Morphing is an interesting topic in not only graphics area but also in geometry area. In geometry area, one of the objects for morphing are graph drawings. This paper is focus on one special cases “orthogonal polygon drawing” of graph drawings. For two planar orthogonal drawings S and T of the same polygon graph G, we want to morph from S to T while preserving planarity and orthogonality and good visual effect. Based on the algorithm flow of Lubiw et al. [27], we improve one of its main operations “twist”. This improvement reduces the number of another main operation “zig-zag elimination”. Also the number of bends decreases. Moreover, we add an operation “MTFP” between each two original operations. “MTFP” moves each vertex to the position of its corresponding vertex in the target drawing, i.e. its final position. By doing “MTFP”, the shape of the intermediate drawings are more alike to the shape of the target drawing. Also the morphing become more smooth. Besides, in their original algorithm, they just give a basic idea about the process of adding non-intersecting orthogonal path but there is no details. Thus we provide a simple algorithm in details to complete the process. Then we input ten thousand pairs of random-generated 15-vertex polygons to do some experiments. These experimentation shows that our work obtain obvious improvements comparing to the previous works.

長久以來，物體間的形體轉換一直都是研究者感興趣的題目。此題目不僅在電腦圖學領域中被重視，在圖形畫法的領域中，也被廣泛的討論。一個良好的圖形畫法間的轉換，能夠讓使用者更容易去瞭解複雜圖形中所隱含的資訊。研究者針對各種特定類別的圖形，提出了許多不同形式的形體轉換演算法。平面正交圖形畫法是其中一種常被使用的圖形畫法。關於這類的圖形，前人提供了一個有良好時間複雜度的演算法。但是此演算法在視覺上並沒有辦法提供平順的形體變換效果。在這篇論文中，我們基於前人的演算法，並做出了三項貢獻。第一項是我們對其中的主要技巧Twist做了改良，使得在形體轉換的過程中，所產生的彎曲點變少。這項改進同時造成所需的步驟數變少。我們的第二項改進是加入了一個新的步驟MTFP，去調整中間圖形的頂點位置，使得中間圖形跟結束圖形的相似度提高。這個改進使得形體轉換的過程變得更加平順。最後我們補完了在原本的演算法的第四階段中，沒被詳述的加入正交路徑的演算法。我們同時實作了原本的演算法與我們新版本的演算法，並進行了一系列的實驗去測試改進的幅度。實驗結果證明，我們的演算法對於平面正交圖形間的形體轉換有了良好的改進。

[1] M. Alexa. Merging polyhedral shapes with scattered features. In Shape Modeling and Applications,
1999. Proceedings. Shape Modeling International ’99. International Conference on,
pages 202 –210, 278, mar 1999.
[2] M. Alexa, D. Cohen-Or, and D. Levin. As-rigid-as-possible shape interpolation. In Proceedings
of the 27th annual conference on Computer graphics and interactive techniques,
SIGGRAPH ’00, pages 157–164. ACM Press/Addison-Wesley Publishing Co., 2000.
[3] B. Aronov, R. Seidel, and D. Souvaine. On compatible triangulations of simple polygons.
Computational Geometry, 3(1):27 – 35, 1993.
[4] T. Beier and S. Neely. Feature-based image metamorphosis. SIGGRAPH Comput. Graph.,
26:35–42, July 1992.
[5] T. Biedl, A. Lubiw, and M. Spriggs. Morphing planar graphs while preserving edge directions.
In P. Healy and N. Nikolov, editors, Graph Drawing, volume 3843 of Lecture Notes in
Computer Science, pages 13–24. Springer Berlin / Heidelberg, 2006.
[6] T. Biedl, A. Lubiw, and M. J. Spriggs. Morphing planar graphs while preserving edge directions.
In 13th Symposium on Graph Drawing (GD).
[7] S. S. Bridgeman, G. D. Battista, W. Didimo, G. Liotta, R. Tamassia, and L. Vismara.
Turn-regularity and optimal area drawings of orthogonal representations. Comput. Geom.,
16(1):53–93, 2000.
[8] S. S. Cairns. Deformations of plane rectilinear complexes. The American Mathematical
Monthly, 51(5):pp. 247–252, 1944.
[9] Carsten and Thomassen. Deformations of plane graphs. Journal of Combinatorial Theory,
Series B, 34(3):244 – 257, 1983.
[10] D. Cohen-Or, A. Solomovic, and D. Levin. Three-dimensional distance field metamorphosis.
ACM Trans. Graph., 17:116–141, April 1998.
[11] W. Didimo and A. Leonforte. Grid: An interactive tool for computing orthogonal drawings
with the minimum number of bends. In Graph Drawing, pages 309–315, 1997.
[12] R. Diestel. Graph Theory. Springer-Verlag, 2010.
[13] M. Eiglsperger, S. Fekete, and G. Klau. Orthogonal graph drawing. In M. Kaufmann and
D. Wagner, editors, Drawing Graphs, volume 2025 of Lecture Notes in Computer Science,
pages 121–171. Springer Berlin / Heidelberg, 2001.
[14] M. S. Floater and C. Gotsman. How to morph tilings injectively. Journal of Computational
and Applied Mathematics, 101(1-2):117 – 129, 1999.
[15] C. Friedrich and P. Eades. Graph drawing in motion. Journal of Graph Algorithms and
Applications, 6(3):353–370, 2002.
[16] E. Goldstein and C. Gotsman. Polygon morphing using a multiresolution representation. In
Proceedings of graphics interface 1995, GI ’95, pages 247–254, 1995.
[17] J. Gomes. Warping and Morphing of Graphical Objects. Morgan Kaufmann, 1999.
[18] C. Gotsman and V. Surazhsky. Guaranteed intersection-free polygon morphing. Computers
and Graphics, 25(1):67 – 75, 2001.
[19] T. Kanai, H. Suzuki, and F. Kimura. 3d geometric metamorphosis based on harmonic map.
In Computer Graphics and Applications, 1997. Proceedings., The Fifth Pacific Conference
on, pages 97 –104, oct 1997.
[20] M. Kaufmann and D.Wagner, editors. Drawing Graphs: Methods and Models (Lecture Notes
in Computer Science). Springer Verlag, 2001.
[21] J. R. Kent, W. E. Carlson, and R. E. Parent. Shape transformation for polyhedral objects.
SIGGRAPH Comput. Graph., 26:47–54, July 1992.
[22] G.W. Klau and P. Mutzel. Optimal compaction of orthogonal grid drawings. In IPCO, pages
304–319, 1999.
[23] F. Lazarus and A. Verroust. Three-dimensional metamorphosis: A survey. The Visual Computer,
14(8-9):373–389, 1998.
[24] A. W. F. Lee, D. Dobkin, W. Sweldens, and P. Schr¨oder. Multiresolution mesh morphing. In
Proceedings of the 26th annual conference on Computer graphics and interactive techniques,
SIGGRAPH ’99, pages 343–350. ACM Press/Addison-Wesley Publishing Co., 1999.
[25] T.-Y. Lee and P.-H. Huang. Fast and intuitive metamorphosis of 3d polyhedral models using
smcc mesh merging scheme. IEEE Transactions on Visualization and Computer Graphics,
9:85–98, 2003.
[26] A. Lubiw and M. Petrick. Morphing planar graph drawings with bent edges. Electronic Notes
in Discrete Mathematics, 31(0):45 – 48, 2008.
[27] A. Lubiw, M. Petrick, and M. Spriggs. Morphing orthogonal planar graph drawings. SODA
’06: Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm,
pages 222–230, 2006.
[28] M. Malkova. A new core-based morphing algorithm for polygons. In Proceedings of CESCG,
2007.
[29] M. Malkova, I. Kolingerova, and J. Parus. Core-based morphing algorithm for triangle
meshes. In SIGRAD 2008, volume 34, pages 39–46, 2008.
[30] K. Misue, P. Eades, W. Lai, and K. Sugiyama. Layout adjustment and the mental map.
Journal of Visual Languages and Computing, 6:183–210, 1995.
[31] M. Mlkov, J. Parus, I. Kolingerov, and B. Bene. An intuitive polygon morphing. The Visual
Computer, 26:205–215, 2010.
[32] T. Nishizeki and M. S. Rahman. Planar Graph Drawing (Lecture Notes Series on Computing).
World Scientific Publishing Company, 2004.
[33] G. L. Roberto Tamassia. Graph drawing. In J. E. Goodman and J. O’Rourke, editors, Handbook
of Discrete Computational Geometry, chapter 52, pages 1163–1185. Springer Verlag, 2
edition, 2004.
[34] T.W. Sederberg, P. Gao, G.Wang, and H. Mu. 2-d shape blending: an intrinsic solution to the
vertex path problem. In Proceedings of the 20th annual conference on Computer graphics
and interactive techniques, SIGGRAPH ’93, pages 15–18. ACM, 1993.
[35] T. W. Sederberg and E. Greenwood. A physically based approach to 2d shape blending.
SIGGRAPH Comput. Graph., 26:25–34, July 1992.
[36] M. Shapira and A. Rappoport. Shape blending using the star-skeleton representation. Computer
Graphics and Applications, IEEE, 15(2):44 –50, mar 1995.
[37] R. Singh, R. M. Voyles, D. Littau, and N. Papanikolopoulos. Shape morphing-based control
of robotic visual servoing. Auton. Robots, 10(3):317–338, 2001.
[38] I. G. Tollis, G. D. Battista, P. Eades, and R. Tamassia. Graph Drawing: Algorithms for the
Visualization of Graphs. Prentice Hall, 1999.
[39] A. P. Toms and A. L. Bajuelos. Generating random orthogonal polygons. In IN POSTCONFERENCE
PROC. OF CAEPIA-TTIA’2003, LNAI, SPRINGER-VERLAG, 2003.
[40] D. B. West. Introduction to Graph Theory. Prentice Hall, 2001.
[41] R. R. Wilcox. Applying Contemporary Statistical Techniques. Academic Press, 2003.
[42] G. Wolberg. Image morphing: A survey. The Visual Computer, 14:360–372, 1998.