## Line Equation

The Cartesian slop-intercept equation for a straight line is

y=mx+b                                                                                                                          ------         (1)
with
m->slope
b->y intercept

The 2 end points of a line segment are specified at a position(x1,y1)

Determine the values for the slope m and y intercept b with the following calculation.

here, slope m:
m = ( y2 - y1) / ( x2 - x1 )

m=  Dy / Dx                                                                                                            ------     ( 2 )

y intercept b
b=y1-mx1                                                                                                              ------     ( 3 )

• Algorithms for displaying straight line based on this equation
• y interval Dy from the equation

m=  Dy / Dx
Dy= m. Dx                                                                                                             ------    ( 4 )
Similarly x interval Dx from the equation

m = Dy / Dx
Dx =  Dy /m                                                                              -------    ( 5 )

• The digital differential analyzer(DDA) is a scan conversion line algorithm based on calculation either      Dy  or   Dx.
• The line at unit intervals is one coordinate and determine corresponding integer values nearest line for the other coordinate.
• Consider first a line with positive slope.

### Step : 1

If the slope is less than or equal to 1 ,the unit x intervals Dx=1 and compute each successive y values.
Dx=1

m =  Dy / Dx
m = ( y2-y1 ) / 1
m = ( yk+1 – yk ) /1

yk+1  = yk + m                                                                                              -------- ( 6 )

• subscript k takes integer values starting from 1,for the first point and increment   by 1 until the final end point is reached.
• m->any real numbers between 0 and 1
• Calculate y values must be rounded to the nearest integer

### Step : 2

If the slope is greater than  1 ,the roles of x any y at the unit y intervals Dy=1  and compute each successive y values.
Dy=1

m= Dy / Dx
m= 1/ (  x2-x1 )
m = 1 / ( xk+1 – xk  )

xk+1   =  xk   +  ( 1 / m )                                                                        -------   ( 7 )

• Equation 6 and Equation 7 that the lines are to be processed from left end point to the right end point.

### Step : 3

If the processing is reversed, the starting point at the right
Dx=-1

m= Dy / Dx
m = ( y2 – y1 ) / -1
yk+1  = yk - m                                                                                          --------  ( 8 )

Iintervals Dy=1  and compute each successive y values.

### Step : 4

Here, Dy=-1

m= Dy / Dx
m = -1 / ( x2 – x1 )
m = -1 / ( xk+1 – xk  )

xk+1   =  xk   +  ( 1 / m )                                                                    --------   ( 9 )

Equation 6 and Equation 9 used to calculate pixel position along a line with –ve slope.

faster method for calculating pixel position then the equation of a  pixel position.
Y=mx+b

The accumulation of round of error is successive addition of the floating point  increments is used to find the pixel position but it take lot of time to compute the pixel position.

dx = x2 - x1
dy = y2 - y1
for x from x1 to x2 {
y = y1 + (dy) * (x - x1)/(dx)
pixel(x, y)
}

## C Programming

void linedda(int xa,int ya,int xb,int yb)              {

int dx=xb-xa,dy=yb-ya,steps,k;
float xincrement,yincrement,x=xa,y=ya;

if(abs(dx)>abs(dy))       steps=abs(dx);
else                       steps=abs(dy);

xincrement=dx/(float)steps;
yincrement=dy/(float)steps;

putpixel(round(x),round(y),2)

for(k=0;k<steps;k++)   {
x+=xincrement;
y+=yincrement;
putpixel(round(x),round(y),2);
}
}

DDA Line Drawing Algorithm Using C Programming

## Example:

xa,ya=>(2,2)
xb,yb=>(8,10)
dx=6
dy=8

xincrement=6/8=0.75
yincrement=8/8=1

1)     for(k=0;k<8;k++)
xincrement=0.75+0.75=1.50
yincrement=1+1=2
1=>(2,2)

2)     for(k=1;k<8;k++)
xincrement=1.50+0.75=2.25
yincrement=2+1=3
2=>(3,3)

it will be incremented upto the final end point is reached.