Edited from http://www.comp.hkbu.edu.hk/~sci3750/lect8-04.ppt The Rasterization Problem: Idealized Primitives Map to Discrete Display Space Edited from http://www.comp.hkbu.edu.hk/~sci3750/lect8-04.ppt 更新时间2019年2月19日星期二5时35分45秒

Solution Involves Selection of Discrete Representation Values Edited from http://www.comp.hkbu.edu.hk/~sci3750/lect8-04.ppt

Scan Converting Lines: Characterizing the Problem ideal line i.e. for each x, choose y i.e. for each y, choose x Edited from http://www.comp.hkbu.edu.hk/~sci3750/lect8-04.ppt

Scan Converting Lines: The Strategy Pick pixels closest to endpoints Select in between pixels “closest” to ideal line Objective: To minimize the required calculations. Pixels can be displayed in multiple shapes and sizes. In OpenGL, the centers of pixels are located at values halfway between intergers. Edited from http://www.comp.hkbu.edu.hk/~sci3750/lect8-04.ppt

Scan Converting Lines: DDA (Digital Differential Analyzer) Algorithm Calculate ideal line equation Starting at leftmost point: for each xi Calculate Select pixel at Selected Not Selected Edited from http://www.comp.hkbu.edu.hk/~sci3750/lect8-04.ppt

Scan Converting Lines: DDA Algorithm, Incremental Form Therefore, rather than recomputing y in each step, simply add m. The Algorithm: void Line(int x0, int y0, int xn, int yn) { int x; float dy, dx, y, m; dy = yn - y0; dx = xn - x0; m = dy/dx; y = y0; for (x = x0; x<=xn,x++){ WritePixel(x, round(y)); y += m; } Edited from http://www.comp.hkbu.edu.hk/~sci3750/lect8-04.ppt

Edited from http://www.comp.hkbu.edu.hk/~sci3750/lect8-04.ppt We assume m ≤1 in the above. For larger slope, the separation between pixels can be large, generating an unacceptable rendering of the line segment. We swap the roles of x and y for line segments with larger slopes. Edited from http://www.comp.hkbu.edu.hk/~sci3750/lect8-04.ppt

Bresenham’s Algorithm: Allowable Pixel Selections DDA requires a floating-point addition. Bresenham derived an algorithm that avoids all floating-point claculations. Option NE Option E Not Allowed Last Selection 0 <= Slope <= 1 Edited from http://www.comp.hkbu.edu.hk/~sci3750/lect8-04.ppt

Bresenham’s Algorithm: Iterating 0 <= Slope <= 1 Select NE Select E Edited from http://www.comp.hkbu.edu.hk/~sci3750/lect8-04.ppt

Bresenham’s Algorithm Decision Function: (implicit equation of the line) Edited from http://www.comp.hkbu.edu.hk/~sci3750/lect8-04.ppt

Bresenham’s Algorithm: Calculating the Decision Function Edited from http://www.comp.hkbu.edu.hk/~sci3750/lect8-04.ppt

Bresenham’s Algorithm: Incremental Calculation of di Option NE Option E Edited from http://www.comp.hkbu.edu.hk/~sci3750/lect8-04.ppt

Bresenham’s Algorithm: The Code void BresenhamLine(int x0, int y0, int xn, int yn) { int dx,dy,incrE,incrNE,d,x,y; dx=xn-x0; dy=yn-y0; d=2*dy-dx; /* initial value of d */ incrE=2*dy; /* decision funct incr for E */ incrNE=2*dy-2*dx; /* decision funct incr for NE */ x=x0; y=y0; DrawPixel(x,y) /* draw the first pixel */ while (x<xn){ if (d<=0){ /* choose E */ d+=incrE; x++; /* move E */ }else{ /* choose NE */ d+=incrNE; x++; y++; /* move NE */ } DrawPixel (x,y); Edited from http://www.comp.hkbu.edu.hk/~sci3750/lect8-04.ppt

Bresenham’s Algorithm An example 12 11 10 9 8 7 4 5 6 7 8 9 Edited from http://www.comp.hkbu.edu.hk/~sci3750/lect8-04.ppt

Bresenham’s Algorithm An example 12 11 10 9 8 7 4 5 6 7 8 9 Edited from http://www.comp.hkbu.edu.hk/~sci3750/lect8-04.ppt

Bresenham’s Algorithm An example 12 11 10 9 8 7 4 5 6 7 8 9 Edited from http://www.comp.hkbu.edu.hk/~sci3750/lect8-04.ppt

Another derivation: comparing d1 and d2 Allowed Option NE 0 <= Slope <= 1 d1 d2 Last Selection Option E Ref. [HA], Section 3.2.2 Edited from http://www.comp.hkbu.edu.hk/~sci3750/lect8-04.ppt