13.7 Scan Converting Lines

Given: p₁ = (x₁,y₁) and p₂ = (x₂,y₂)

Objective: Find pixels which approximate the line segment (p₁,p₂)

o o o o o o o o

o o o o o o o----o
---------
o o o --o---o-- o o o
---------
o----o o o o o o o

o o o o o o o o

Basic Algorithm (Digital Differential Analyzer, DDA):

slop := (y2 - y1) / (x2 -x1)
y := y1
FOR x := x1 TO x2 DO
PaintPixel(x,round(y))
y := y + slop
END

o o o o o o o o

o o o o o o oo----oo
--------
o o oo -oo----oo- oo o o
---------
oo----oo- o o o o o o

o o o o o o o o

o o |oo o o o o o
||
o o || o o o o o o
||
o oo| o o o o o o
|||
o || o o o o o o o
||
oo| o o o o o o o

Gaps occur for slop > 1

Bresenham’s Midpoint Line Algorithm

$dy 0 < slop = dx < 1$

o----o----o----o----o----o----o----o
----------------------------------------
o----o----o----o----o----o----o----o
----------------------------------------
o----o----o----o----o----o----o----o
----------------------------------------
o----o----o----o----o----o----o----o
----------------------------------------
o----o----o----o----o----o----o----o

(x,y) -- intersection point of the given line with a vertical grid line.
-- vertical distance of current point (x,y) from the “midpoint” above it.
Y -- pixel (x,Y) corresponding to (x,y)
For the next intersection point we have
- _x+1 := $Dx + dy- dx$
- If (_x+1 > 0) $Y := Y+ 1; Dx+1 := Dx+1 - 1$

real variable: D
dx := x2 - x1
dy := y2 - y1
D := -0.5
x := x1
Y := y1
FOR i := x1 TO x2 DO
   PaintPixel(x,Y)
   x := x + 1
   D := D + dy / dx
   IF  D>0  THEN
     Y := Y + 1
     D := D - 1
   END
END

Remove division and floating point arithmetics

dx := x2 - x1
dy := y2 - y1
D := -dx               * (2 * dx)
x := x1
Y := y1
FOR i := x1 TO x2 DO
   PaintPixel(x,Y)
   x := x + 1
   D := D + 2 * dy      * (2 * dx)
   IF  D>0  THEN
     Y := Y + 1
     D := D - 2 * dx    * (2 * dx)
   END
END

Generalization

|||||
---|| | ||---
----3 |2-----
-----4------1-----
-- 5------8 --
----6 |7 ----
--|||||||||--
|

1
2	switch the roles of x & y
...	...