This document discusses algorithms for drawing 2D graphics primitives like lines, triangles, and circles in computer graphics. It begins by introducing basic concepts like coordinate systems, pixels, and graphics APIs. It then covers algorithms for drawing lines, including the slope-intercept method, DDA algorithm, and Bresenham's line drawing algorithm, which uses only integer calculations for better performance. Finally, it briefly mentions extending these techniques to draw other shapes like circles and curves, as well as filling shapes.

1. CAP4730: Computational
Structures in Computer Graphics
2D Basics, Line Drawing, and
Clipping
Chapter 3 Hearn & Baker
Portions obtained from Leonard McMillan’s COMP136
Notes:
www.cs.unc.edu/~mcmillan/comp136/Lecture 6

2. Definitions
• CG API – Computer Graphics
Application Programming Interface
(OpenGL, DirectX)
• Graphics primitives – functions in the
API that describe picture components
• How could we describe an object?
– Typically focus on object shape
– Define an object’s shape with geometric
primitives
– Span of primitives are defined by the API
– What are some types?
• Lines, Triangles, Quadrics, Conic sections,
Curved surfaces

3. Two Dimensional Images
• Use Cartesian coordinates
• We label the two axes as
– X (horizontal)
– Y (vertical)
• Origin is in the lower left
• How big is the space?
– So what is the image we see
on a screen?
– We call this space the world
coordinate system
X Axis
+Y
Y
Axis
(0,0) +X

4. Partition the space into pixels
1. Define a set of points
(vertices) in 2D space.
2. Given a set of vertices,
draw lines between
consecutive vertices.
3. If you were writing
OpenGL yourself, let’s
talk about low level calls
4. What about 2D vs 3D?
(9,7)
(9,1)
+X
+Y
(2,7)
(2,1)
Screen Coordinates – references to frame buffer locations
Q: True or Flase: Screen Coordinates == World Coordinates

5. Pixels
• ?=glSetPixel(?)
• ?=glGetPixel(?)
• Scan line number – y
• Column number – x

6. Absolute and Relative Coordinate
Specifications
• Absolute coordinates –
location specified as a
relationship to the origin
• Relative coordinates –
location specified as a
relationship to other points
– Good for pen/plotters
– Publishing/layout
– Allows for a very object
oriented approach
• For this class we will
always use absolute
coordinates
+Y
(0,6)
(2,1)
(7,0)
(0,-6)

7. Specifying a World Coordinate System
in OpenGL
+X
+Y
gluOrtho2D (xmin, xmax, ymin, ymax)
What should our xmin, xmax, ymin,
ymax values be?
Equivalent to the size of the
framebuffer

8. From a geometry point of view, a pixel is a point. Q: Where is (2,1)?
What is a “pixel”
3
2
2
1
0 1 3 4
5
Q: What is a pixel? A square or a point?

9. But when we think about images, a pixel is a rectangle.
Q: Where is (2,1)? A. The center of a pixel
1
2
0
0 1 3 4
2

10. Basic OpenGL Point Structure
• In OpenGL, to specify a point:
– glVertex*();
• In OpenGL, some functions require both a dimensionality and a data
type
– glVertex2i(80,100), glVertex2f(58.9, 90.3)
– glVertex3i(20,20,-5), glVertex3f(-2.2,20.9,20)
• Must put within a ‘glBegin/glEnd’ pair
– glBegin(GL_POINTS);
– glVertex2i(50,50);
– glVertex2i(60,60);
– glVertex2i(60,50);
– glEnd();
• Let’s draw points in our assignment #1
• Next up? Lines

11. Draw a line from 0,0 to 4,2
2
How do we choose between 1,0 and 1,1? What would be a good heuristic?
2
1
0 1
0
3 4
(0,0)
(4,2)

12. What are lines composed of?
Write glBegin(GL_LINES)
2
2
1
0 1
0
3 4
(0,0)
(4,2)

13. What we are working with
V1: (6,8)
V0: (6,2)
V2: (13,8)
V3: (13,2)
• We are still dealing with vertices
• Draws a line between every pair of vertices
• glBegin(GL_LINES);
• glVertex2i(6,2);
• glVertex2i(6,8);
• glEnd();

14. Let’s draw a triangle
(0,2) (4,2)
(2,0)
2
2
1
0 1
0
3 4

15. Consider a translation
(-0.2,2) (3.8,2)
(1.8,0)
2
2
1
0 1
0
3 4

16. The Ideal Line
• Continuous
appearance
• Uniform
thickness and
brightness
• Pixels near the
ideal line are
“on”
• Speed
(2,2)
(17,8)
Discretization - converting a continuous signal
into discrete elements.
Scan Conversion - converting vertex/edges
information into pixel data for display
What do we want?

17. Slope-Intercept Method
• From algebra: y = mx + b
– m = slope b = y intercept Let’s write some code
class Point
{
public:
int x, y;
int r,g,b;
};
unsigned byte framebuffer[IMAGE_WIDTH*IMAGE_HEIGHT*3];
DrawLine (Point point1, Point point2)
{
}

18. Slope-Intercept Method
• From algebra: y = mx + b
– m = slope b = y intercept Let’s write some code
DrawLine (Point point1, Point point2){
m=(point2.y-point1.y) / (point2.x-point2.x);
b=point1.y + (-point1.x) * m;
for i=point1.x to point2.x
SetPixel(i , round(m*i+b)), pixel1.r, pixel1.g,
pixel1.b;}
SetPixel(int x, int y, int r, int g, int b){
framebuffer[(y * IMAGE_WIDTH+x) * 3 + 0]=r;
framebuffer[(y * IMAGE_WIDTH+x) * 3 + 1]=g;
framebuffer[(y * IMAGE_WIDTH+x) * 3 + 2]=b;}

19. Example 1:
Point1 V:(2,2) C:(255,102,0)
Point2 V:(17,8) C:(255,102,0)
What if colors were different?
(17,8)
(2,2)
(0,9)
(0,0) (18,0)

20. How do we change the
framebuffer?
(17,8)
(0,9) What’s the index into GLubyte framebuffer[]? Point is 9,5
(2,2)
(0,0) (18,0)

21. Example:
(7,9)
(12,0)
(0,9)
Example 2:
Point1 V:(7,9 ) C:( 0,255,0)
Point2 V:(12,0) C:(0,255,0)
(0,0) (18,0)
What are the problems with this method? Slope>1

22. Revised Slope Intercept
DrawLine (Point point1, Point point2){
m=(point2.y-point1.y) / (point2.x-point2.x);
b=point1.y + (-point1.x) * m;
if (m>1)
{
for i=point1.x to point2.x
SetPixel(i , round(i*m+b));}
}
else
{
for i=point1.y to point2.y
SetPixel(i , round(i-b/m));}
}
Which one should we use if m=1?
What is the cost per pixel?

23. Optimization (DDA Algorithm)
• Since we increment y by the same amount,
we can also inline rounding:
DrawLine (Point point1, Point point2){
m=(point2.y-point1.y) / (point2.x-point2.x);
j=point1.y + (-point1.x) * m + 0.5;
for i=point1.x to point2.x
SetPixel(i , (int)j+=m));}
• New cost: one floating point addition, one
integer addition, one cast.

24. Bresenham’s Line Drawing
• In general:
– Addition and Subtraction are faster than
Multiplication which is faster than Division
– Integer calculations are faster than Floating
point
• Made all math just integers (Section 3.5)
• How?

25. What you need to know about Bresenham LDA
1) Why we use it
2) Major idea of integer-izing a decision point
3) How this reduces things to just integer form.
(17,8)
(2,2)
(0,9)
(0,0) (18,0)

26. Recap
• DDA/Line Intercept Algorithm
– Slope Intercept y = mx + b
– Easy to implement
– Slow
• Bresenham
– No floating point math
– Fast
• Why do we spend so much time optimizing this?

27. Other Primitive Drawing Solutions
• What other shapes might we want to draw
quickly?
– Circles (and thus) Ovals
– Curves
• Fill?