EXPLANATION OF THE ALGORITHMS FOR DISPLAYING 3D FIGURES ON THE COMPUTER SCREEN

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1 Original Research Article: full paper [13] Galárraga, L., Heitz, G., Murphy, K., Suchanek, F. M. (2014). Canonicalizing Open Knowlege Bases. Proceeings of the 23r ACM International Conference on Conference on Information an Knowlege Management CIKM 14. oi: [14] Bollacker, K., Evans, C., Paritosh, P., Sturge, T., Taylor, J. (2008). Freebase. Proceeings of the 2008 ACM SIGMOD International Conference on Management of Data SIGMOD 08. oi: oi.org/ / [15] Shin, J., Wu, S., Wang, F., De Sa, C., Zhang, C., Ré, C. (2015). Incremental knowlege base construction using DeepDive. Proceeings of the VLDB Enowment, 8 (11), oi: / [16] Huynh, T. N., Mooney, R. J. (2008). Discriminative structure an parameter learning for Markov logic networks. Proceeings of the 25th International Conference on Machine Learning ICML 08. oi: [17] Richarson, M., Domingos, P. (2006). Markov logic networks. Machine Learning, 62 (1-2), oi: [18] Niu, F., Zhang, C., Re, C., Shavlik, J. (2012). Scaling Inference for Markov Logic via Dual Decomposition IEEE 12th International Conference on Data Mining. oi: [19] Singla, P., Domingos, P. (2006). Entity Resolution with Markov Logic. Sixth International Conference on Data Mining (ICDM 06). oi: [20] Van Haaren, J., Davis, J. (2012). Markov network structure learning: A ranomize feature generation approach. Proceeings of the Twenty-Sixth AAAI Conference on Artificial Intelligence, [21] Niu, F., Ré, C., Doan, A., Shavlik, J. (2011). Tuffy. Proceeings of the VLDB Enowment, 4 (6), oi: EXPLANATION OF THE ALGORITHMS FOR DISPLAYING 3D FIGURES ON THE COMPUTER SCREEN Alexaner Zhyrytovskyi Institute of Computer Technologies, Grauate School Open International University of Human Development Ukraine 23 Lvivs ka str., Kyiv, Ukraine i.am.zhirik@gmail.com Abstract Not long time ago people can only ream about the thing that they can see 3-imensional worl insie of the screen of the computer. Computers at that time can isplay only 256 colors on their screens, they can work only with integer numbers, an their computation spee was not fast enough. This ream comes true with the help of I Software company. This company starte to buil its games by using of simplifie 3D graphics. All these simplifie 3D graphics were calculate by the software 3D renering algorithms which were not publishe by these company. At that time there was a global goal to speeup calculations for 3D graphics an to make it look more realistic. The solution for this comes from harware companies. Harware companies starte to work on harware 3D accelerators, which can rener 3D graphics much faster than software algorithms. These 3D accelerators are fully patente, so that no one knows how they are implemente insie. At that time there were propose two 3D renering interfaces: Direct3D an OpenGL. An manufacturers of graphic cars starte supporting both of these interfaces. Lots of game evelopment companies starte to use these interfaces. An also I Software company starte to reesign its game engine to support these interfaces to take place in game market. From that time, everyboy starte to use 3D accelerators with built-in 3D renering features an as a result everyboy forgot about software 3D renering algorithms. Nowaays the situation come in the following way, that most of the algorithms which are use in 3D games are not publishe on paper, an all the 3D accelerator internal algorithms are patente. All moern 3D graphics are fully conserve by the vieo cars, so that 3D renering coul not be evelope in other way. The goal of this article is explanation of how 3D graphics can be isplaye on the screen of the computer Keywors: 3D, graphics, renering, mesh, z-buffer, texture, projection, perspective, triangulation, triangle rawing. DOI: / Alexaner Zhyrytovskyi 35

2 1. Introuction 3D graphics nowaays uses in games, movies, cartoons an for engineering process. Most of things such as cars, planes, houses an portable evices before being implemente shoul have 3D moel of itself an also 3D moels of its internal parts. Therefore, 3D graphics will be in eman for the near future. Toay probably every person seen the 3-imensional figures insie the computer, but noboy can figure out what formulas an what operations were performe to buil these figures. During this article, it is possible to see all the basic steps that nee to be one to buil 3D figures on the screen of the computer. 2. 3D rotation formulas The most funamental part in 3D graphics is 3-imensional rotation formulas, which looks the following: x' = xcosa ysina; y' = (xsina + ycosa)cosb zsin b; z' = (xsina + ycosa)sin b + zcosb, here x, y, z coorinates of the initial source point; x', y', z' coorinates of the rotate final point; a, b horizontal an vertical angles of the rotation Flat projection To project 3D worl point coorinates x', y', z' into the screen point coorinates x", y", it is possible to use the following formula: x" = x'; y" = y'. Here for projection just exclue z' variable. As an example on Fig. 1, it is possible to see the figure that was previously rotate by 3D rotation formulas, an now it is projecte own to the screen using flat projection formulas. Fig. 1. Flat projection 3. Projection with perspective In real worl, you can look at the roa an you can see that this roa becomes smaller as far it is locate from us (Fig. 2). 36

3 Fig. 2. Perspective In some istance from us, the object becomes really small so that we stop seeing it. The istance from the screen to the center of perspective comes here as value of perspective (Fig. 3). Fig. 3. Projection with perspective (as far the object locates from us, as smaller it is) Such projection is escribe using following formulas: x" = x' ; + z' y" = y', + z' here x', y', z' coorinates of the initial source point; x", y" coorinates of the final rotate point; value of perspective. 4. Drawing 3D figure using ots The simplest 3D figures can be built using ot-by-ot projection to the screen. The example of such figure is isplaye on Fig. 4. Fig. 4. Drawing 3D figure using ots Drawing 3D figure using mesh Other approach is to use lines instea of ots. In this case, all the projecte points we can connect into mesh of lines (Fig. 5). 37

4 Fig. 5. Drawing 3D figure using mesh Triangulation To make the figure soli it is possible to buil it using mesh of soli triangles (Fig. 6). Fig. 6. Soli triangle-base mesh Triangle rawing algorithm To raw the triangle itself, there is triangle rawing algorithm. On Fig. 7, it is possible to see how triangle rawing algorithm works. Fig. 7. Triangle rawing algorithm Algorithm: 1. Sort triangle vertices so that: Y 1 Y 2 Y 3 ; 2. Create two points P 1 (Fig. 7). At the beginning both of these points will have coorinates (X 1, Y 1 ); 3. Calculate horizontal increments 1 an 2, an also number of iterations N. It is possible to o this by the following formulas: X X = ; Y2 Y1 38

5 X X = ; Y3 Y1 N= Y2 Y Repeat it N times: move the points P 1 by the one step to the bottom. Also, it is necessary to move horizontally P 1 by the increment 1. An also to move horizontally P 2 by the increment 2. An fill empty horizontal place between P 1 using ots; 2. Recalculate first increment 1 an number of iterations N: X X = ; Y3 Y2 N= Y3 Y Repeat it N times: move the points P 1 by the one step to the bottom. Also it is necessary to move horizontally P 1 by the increment 1. An also to move horizontally P 2 by the increment 2. An fill empty horizontal place between P 1 using ots Z-buffer When rawing soli triangle mesh on the screen if these triangles are collapse, it is possible to see visual orering problem (Fig. 8). Fig. 8. Triangle orer isplaying problem Basically this problem can happen if on the same place we nee to raw 2 points with ifferent colors an ifferent z-istances. An the problem is how to keep the point with minimal z-istance. To solve this problem, it is necessary to use extra image which will contain z-istances for all the points that are isplaye on the screen, this image is calle z-buffer, it is possible to see it on Fig. 9. Fig. 9. Normal image an z-buffer image Let s suppose that at the beginning z-buffer fille with extremely big numbers. When we want to put a point on the screen, we can o this only if its z-istance is smaller than z-istance of corresponing point of z-buffer. After when we put the point on the screen we also nee to put its z-istance to z-buffer. 39

6 5. Filling the triangle with 3 color graient Triangle has 3 vertices. In this approach, each triangle vertex has its coorinates an color. To fin the color of the current point insie triangle it is necessary to fin the istances to each of the vertices (Fig. 10). Fig. 10. Filling triangle with 3 color graient Let s fin these istances by the following formulas: L = (X X ) + (Y Y ) ; L = (X X ) + (Y Y ) ; L = (X X ) + (Y Y ), here X, Y coorinates of current point insie triangle; X 1, Y 1 coorinates of first vertex of the triangle; X 2, Y 2 coorinates of secon vertex of the triangle; X 3, Y 3 coorinates of thir vertex of the triangle; L 1, L 2, L 3 istances from current point to each of vertices. Using these istances, it is possible to calculate the color of current point by the following formulas: R L + R L + R L R = ; L + L + L G L + G L + G L G = ; L + L + L B L + B L + B L B =, L + L + L here R 1, G 1, B 1 re, green an blue components of the vertex 1 color; R 2, G 2, B 2 re, green an blue components of the vertex 2 color; R 3, G 3, B 3 re, green an blue components of the vertex 3 color; R, G, B re, green an blue components of the estination color for the current point Texture overlay on 3D figure Texture overlaying on triangle is very similar to filling it with 3 color graient, the only ifference is that instea color component variables it is necessary to use texture coorinate variables (Fig. 11). To raw 3D figure with texture overlay, first wit is necessary to take some texture image an ivie it into triangles. This iving means that for each triangle we nee to calculate its coorinates on texture (Fig. 12). 40

7 Fig. 11. Drawing texture on the triangle Fig. 12. Overlaying the texture over the mesh of triangles After that, it is possible to change coorinates of the mesh with triangles, an prouce any 3D figure with texture (Fig. 13). Fig. 13. Wrapping mesh of triangles into ifferent shapes 6. Conclusion Theoretical algorithms for builing 3D figures, that were provie in this article, were implemente an teste. It is possible to see their implementation in software program, which can be viewe by the following link: As the result using these algorithms, it is possible to buil harware inepenent 3D graphics which can be calculate in real time without usage of any graphic car interfaces such as DirectX or OpenGL. Also nowaays, there looks to be no literature that escribing 3D renering process, an possibly this article can open eyes of the reaer on 3D renering process an will help him to unerstan how it performs. 41

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