer Vision, Graphics, and Image Processing: Image Understanding, Vol. 54, No. 1, July. 1991, pp. 119-144.
3. S. Pinker, "Visual Cognition: An Introduction,"Visual Cognition, S. Pinker, ed., A Bradford Book, Cambridge, Mass., 1986.
4. F.P. Kuhl and C.R. Giardina, "Elliptic Fourier Features of a Closed Contour,"Computer Graphics and Image Processing, Vol. 18, No. 3, Mar. 1982, pp. 236-258.
5. R. Chellappa and R. Bagdazian, "Fourier Coding of Image Boundaries,"IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. PAMI-6, No. 11, Jan. 1984, pp. 102-105.
6. C.C. Lin and R. Chellappa, "Classification of Partial 2-D Shapes Using Fourier Descriptors,"IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. PAMI-9, No. 5, Sept. 1987, pp. 686-690.
7. R. Rubinstein,Digital Typography--An Introduction to Type and Composition for Computer System Design, Addison Wesley, Reading, Mass., 1988.
8. P.K. Ghosh, "Study of Fourier Discrimination of Digitized Letterforms,"Proc. Int'l Conf. Computers, Systems, and Signal Processing, Indian Inst. Science, Bangalore, India, 1984, pp. 72-76.
9. E. Kreyszig,Advanced Engineering Mathematics, Addison Wesley, Reading, Mass., 1983.
10. R.M. Bozinovic and S.N. Srihari, "Off-Line Cursive Script Word Recognition,"IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. PAMI-11, No. 1, Jan. 1989, pp. 68-83.
11. J. Andre and B. Borghi, "Dynamic Fonts," inRaster Imaging and Digital Typography, J. Andre and R.D. Herch, eds., Cambridge Univ. Press, New York, 1989, pp. 198-204.

Citation:
Pijush Ghosh, Pradeep K. Jain, "An Algebra of Geometric Shapes," IEEE Computer Graphics and Applications, vol. 13, no. 5, pp. 50-59, Sept.-Oct. 1993, doi:10.1109/38.232114
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