AN EXTENDED STIPPLE LINE ALGORITHM AND ITS CONVERGENCE ANALYSIS

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1 Journal of Theoretical and Alied Information Technology AN EXTENDED STIPPLE LINE ALGORITHM AND ITS CONVERGENCE ANALYSIS ZHANDONG LIU, HAIJUN ZHANG, YONG LI, XIANGWEI QI, MUNINA YUSUFU Deartment of Comuter Science and Technology, Xinjiang Normal University, NO. 12, New Medicine Road, Urumqi City, Xinjiang Province, China, 8354 ABSTRACT Not only does the stile line algorithm of revious have characteristics of a single tye, ut also the henomenon, that the osition of calculated ixel deviates from the actual line, aears easily when we solve ractical rolems. This aer introduces the revious stile line algorithm. The other conditions of the stile line algorithm are analyzed in detail. Last, the convergence of the stile line algorithm is roved. The result shows that the vertical coordinate s location, which the next ixel oint of the current ixel oint, converges wealy to the ideal line. Keywords: Stile Line Algorithm, Algorithm's Alication, Convergence Analysis 1. INTRODUCTION At resent, the technology of comuter grahics is widely used, and the research of comuter grahics method is ecoming more and more imortant. Curve drawing is the asic content of comuter grahics, and the drawing of a straight line is ased on the curve drawing [1]. In mathematics, the straight line, has no width, is a collection, which is comosed of countless oints. When the straight line is rasterized, we determine the est aroximation to the straight line of a grou of ixels, in which the dislay of the finite ixel matrix. According to the scan order, these ixels are oerated y the current writing attern. A icture may contain thousands of lines, so the seed of drawing algorithms must e as fast as ossile. The generating algorithm of classic line includes: Digital Differential Analysis (DDA) algorithm, stile line algorithm and Bersenham algorithm[2]. DDA algorithm[3] need use the floating-oint to calculate, so the accumulative error will mae the calculated ixel osition deviate from the actual line segment. Because the seed of ixel grid coordinate s integer oerations and floating-oint calculations is very slow, the algorithm efficiency is very low, and DDA algorithm is rarely used in ractice. The algorithms of stile line and Bresenham overcome the disadvantages of DDA algorithm, are the most effective algorithm to create line. Literature [2], descries the stile line algorithm, ut its tye is single. The henomenon, that the osition of calculated ixel deviates from the actual line, aears easily when we use the algorithm which is introduced in the literature [2] to solve the ractical rolems. This aer introduces other situations of the stile line algorithm, and analyses the convergence of the algorithm. 2. STIPPLE LINE ALGORITHM 2.1 Two Dimensional Line In the lane coordinate, straight line can e exressed as ax + y + c =. For the straight segment whose starting coordinate is (x s, y s ), the end coordinate is (x e, y e ). Its sloe can e exressed as: ( y y m s ) = ( x x ) e = e s dy dx For any x, along the linear, whose increment is x, the increment of y is y = m * x. For any y, x = y / m. x and y constitute the foundations of mathematics which determines the CRT(Cathode Ray Tue) deflection voltage. For the line segment which m 1, firstly, we set the level comonent as x. Then, according to the exression of y = m * x, we can get the corresonding vertical comonent of y. For the line segment which m >1, we set the vertical comonent as y. Then, according to the exression of x = y / m, we can get the corresonding level comonent of x. For the raster scan dislay, in order to mae the generating line smooth, line of the show is always 665

2 Journal of Theoretical and Alied Information Technology in the direction of coordinate where the adjacent ixels are taen[3-4]. 2.2 Previous Stile Line Algorithm Let F(x, y) say the line segment of L(P (x, y ), P 1 (x 1, y 1 )), the current ixel oint is (x, y ). For the next ixel, there are two otions to choose: P 1 (x +1, y ) and P 2 (x + +1). Let M(x + +.5) e the midoint of P 1 and P 2, Q is the crossing oint of the ideal line and x x + 1. We comare the osition = of Q and M. (a) When the oint of M is located eneath the oint of Q, P 2 should e the next ixel; () When the oint of Q is located eneath the oint of M, P 1 should e the next ixel. As shown in Figure 1. Figure 1: Previous Stile Line Algorithm In order to judge the osition relationshi Q and M, we need to ut the M into the exression: F(x, y) = ax + y + c. Then judge the symols of F(M). Now, we constructed discriminant: d = F(M) = F(x + +.5) + 1) + (y +.5) + c which a = y 1 = x 1 y1. (1) Because d is a line function of x and y, we use incremental comuting to imrove the oerational (a) When d, we select the right oint of P 1 as the next ixel. When the next ixel is determined, we calculate the exression: d1 = F(x = d + a + 2) + (y +.5) +.5) + c the increment is a. () When d<, we select the to right oint of P 2 as the next ixel. When the next ixel is determined, we calculate the exression: = d + a + the increment is a+. + 2) + (y ) + 1.5) + c Let s egin to draw line from the oint (x, y ). By Eq. 1, we now d =a+.5. In order to eliminate floating-oint arithmetic, we use 2d to relace d. Algorithm rogram is shown as follows. void Midointline(int x, int x1, int y, int y1) dx=x1-x; dy=y1-y; x=x, y=y; drawixe1(x, y, color); if(dy/dx>&&dy/dx<=1) d=dx-2*dy; d1=-2*dy; d2=2*(dx-dy); while(x<x1) if(d >=) x=x+1; d=d+d1; elseif(d<) x=x+1; y=y+1; d=d+d2; drawixe1(x, y, color) 2.3 Extended Stile Line Algorithm Let F(x, y) show the line segment of L(P (x, y ), P 1 (x 1, y 1 )), the current ixel oint is (x, y ). According to the sloe size of the straight segment L, for the next ixel oint selection, we divide three cases to analyse[5-9]. (1)When a 1 For the next ixel, there are two otions to choose: P 1 (x + ) and P 2 (x + -1). Let M(x +1, y -.5) e the midoint of P 1 and P 2, Q is the crossing oint of the ideal line and x=x +1. We comare the osition of Q and M. (a) When the oint of M is located eneath the oint of Q, P 2 should e the next ixel; () When the oint of Q is located eneath the oint of M, P 1 should e the next ixel. As shown in Figure

3 Journal of Theoretical and Alied Information Technology P=(x,y ) M P 1 P 2 Q Figure 2: The First Case Of The Stile Line Algorithm Now, we constructed discriminant: d = F(M) = F(x + 1) + (y +.5) + c P 1.5) which a = y 1 = x 1 y1. (2) Because d is a line function of x and y, we use incremental comuting to imrove the oerational (a) When d>, we select the right-ottom oint of P 1 as the next ixel. When the next ixel is determined, we calculate the exression: d1= F(x + 2) + (y = d + a the increment is a-..5-1) 1.5) + c () When d, we select the right oint of P 2 as the next ixel. When the next ixel is determined, we calculate the exression: = d + a the increment is a. + 2) + (y.5).5) + c Let s egin to draw line from the oint (x, y ). By Eq. 2, we now d = a. 5. In order to eliminate floating-oint arithmetic, we use 2d to relace d. Algorithm rogram is shown as follows. void Midointline(int x, int x1, int y, int y1) dx=x1-x; dy=y1-y; x=x, y=y; drawixe1(x, y, color); if((dy/dx>=-1)&&(dy/dx<=)) d=-dx-2*dy; d1=-2*(dx+dy); d2 =-2*dy; while(x<x1) if(d > ) x=x+1; y=y-1; d=d+d1; elseif(d<= ) x=x+1; d=d+d2; drawixe1(x, y, color); a (2) When 1 < For the next ixel, there are two otions to choose: P 1 (x, y +1) and P 2 (x + +1). Let M(x +.5, y +1) e the midoint of P 1 and P 2, Q is the crossing oint of the ideal line and y=y +1. We comare the osition of Q and M. (a) When the oint of M is located left the oint of Q, P 2 should e the next ixel; () When the oint of Q is located right the oint of M, P 1 should e the next ixel. As shown in Figure 3. Figure 3: The Scond Case Of The Stile Line Algorithm Now, we constructed discriminant: d = F(M) = F(x +.5, y + 1) +.5) + (y + 1) + c (3) which a = y 1 = x 1 y1. Because d is a linear function of x and y, we use incremental comuting to imrove the oerational (a) When d>, we select the uer right oint of P 2 as the next ixel. When the next ixel is determined, we calculate the exression: d1 = F(x = d + a + the increment is a ) + (y 1) + 2) + c 667

4 Journal of Theoretical and Alied Information Technology () When d, we select the right aove oint of P 1 as the next ixel. When the next ixel is determined, we calculate the exression: = d + +.5, y +.5) + (y 1) + 2) + c the increment is. Let s egin to draw line from the oint (x, y ). By Eq. 3, we now d =. 5a +. In order to eliminate floating-oint arithmetic, we use 2d to relace d. Algorithm rogram is shown as follows. void Midointline(int x, int x1, int y, int y1) dx=x1-x; dy=y1-y; x=x, y=y; drawixe1(x, y, color); if(dy/dx>1) d=2*dx-dy; d1=2*(dx-dy); d2=2*dx; while(y<y1) if(d >) x=x+1; y=y+1; d=d+d1; elseif(d<=) y=y+1; d=d+d2; drawixe1(x, y, color); (3)When a < 1 For the next ixel, there are two otions to choose: P 1 (x, y -1) and P 2 (x + -1). Let M(x +.5, y -1) e the midoint of P 1 and P 2, Q is the crossing oint of the ideal line and y y + 1. = We comare the osition of Q and M. (a) When the oint of M is located left the oint of Q, P 2 should e the next ixel; () When the oint of Q is located right the oint of M, P 1 should e the next ixel. As shown in Figure 4. Figure 4: The Third Case Of The Stile Line Algorithm Now, we constructed discriminant: d = F(M) = F(x +.5, y 1) +.5) + (y 1) + c which a = y 1 = x 1 y1. (4) Because d is a linear function of x and y, we use incremental comuting to imrove the oerational (a) When d>, we select the elow oint of P 1 as the next ixel. When the next ixel is determined, we calculate the exression: d1 = F(x = d the increment is , y +.5) + (y 2) 2) + c () When d, we select the right elow oint of P 2 as the next ixel. When the next ixel is determined, we calculate the exression: = d + a the increment is a ) + (y 2) 2) + c Let egin to draw line from the oint of (x, y ). By Eq. 4, we now d =. 5a. In order to eliminate floating-oint arithmetic, we use 2d to relace d. Algorithm rogram is shown as follows. void Midointline(int x, int x1, int y, int y1) dx=x1-x; dy=y1-y; x=x; y=y; drawixe1(x, y, color); if(dy/dx<-1) d=-2*dx-dy; d1=-2*dx; 668

5 Journal of Theoretical and Alied Information Technology d2=-2*(dx+dy); while(y>y1) if(d>) y=y-1; d =d+d1; elseif(d <= ) y=y-1; x=x+1; d =d+d2; drawixe1(x, y, color); 3. THE APPLICATION OF EXTENDED STIPPLE LINE ALGORITHM Examle 1. Use the Extended Stile Line Algorithm to find the ixel oints, which is assed y the line segment, that connects the oints P 1 (2, 11) and P 2 (13, 1). Solution: Let L connect the oints P 1 (2, 11) and P 2 (13, 1). At this time, a 1 = 1. By the Extended Stile Line Algorithm, we now that L asses the ixel oints which is shown as Figure 5. Examle 2. Use the Extended Stile Line Algorithm to find the ixel oint, through which the line segment connections the oints P 1 (1, ) and P 2 (8, 14). Solution: Let L connect the oints P 1 (1, ) and P 2 (8, 14). At this time, a 1 < = 2. By the Extended Stile Line Algorithm, we now that L asses the ixel oints which is shown as Figure 6. Examle 3. Use the Extended Stile Line Algorithm to find the ixel oint, through which the line segment connections the oints P 1 (1, 14) and P 2 (6, 4). Solution: Let L connect the oints P 1 (1, 14) and P 2 (6, 4). At this time, a = 2 < 1. By the Extended Stile Line Algorithm, we now that L asses the ixel oints which is shown as Figure 7. Figure 5: -1 -a/ Figure 6: 1<-a/ Figure 7: -a/<-1 4. THE CONVERGENCE OF STIPPLE LINE ALGORITHM Definition: In the rocess of drawig line, let the current ixel oint e (x, y ), and the vertical coordinate e y, horizontal coordinate e x as the next ixel osition of (x, y ). For any interval of [x, x ], g( is a continuous function. When >N, there is lim x [ y f ( ] g( dx =, x 669

6 Journal of Theoretical and Alied Information Technology where and is a natural numer, in the interval of [x, x ], we say that y wealy converges to the function of f(. Theorem: In the interval [x, x ], if the sequence of y converges to the function of f(, and each y is continuous, then y wealy converges to the function of f(. Proof: In the interval of [x, x ], let y converge to f(, so f( is continuous. In the interval of [x, x ], let g( e a continuous function. By the roem set, in the interval of [x, x ], we now g(.y and g(.f( are integrale[1-13]. In the interval of [x, x ], g( is continuous, so g( has a maximal value, it is set to M(M>). In the interval of [x, x ], y is uniform convergence to f(. So for any ositive numer ε, there is N, when P>N, for any x [x, x ], the exression: y ε f ( < ( x x ) M will e estalished. According to the roerties of definite integral, when P>N, the exression: = M g( y dx g( [ y will e estalished. So lim x x y g( f ( dx f ( ] dx f ( dx < ε y g( dx = f ( g ( dx. That is to say y wealy converges to f(. 5. CONCLUSION In this aer, the revious stile line algorithm is introduced. Based on this, we exand the stile line algorithm. Finally, we rove the convergence of the stile line algorithm. The result showes that the stile line algorithm has ovious wea convergence. The next ste of the research is focused on the design of three dimensional midoint algorithm, the efficiency of the algorithm and its alication. ACKNOWLEDGEMENTS This wor is suorted y the University Scientific Research Project of Xinjiang Uyghur Autonomous Region(XJEDU212S29), the x Humanities and Social Science Project of China Ministry Education(11YJC8714), the National Nature Science Foundation of China ( ), the National Nature Science Foundation of China ( ), the Comuter Alication Technology Key Discilines of Xinjiang Normal University. REFERENCES: [1] Q. F. Zhang, H. L. Wang, L. Zhang, Study on an algorithm of linear scan conversion, Guizhou University of Technoogy Journal, Vol. 32, No. 2, 23, [2] Andreas Löhne, Constantin Z alinescu, On convergence of closed convex sets, Journal of Mathematical Analysis and Alications, Vol. 319, 25, [3] Z. X. Wang, R. Y. Wang, Generalizing Midoint Algorithm to 3D Straight-Line, Comuter Simulation Journal, Vol. 24, No. 4, 24, [4] S. G. Zhao, H. X. Cao, Piontwise convergence of artial sums of wavelet exansions, Pure and Alied Mathematics Journal, Vol. 23, No. 1, 27, [5] Huaiyi Gao, Zhongdi Cen, The Convergence of a Midoint Iterative Method with Milder Conditions, Ning Xia University(Natural Science Edition)Journal, Vol. 26, No. 1, 25, [6] Zhandong Liu,Yugang Dai, An Imroved Immune Algorithm and Its Convergence Analysis, Proceedings of International Conference on Comuter Design and Alications, IEEE Conference Pulishing Services, June 25-27, 21, Vol. 1, [7] Zhandong Liu, Tao Fu, Yugang Dai, Qinghua Zhao, A Study on the Convergence of the Clonal Select ion Algorithm Based on the Interval Sheath Theorem, Journal of Comuter Engineering & Science, Vol. 32, No. 5, 21, [8] Li Ma, Licheng Jiao, Lin Bai, Polyclonal Clusterring Algorithm and Its Convergence, China Universities of Posts and Telecommunications Journal, Vol. 15, No. 3, 28, [9] Juan A. Cljesta, Carlos Matran, Strong Convergence of Weighted Sums of Random Elements through the Equivalence of Sequences of Distriutions, Journal of Multivariate Analysis, Vol. 25, No. 2, 1988,. 31l

7 Journal of Theoretical and Alied Information Technology [1] Xiaojun Chen, Suerlinear convergence of smoothing quasi-newton methods for nonsmooth equations, Journal of Comutational and Alied Mathematics, 1997, Vol. 8, No. 1, 1997, [11] Moon-Soo Kim, Chulhyun Kim, On A Patent Analysis Method for Technological Convergence, Journal of Procedia Social and Behavioral Sciences, Vol. 4, 212, [12] Mare Balcerza, Katarzyna Dems, Andrzej Komisarsi, Statistical convergence and ideal convergence for sequences of functions, Journal of Mathematical Analysis and Alications, Vol. 328, No. 1, 27, [13] Jianing Luo, Xia Li, Mintong Chen, The Marov Model of Shuffled Frog Leaing Algorithm and Its Convergence Analysis, Acta Electronica Sinica Journal, Vol. 38, No. 12, 21,