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Lecture 5 Rendering lines

Lecture 5 Rendering lines. Steps of line rendering Scan-conversion for line segments A1 tutorial. Description of a line in WC A line is projected to view plane and clipped by clipping window. It becomes a line segment, with two ends within the clipping window

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Lecture 5 Rendering lines

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  1. Lecture 5 Rendering lines Steps of line rendering Scan-conversion for line segments A1 tutorial CP411 Computer Graphics Fall 2007 Wilfrid Laurier University

  2. Description of a line in WC • A line is projected to view plane and clipped by clipping window. It becomes a line segment, with two ends within the clipping window • Line segment is transformed to normalized coordinate system • Line segment is mapped to screen coordinates, then converted to integers in view port ordinate system of the display • A line segment in SC is a graphics output primitives. It needs to be converted to pixels in SC by scan-conversion or rasterization algorithm. i.e. to set up pixels in the frame buffer. Each pixel position (x, y) has a location in frame buffer setPixel(x, y) is to set the color value at pixel (x, y) 1. Line-Drawing Steps CP411 Computer Graphics

  3. (x2,y2) (x2,y2) (x1,y1) (x1,y1) CP411 Computer Graphics Fall 2007 Wilfrid Laurier University

  4. Scan conversion • From SC description to pixels by scan conversion • Transforming the continuous into this discrete by sampling. Can never be ideal. • The best we can do is a discrete approximation of an ideal line. CP411 Computer Graphics

  5. The Basic idea of drawing lines • Let (x0, y0) and (x1, y1) be the screen coordinates of the two end points of a line segment. x0 x1 • Step through the x-coordinate from x0 to x1. Set the pixels closest to the line. (x2,y2) (x1,y1) CP411 Computer Graphics

  6. Important line qualities • Continuous appearance • Uniform thickness and brightness • Turn on the pixels nearest the ideal line • How fast is the line generated • Preferred algorithms • Use integers rater than floating point • Use addition, subtraction rather than multiplication and division • Use iteration or recurrence • Use increment CP411 Computer Graphics

  7. Simple Line Algorithm I int x0= , y0= , x1= , y1= ; int dx = x1 - x0; int dy = y1 - y0; setPixel(x0, y0); if (dx != 0) { float m = (float) dy / (float)dx; float b = y0 - m*x0; dx = (x1 > x0) ? 1 : -1; while (x0 != x1) { x0 += dx; y0 = Math.round(m*x0 + b); setPixel(x0, y0); } } y = m x + b m = (y1-y0)/(x1-x0) x = x0, x0+1, …, x1 y = mx + b CP411 Computer Graphics

  8. The above algorithm only works well when the angle between the line and the x-axis, , is less than 45o • The left figure is resulted for a line with  = 70o . The pixels on the line are too sparsely set. A more desirable one is shown on the right.  = 70o CP411 Computer Graphics

  9. Algorithm I improvement int x0 = , y0=,x1= , y1=..; int dx = x1 - x0; int dy = y1 - y0; setPixel(x0, y0); if (Math.abs(dx) > Math.abs(dy)) { float m = (float) dy / (float) dx; float b = y0 - m*x0; dx = (dx < 0) ? -1 : 1; while (x0 != x1) { x0 += dx; setPixel(x0, Math.round(m*x0 + b)); } } else if (dy != 0) { float m = (float) dx / (float) dy; float b = x0 - m*y0; dy = (dy < 0) ? -1 : 1; while (y0 != y1) { y0 += dy; setPixel(Math.round(m*y0 + b), y0); }} CP411 Computer Graphics

  10. d y dx=1 Algorithm II – by increment, remove multiplication int x0 = …, y0 = … , x1 = …, y1 = …; int x = x0; float y = y0, dy = (y1 – y0) / (x1 – x0); for (x = x0; x  x1; ++x) { setPixel( x, round(y) ); y = y + dy; } Round operation is slow as well CP411 Computer Graphics

  11. Algorithm III • error is the distance between the centre of a pixel and the line) int x0 = …, y0 = … , x 1 …, y1 = …; int x, y = y0; float error = 0.0, dy = (y1 – y0) / (x1 – x0); for (x = x0; x  x1; ++x) { setPixel( x, y ); error = error + dy; if (error > 0.5) { ++y; error = error - 1.0; } } CP411 Computer Graphics

  12. Algorithm III _ improvement • The floating point arithmetic can be entirely eliminated by scaling up error and y by 2(x2-x1) int x0 = …, x =1 …, y0 = … , y1 = …; int x, y = y0, dx = 2*(x1 – x0), dy = 2*(y1 – y0), error = 0; for (x = x1; x  x1; ++x) { setPixel( x, y ); error = error + dy; if (error > dx/2) { ++y; error = error - dx; } } CP411 Computer Graphics

  13. Algorithm III _improvement • The program runs even faster if we offset error by dx/2 • int x0 = …, x1 = …, y0 = … , y1 = …; • int x, y = y0, dx = 2*(x1 – x0), dy = 2*(y1 – y0), error = -dx / 2; • for (x = x0; x  x1; ++x) { • setPixel( x, y ); • error = error + dy; • if (error > 0) { ++y; error = error - dx; } • } CP411 Computer Graphics

  14. Algorithm V The Bresenham’s line drawing algorithm Combine the operations error = error + dy and error = error – dx into one when error > 0; int x0 = …, x1 = …, y0 = … , y1 = …; int x, y = y0, dx = 2*(x1 – x0), dy = 2*(y1 – y0), error = -dx / 2 - dy , dymx = dy - dx; for (x = x1; x  x2; ++x) { setPixel( x, y ); if (error < 0) error = error + dy; else { ++y; error = error + dymx; } } CP411 Computer Graphics

  15. void lineBres (int x0, int y0, intxEnd, intyEnd) { intdx = fabs (xEnd - x0), dy = fabs(yEnd - y0); int p = 2 * dy - dx; inttwoDy = 2 * dy, twoDyMinusDx = 2 * (dy - dx); int x, y; /* Determine which endpoint to use as start position. */ if (x0 > xEnd) { x = xEnd; y = yEnd; xEnd = x0; } else { x = x0; y = y0; } setPixel (x, y); while (x < xEnd) { x++; if (p < 0) p += twoDy; else { y++; p += twoDyMinusDx; } setPixel (x, y); } } CP411 Computer Graphics

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