1. Lecture 8 Shear and Line Drawing Algorithms
Computer Graphics
Lecture 8
Shear
and
Line Drawing Algorithms

2. Shear It is a transformation that distorts the shape of an object.
It means “pulling or elongating” a graphical object to right, left, up or down.
It can be done along x-axis or along y-axis.
Shear along x-axis
x’ shx x
y’ = y
x’ = x + shx.y
y’ = y

3. shx can be assigned any real number.
y y
x x
Positive values of shx shift the coordinate positions towards right and negative values shift them to left.

4. Example Shear the following object along x-axis, shx = 2
(0,1) (1,1) (2,1) (3,1)
(0,0) (1,0) (0,0) (1,0)

5. For point (0,1)
x’ = x + shx.y
x’ = = 2
(0,1) = (2,1)
For point (1,1)
x’ = = 3
=> (1,1) = (3,1)

6. For shearing in x-direction relative to other reference lines, that is, shifting coordinate positions by an amount equal to its distance from reference line y = yref
x’ = x + shx (y – yref) ,
(0,1) (1,1) (1,1) (2,1)
(0,0) (1,0) (1/2, 0) (3/2,0)
yref -1

7. Shear in y-direction
In this case x’ = x
y’ = y + shy (x – xref)

8. Line Drawing Algorithms
In random scan systems, a straight line can be drawn from one endpoint to other by using line drawing commands from frame buffer.
In raster scan systems the color intensities for a line are stored in the frame buffer and are read from the frame buffer.
In order to draw lines, we use the equation of straight line which is given as
y = mx + b
m= slope of line (that is, how steep line is)
b = y intercept, y= how far up, x = how far along

9. y m b x
y (x2,y2)
y1 (x1,y1)
x x2

10. The slope m of a line is equal to the change in y divided by change in x, that is:
m = change in x = y2 - y1
change in y x2 - x1
m = δy δx
=> δy = m . δx
=> δx = δy m

11. Positive Slope : On increasing x if y also increases then slope is positive.
Negative Slope : On increasing x if y decreases then the slope is negative.
y y
m m
positive slope negative slope
x x

12. Digital Differential Analyzer (DDA)
It is a scan conversion line drawing algorithm.
It is based on calculating either δy or δx using equations δy = m . δx and δx = δy / m
In this algorithm, the line is sampled at unit intervals in one coordinate and the integer nearest to the line path for other coordinate are determined.
We will consider a line with positive slope.

13. Line with positive slope

14. Case 1: slope is less than or equal to 1 (m<=1)
In this case, the line is sampled at unit x intervals and then the y-values are computed.
δy = m . δx
δy = m (δx = 1)
=> yk+1 – yk = m
=> yk+1 = yk + m [This means the next point in x is +1 and next point in y is +m]

15. Case 2: slope is greater than 1 (m > 1)
In this case the roles of x and y are exchanged. Now number of steps is determined by y (δy = 1) and the line is sampled at unit y intervals and values of x are computed.
δx = δy / m
δ x = 1 / m (δy = 1)
xk+1 – xk = 1 / m
xk+1 = xk + 1 / m

16. DDA Algorithm
Input x1, y1, x2, y2 dx = x2 – x1 dy = y2 – y1 If dx > dy steps = dx Else steps = dy xincrement = dx / steps yincrement = dy / steps x = x1 y = y1 for (i=0; i< steps; i++) x = x + xincrement y = y + yincrement