由於現今的影像系統已相當的進步，故目前相關的研究，大都把焦點放在高解析度上。然而隨著對速率要求的增加，相對低解析度所帶給我們的便利如：處理速度的增加，低儲存容量、低耗電量等，也就更形重要了。由以往的我們對六角格子的研究上可以了解到，在微觀的角度下，六角格子的角解析較佳，影像會有較好的品質，因此對六角格子而言，低解析度的領域將是其爾後發展的重心。
就目前而言，在影像系統中，無論是輸出或輸入大都是使用矩形格子系統，因此相關的顯示技術及原理，自然也都是以矩形格子系統為主。而對於一個顯示系統而言，文字及圖形可以說是其構成的兩大元素，也因此對於我們欲以六角形格子系統取代矩形格子系統而言，這也是首要考慮的因素。
本研究是以矩形格子系統為基礎，將其平面參數式曲線及充填演算法應用至六角形格子系統中，並進而探討文字及圖形在六角形格子系統上的顯示方法。

Due to the advances of image system, most researches are developed on high-resolution system. However, the low-resolution system have an advantage over the high-resolution system on processing speed, saving space and power consumption. From the research of hexagonal grid, we know that from the view of microcosmic, the angle resolution and connection of hexagonal grid are better than rectangular grid, so images on hexagonal grid also have the better quality.
Almost input and output systems of image device are on rectangular grid, so its technology and theory are developed on rectangular grid system. For a displaying system, fonts and images are the main elements. If we want to substitute hexagonal grid system for rectangular grid system, they will be primary factories that we consider.
In this research based on rectangular grid system, we will apply the plane parametric curves and fill algorithm on hexagonal grid system, and probe into the displaying method of fonts and images on hexagonal grid system.

第一章 緒論……………………………………………………………1
第二章 對稱性六角座標系平面參數式曲線的產生…………………6
2-1 對稱性六角座標系………………………………………….6
2-2 對稱性六角座標系的三角函數…………………………….8
2-3 平面參數式曲線的產生…………………………………….9
2-3.1 直線的參數式…………………………………………..10
2-3.2 圓的參數式……………………………………………..11
2-3.3 橢圓的參數式…………………………………………..15
2-3.4 貝茲曲線的參數式……………………………………..20
2-3.5 像素點的選取…………………………………………..22
第三章 六角格子上多邊形的充填方法……………………………..29
3-1 邊界充填演算法…………………………………………...30
3-2 掃描線充填演算法………………………………………...33
3-3 奇偶充填演算法…………………………………………...36
3-4 非零纏繞充填演算法……………………………………...37
3-5 奇異性……………………………………………………...38
第四章 六角格子上的字型產生……………………………………..40
4-1 點陣字型…………………………………………………...42
4-2 外框字型…………………………………………………...44
4-2.1 建構外框字型的基本原理……………………………..45
4-2.2 平滑化處理……………………………………………..60
第五章 六角格子上的圖形…………………………………………..68
5-1 六角形格子取樣…..……………………………………….69
5-1.1 六角形格子垂直交錯的排列方式……………………..69
5-1.2 六角形格子水平交錯的排列方式……………………..76
5-2 六角形格子排列方式之比較……………………………...81
5-3 矩形格子系統及六角形格子系統的顯示探討…………...84
5-3.1 矩形格子系統…………………………………………..84
5-3.2 六角形格子系統………………………………………..88
5-4 範例比較…………………………………………………...90
第六章 結論與建議…………………………………………………..92

A. P. Fitz and R. J. Green, “Fingerprint Classification using a Hexagonal Fast Fourier Transform,” Pattern Recognition, vol. 29, no. 10, pp. 1587-1597, 1996.
D. B. Olfe, Computer Graphics for Design. Englewood Ciffs, New Jersey: Prentice Hall, 1995.
D. F. Rogers and J. A. Adams, Mathematical Elements for Computer Graphics, New York: McGraw-Hill, 1990.
E. Bribiesca, “A new chain code,” Pattern Recognition, vol. 32, no. 2, pp. 235-251, Feb. 1999.
E. S. Deutsch, “Thinning Algorithms on Rectangular, Hexagonal, and Triangular Arrays,” Commun. ACM, vol. 15, pp. 827-837, 1972.
H. Ballard and C. M. Brown, Computer Vision, Englewood Ciffs, New Jersey: Prentice Hall, 1982.
I. Her, “A Symmetrical Coordinate Frame on the Hexagonal Grid for Computer Graphics and Vision,” ASME J. Mech. Design, vol. 115, pp. 447-449, Sept. 1993.
I. Her and C.-T. Yaun, “Resampling on a Pseudo-hexagonal Grid,” CVGIP: Graphical Models Image Processing, vol. 56, no. 4, pp. 336-347, July 1994.
I. Her, “Geometric transformations on the Hexagonal Grid,” IEEE Trans. Image Processing, vol. 4, no. 9, pp. 1213-1222, Sept. 1995.
M. J. E. Golay, “Hexagonal parallel pattern transformations,” IEEE Transactions on Computer, vol. C-18, no. 8, pp.733-740, Aug. 1969.
Narendra Ahuja, “On Approaches to Polygonal Decomposition for Hierarchical Image Representation,” Computer Vision Graphics and Image Processing, vol. 24, pp. 200-214, 1983.
R. C. Staunton, “The design of hexagonal sampling structure for image digitization and their use with local operators,” Image and Vision Computing, vol. 7, pp. 162-166, 1989.
R. C. Staunton, “An Analysis of Hexagonal Thinning Algorithms and Skeletal Shape Representation,” Pattern Recognition, vol. 29, no. 7, pp. 1131-1146, 1996.
R. D. Hersch, C. Betrisey, J. Bur and A. Gurtler, “Perceptually Tuned Generation of Grayscale Fonts,” IEEE Computer Graphics and Applications, vol.15, no.6, pp. 78-88, 1995.
R. M. Mersereau, “The processing of hexagonally sampled two dimensional signals,” Proceeding of the IEEE, vol. 67, pp. 930-949, 1979.
T. Wright, “History and Technology of computer Fonts,” IEEE Annals of the History of Computing, vol. 20, no. 2, pp. 30-34, 1998.
Ulichney, Robert, Digital Halftoning, Cambridge, MA: MIT Press, 1987.
W. K. Pratt, Digital Image Processing, New York: Wiley Interscience, 1991.
袁繼曾, “六角形格子上之直線產生及影像縮放技術,” 碩士論文, 國立中山大學, 高雄, 1993.
麥英佺, “六角形格子上之幾何轉換與二次曲線之產生,” 碩士論文, 國立中山大學, 高雄, 1994.
李祐魁, “利用次像素在低解析六角格子影像上作輪廓辨識,” 碩士論文, 國立中山大學, 高雄, 2000.
黃耀明, 電腦繪圖學, 全華科技圖書, 台北, 1989.
Microsoft typography, http://www.microsoft.com/typography.
TrueType Reference Manual, http://fonts.apple.com