1. CGMB214: Introduction to Computer Graphics
Topic 4
Graphics Output Primitives (I)

2. Objectives To be able to define the term output primitives
To be able to define and create points and lines
To understand line-drawing algorithms and line function

3. Output Primitives
“where scene been represented by basic geometric structure which map into rectangular grid of pixel”
In short: Basic objects that create computer drawn pictures (points, lines, circles etc)
Pixel: little rectangles with same size and shape at each screen points
Image: is a rectangular grid or array of pixels.

4. The simplest form of output primitives are:
Points
Straight Lines
These primitives are defines in the graphics library (high level)
Learn the device-level algorithm for displaying 2D output primitives

5. Coordinate Reference Frames
Screen coordinate
Pixel position in the frame buffer
Absolute coordinate
The actual position within the coordinate in use
Relative coordinate
Coordinate position as an offset from the last position that was referenced (current position)

6. Points
Accomplished by converting a single coordinate position by an application program into appropriate operations for output device in use (i.e. monitor)
Electron beam is turned on to illuminate the screen phosphor depending on the display technology (Random-scan/Raster-scan)

7. OpenGL Point Function
Default colour for primitives is white and the default point size is equal to the size of 1 pixel.
Use glVertex*(); to state the coordinate value of a single position, where (*) determine the spatial dimension and data type of the coordinate value.
glVertex must be placed in between glBegin (**); and glEnd();, where (**) refers to the kind of output primitive to be displayed

8. The type of output primitive to be displayed
OpenGL Point Function
glBegin(GL_POINTS);
glVertex2i(40,50);
glEnd();
The type of output primitive to be displayed
The coordinate of the output primitive created
Two dimensional,
Integer data type

9. Lines
Most common 2D primitive - done 100s or 1000s of times each frame
Even 3D wire-frames are eventually 2D lines!
Optimized algorithms contain numerous tricks/techniques that help in designing more advanced algorithms

10. Lines Converting a line segment
Consider a line segment with endpoint (0,4) and (21,21) . Which pixel do we select?

11. Lines Naïve line conversion
The naïve method select every pixel that the mathematical line segment intercept.

12. Lines
Requirements:
Must compute integer coordinates of pixels which lie on or near a line or circle.
Pixel level algorithms are invoked hundreds or thousands of times when an image is created or modified – must be fast!
Lines must create visually satisfactory images.
Lines should appear straight
Lines should terminate accurately
Lines should have constant density
Line algorithm should always be defined.

13. Line Drawing Algorithm
Lines
Line Drawing Algorithm
Line Equation
DDA (Digital Differential Algorithm)
Bresenham’s Line Algorithm

14. Line Equation To create a line, an equation is used y = mx + c
where: m is slope
c is y intercept
For example, given 2 points P1 (5,2) and P2 (10,12)

15. Line Equation
In raster scan system, it will check for every pixel → need to calculate for all y for x = 5 until x = 10 y
Plot P1 and P2
Calculate m and c
Plot points from P1 to P2, based on x = 5 until x = 10 12
x = 6, y = 4
x = 7, y = 6
x = 8, y = 8
x = 9, y = 10
As you can see, the points are not connected, therefore the nearest pixel to the point is taken -8 x 5 10

16. Digital Deferential Algorithm
A scan conversion line algorithm based on calculating either △x or△y, where △x = x2 – x1 and △y = y2 - y1
m = △y/ △x
For line with positive slope greater than1
For line with positive slope smaller than1

17. Digital Deferential Algorithm
For example, draw the line between P1(5,2) and P2(10,5) using DDA
△x = 5, △y= 3, m = 3/5
m is positive and smaller than 1, therefore use
k starts at 0
y0 = 2
y0 will take the value of y for P1

18. Digital Deferential Algorithm
Since we are moving on y axis, we are going to move 5 steps in x axis (△x = 5)
y1 is the point of y at x1
The value of y1 is then rounded to the nearest pixel position.

19. Digital Deferential Algorithm
Move x, 1 step towards P2, plot y based on the rounded value of y1
Repeat the process until you reach P2
Plot P1 and P2 3 y 3 7 6 4 5 4 3 4 2 5 6 7 8 9 10 11 x

20. Digital Deferential Algorithm
Advantages:
Faster than using y = mx +c
Eliminates multiplication (mx) by using screen characteristics, i.e. incrementing pixel
Disadvantages:
Floating point arithmetic is slower than integer arithmetic
Rounding down take time
Long lines can drift from the true line due to floating point round-off errors

21. Bresenham’s Line Algorithm
Input the two line endpoints and store the left endpoint in (x0,y0)
Load (x0,y0) into the frame buffer; that is, plot the first point.
Calculate constants x, y, 2y, and 2y - 2x, and obtain the starting value for the decision parameter as
P0 = 2y - x
At each Xk along the line, starting at k = 0, perform the following test:
If Pk < 0, the next point to plot is (xk + 1, yk) and
Pk+1 = Pk + 2y
Otherwise, the next point to plot is (xk + 1, yk + 1) and
Pk+1 = Pk + 2y - 2x
Repeat step 4, x times.

22. Bresenham’s Line Algorithm
Given P1(5,2) and P2(10,5). Draw a line from P1 to P2 using Bresenham’s algorithm:
△x = 5, △y= 3
P0 = 2△y-△x
= 2(3) – 5
= 1
If Pk > 0, the next point to plot is (Xk + 1, Yk + 1)
Therefore, plot
(6,3)

23. Bresenham’s Line Algorithm
△x=5, △y=3
P0=2△y-△x=2(3)-5=1
P0+1=P0+2△y-2△x=1+2(3)-2(5) =-3
P1+1=P1+2△y=-3+2(3) = 3
P2+1=P2+6-10=-1
P3+1=P3+6=5
P0 > 1, therefore (Xk + 1, Yk + 1)
And use
Pk+1 = Pk + 2y - 2y
Plot (6,3)
Plot (7,3)
P0 < 1, therefore (xk + 1, yk)
Plot (8,4)
Plot (9,4)
And use
Pk+1 = Pk + 2y
Stop when the value of Pk is the same as the value of y for P2

24. Bresenham’s Line Algorithm
△x=5, △y=3
P0=2△y-△x=2(3)-5=1
P0+1=P0+2△y-2△x=1+2(3)-2(5) =-3
P1+1=P1+2△y=-3+2(3) = 3
P2+1=P2+6-10=-1
P3+1=P3+6=5 y 7
Plot (6,3) 6 5 4
Plot (7,3) 3 2
Plot (8,4) 5 6 7 8 9 10 11 x
Plot (9,4)

25. Bresenham’s Line Algorithm
Advantages
Accurate and efficient – incremental integer calculations
Can be adapted to draw circles and curves.
Generalization:
Increment (x or y) depending on slope (m) starting endpoints (left or right) special case (y=0 or x =0)

26. Optimisation
Speed can be increased even more by detecting cycles in the decision variable.
These cycles correspond to a repeated pattern of pixel choices.
The pattern is saved and if a cycle is detected it is repeated without recalculating. 11 12 9 10 13 14 15 16 6 7 8