Representations for Lines and Curves

[Hill: p. 555-560]
A preliminary step to drawing lines and curves is choosing a suitable representation for them. There are three possible choices which are potentially useful.
Explicit: y = f(x)
line
circle
Parametric:   x = f(t),   y = f(t)
line
circle
Implicit:  f(x,y) = 0
line
circle

 Optimal Line Drawing

Drawing lines on a raster grid implicitly involves approximation. The general process is called rasterization or scan-conversion. What is the best way to draw a line from the pixel (x1,y1) to (x2,y2)? Such a line should ideally have the following properties. 



  
  

    
      
    
        
  • straight 
        
  • pass through endpoints 
        
  • smooth 
        
  • independent of endpoint order 
        
  • uniform brightness 
        
  • brightness independent of slope 
        
  • efficient 

    
      


A Straightforward Implementation
DrawLine(x1,y1,x2,y2)
int x1,y1,x2,y2;
{
  float y;
  int x;

  for (x=x1; x<=x2; x++) {
    y = y1 +  (x-x1)*(y2-y1)/(x2-x1)
    SetPixel(x, Round(y) );
  }
}


A Better Implementation
DrawLine(x1,y1,x2,y2)
int x1,y1,x2,y2;
{
  float m,y;
  int dx,dy,x;

  dx = x2 - x1;
  dy = y2 - y1;
  m = dy/dx;
  y = y1 + 0.5;
  for (x=x1; x<=x2; x++) {
    SetPixel(x, Floor(y) );
    y = y + m;
  }
}






The Midpoint Algorithm
The midpoint algorithm is even better than the 
above algorithm in that it uses only integer calculations. It treats line 
drawing as a sequence of decisions. For each pixel that is drawn, the next pixel 
will be either E or NE, as shown below. 



The midpoint algorithm makes use of the the implicit definition of the line, 
F(x,y) = 0. The E/NE decisions are made as follows. 

define 
 
if E is chosen, 
 
if NE is chosen, 
 
The process repeats, stepping along x, making 
decisions whether to set E or NE pixel. 







Initialization


 

Now we would like an integer-only algorithm. 
Define 
 

Midpoint Algorithm
The following algorithm produces the desired results 
for lines having x1 less than x2 and a slope less or equal to 1. 
drawline(x1, y1, x2, y2, colour)
int x1, y1, x2, y2, colour;
{
  int dx, dy, d, incE, incNE, x, y;

  dx = x2 - x1;
  dy = y2 - y1;
  d = 2*dy - dx;
  incE = 2*dy;
  incNE = 2*(dy - dx);
  y = y1;
  for (x=x1; x<=x2; x++) {
    setpixel(x, y, colour);
    if (d>0) {
      d = d + incNE;
      y = y + 1;
    } else {
      d = d + incE;
    }
  }
}