computer graphics -...

Computer Graphics: Graphics Output Primitives

Line Drawing Algorithms

By: A. H. Abdul Hafez

[email protected],

March 8, 2017 1 CG, by Dr. A.H. Abdul Hafez, CE Dept. HKU

mailto:[email protected]

Outlines

March 8, 2017 CG, by Dr. A.H. Abdul Hafez, CE Dept. HKU 2

1. Basic concept of lines in OpenGL

2. Line Equation

3. DDA Algorithm

4. DDA Algorithm implementation

5. Bresenham's Line Algorithm

6. Circle generation algorithms

7. Circle midpoint algorithm

8. End

Basic concept of lines in OpenGL


To display the line on a raster monitor, the graphics system must first project the

endpoints to integer screen coordinates

Next, it determines the nearest pixel positions along the line path between the

two endpoints.

Then the line color is loaded into the frame buffer at the corresponding pixel

coordinates.

Basic concept of lines in OpenGL


Reading from the frame buffer, the video controller plots the screen pixels.

This process digitizes the line into a set of discrete integer positions that, in

general, only approximates the actual line path.

This rounding of coordinate values to integers causes all but horizontal and

vertical limes to be displayed with a stair-step appearance ("the jaggies").

Line Equation


The Cartesian slope-intercept equation for a straight line is

we can determine values for the slope m and y intercept b with the following calculations:

For any given x interval δx along a line, we can compute the corresponding y interval δy

On raster systems, lines are plotted with pixels. That is, we must "sample" a line at discrete positions and determine the nearest pixel to the line at each sampled position.

Discrete sample positions along the x axis is shown.

general Line drawing


Case 1 and 2: m>1, while Case 3 and 8: m<1.

Case 1: unit step through +x, increment y.

Case 2: unit step through +y, increment x.

Case 3: unit step through +y, decrement x.

Case 8: unit step through +x, decrement y.

Cases 4,5,6,7: swap end coordinates & use the algorithm for the symmetric quadrants.

DDA Algorithm


We consider first a line with positive slope; start is at left:

1. If the slope is less than or equal to 1, we sample at unit x intervals (δx = 1)

and compute successive y values as

2. For lines with a positive slope greater than 1, we reverse the roles of x and y.

That is, we sample at unity intervals (δy = 1) and calculate consecutive x

values as

x

y

δx = 1

δy = 1

DDA Algorithm


We consider second a line with positive slope; start is at right:

If the slope m <= 1, we sample at unit x intervals (δx = -1) and compute

successive y values as

If the slope m is greater than 1, we sample at unity intervals (δy = -1) and

calculate consecutive x values as

x δx =-1

δy =-m

y

δy = -1 δy =-1

δx =-1/m

DDA Algorithm


We consider first a line with negative slope; start is at left:

1. If the absolute slope |m| is less than or equal to 1, we sample at unit x

intervals (δx = 1) and compute successive y values as

2. For lines with absolute slope |m| greater than 1, we reverse the roles of x and

y. That is, we sample at unity intervals (δy = -1) and calculate consecutive x

values as

x

y

δx = 1

δy = m

δy = - 1

δx = -1/m

DDA Algorithm


We consider second a line with negative slope; start is at right:

If the slope m <= 1, we sample at unit x intervals (δx = -1) and compute

successive y values as

If the slope m is greater than 1, we sample at unity intervals (δy = +1) and

calculate consecutive x values as

x δx =-1

δy =-m

y

δy = 1

δx =1/m

DDA Algorithm implementation


This algorithm is summarized in the following procedure, which accepts as input two integer screen positions for the endpoints of a line segment.

Horizontal and vertical differences between the endpoint positions are assigned to parameters dx and dy. The difference with the greater magnitude determines the value of parameter steps.

Starting with pixel position (x0, y0), we determine the offset needed at each step to generate the next pixel position along the line path.

We loop through this process steps times.

1. If the magnitude of dx is greater than the magnitude of dy and x0 is less than xEnd, the values for the increments in the x and y directions are 1 and m=dy/dx, respectively.

2. If the greater change is in the x direction, but x0 is greater than xEnd, then the decrements -1 and –m=dy/dx are used to generate each new point on the line.

3. Otherwise, we use a unit increment (or decrement) in the y direction and an x increment (or decrement) of 1/m=dx/dy.

DDA Algorithm implementation


DDA Algorithm advantages and dis.


Advantage

Does not calculate coordinates based on the complete equation (uses offset

method)

Disadvantage

Round-off errors are accumulated, thus line diverges more and more from

straight line

Round-off operations take time

Perform integer arithmetic by storing float as integers in numerator and

denominator and performing integer arithmetic.

Bresenham's Line Algorithm


Motivation:

A section of the screen showing a pixel in column xk on scan line yk that is to be plotted along the path of a line segment with slope 0 < m < 1.

Assuming we have determined that

the pixel at (xk, yk) is to be displayed, we next need to decide which pixel to plot in column xk+l = xk + 1. Our choices are the pixels at positions (xk + 1, yk) and (xk + 1, yk+1).



To determine which of the two

pixels is closest to the line path, we

can set up an efficient test that is

based on the difference between the

two pixel separations, and defining

the decision parameter as

Vertical distances between pixel

positions and the line y coordinate

at sampling position xk + 1.



EXAMPLE 3-1 from the text



EXAMPLE 3-1 cont.



Pixel positions along the line path between endpoints (20, 10) and

(30, 18), plotted with Bresenham’s line algorithm.

Bresenham's Line Algorithm Code


The end of the Lecture


Thanks for your time

Questions are welcome

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