Line Drawing Algorithms
CS-321
Dr. Mark L. Hornick
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Line Drawing Algorithms
(x1, y1)
(x0, y0)
Which pixels should be set
to represent the line?
CS-321
Dr. Mark L. Hornick
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Line Equations
(x1, y1)
(x0, y0)
y  mx  b
y1  y0
m
x1  x0
y  y0  m  x  x0 
y  m  x  x0   y0
CS-321
Dr. Mark L. Hornick
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Simple Algorithm
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Start x at origin of line
Evaluate y value
Set nearest pixel (x,y)
Move x one pixel
toward other endpoint
Repeat until done
CS-321
Dr. Mark L. Hornick
y  m  x  x0   y0
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Implementation
// SimplePixelCalc - elementary pixel calculation using
// variation of y=mx+b
void SimplePixelCalc( int x0, int xn, int y0, int yn )
{
for( int ix=x0; ix<=xn; ix++ )
{
float fy = y0 + float(yn-y0)/float(xn-x0)*ix;
// round to nearest whole integer
int iy = (int)(fy+0.5);
SetPixel(ix, iy, 1); // write pixel value
}
}
How well
does this
work?
How does this work?
CS-321
Dr. Mark L. Hornick
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Use the lab1 program on the
following coordinates
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Example 1
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Example 2
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(x0, y0) = (0,1)
(x1, y1) = (10,4)
(x0, y0) = (0,8)
(x1, y1) = (9,0)
Example 3
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(x0, y0) = (0,1)
(x1, y1) = (2,7)
CS-321
Dr. Mark L. Hornick
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m=0.3
Example 1 Results?
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CS-321
Dr. Mark L. Hornick
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m=-0.89
Example 2 Results?
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CS-321
Dr. Mark L. Hornick
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m=3
Example 3 Results?
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CS-321
Dr. Mark L. Hornick
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Simple Algorithm
Adjustment for m
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If |m|<1
y  m  x  x0   y0
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If |m|>1
x  (1/ m)  y  y0   x0
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If |m|=1
x=y
CS-321
Dr. Mark L. Hornick
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Simple Algorithm Summary
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Evaluate y = f(x) at each x position
Requires floating multiplication for each
evaluation of y
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Digital Differential Analyzer
(DDA) Algorithm
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Step through either x or y based on slope
Build the line parametrically
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If |m|<1
 xk+1 = xk + 1
 yk+1 = yk + m
If |m|>1
 xk+1 = xk + 1/m
 yk+1 = yk + 1
If |m|=1
 xk+1 = xk + 1
 yk+1 = yk + 1
CS-321
Dr. Mark L. Hornick
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DDA Implementation
void dda( int x0, int xn, int y0, int yn )
{
float dx = float(xn - x0); // total span in x
float dy = float(yn - y0); // total span in y
float y = float(y0);
float x = float(x0);
float Dx, Dy; // incremental steps in x & y
// determine if slope m GT or LT 1
if( dx > dy )
{
Dx = 1;
Dy = dy/dx;// floating division, but only done once per line
}
else
{
Dx = dx/dy;
Dy = 1;
}
int ix, iy; // pixel coords
for( int k=0; k<=((dx>dy)? dx:dy); k++ )
{
ix = int(x + 0.5); // round to nearest pixel coordinate
iy = int(y + 0.5);
x += Dx; // floating point calculations
y += Dy;
}
}
CS-321
Dr. Mark L. Hornick
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DDA Results?
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CS-321
Dr. Mark L. Hornick
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DDA Algorithm Highlights
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Reduction in strength
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Replaces multiplication with addition
Still some limitations
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Round-off error accumulates
Floating point addition (e.g. not on PPC)
CS-321
Dr. Mark L. Hornick
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Line-Drawing Algorithms
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Simple algorithm
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DDA algorithm
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Evaluate y = f(x) at each x position
Floating multiplication
Floating addition only
Can we do better?
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Tomorrow: Bresenham’s algorithm!
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