1. Graphics Output Primitives Hearn & Baker Chapter 3
COMPUTER GRAPHICS
Graphics Output Primitives
Hearn & Baker Chapter 3
Some slides are taken from Robert Thomson’s notes.

2. OVERVIEW Coordinate reference frames Line drawing algorithms
Curve drawing algorithms
Fill-area primitives
Pixel-array primitives
Character primitives
Picture Partitioning

3. Develop mathematical operations among them Define basic primitives:
Basic Elements
Three basic elements
Scalars
Vectors
Points
Develop mathematical operations among them
Define basic primitives:
Line segments

4. Points are associated with locations in space
Basic Elements
Points are associated with locations in space
Vectors have magnitude and direction
represent displacements between points, or directions
Points, vectors, and operators that combine them are the common tools for solving many geometric problems that arise in
Geometric Modeling,
Computer Graphics,
Animation,
Visualization, and
Computational Geometry.

5. 3.1 COORDINATE REFERENCE FRAMES
To describe a picture, we first decide upon
A convenient Cartesian coordinate system, called the world-coordinate reference frame, which could be either 2D or 3D.
We then describe the objects in our picture by giving their geometric specifications in terms of positions in world coordinates.
e.g., we define a straight-line segment with two endpoint positions, and a polygon is specified with a set of positions for its vertices.
These coordinate positions are stored in the scene description along with other info about the objects, such as their color and their coordinate extents
coordinate extents are the minimum and maximum x, y, and z values for each object.
A set of coordinate extents is also described as a bounding box for an object.
For a 2D figure, the co­ordinate extents are sometimes called its bounding rectangle.

6. 3.1 COORDINATE REFERENCE FRAMES
Objects are then displayed by passing the scene description to the viewing routines
which identify visible surfaces and map the objects to the frame buffer positions and then on the video monitor.
The scan-conversion algorithm stores info about the scene, such as color values, at the appropriate locations in the frame buffer, and then the scene is displayed on the output device.
locations on a video monitor
are referenced in integer screen coordinates, which correspond to the integer pixel positions in the frame buffer.

7. 3.1 COORDINATE REFERENCE FRAMES
Scan-line algorithms for the graphics primitives use the coordinate descriptions to determine the locations of pixels
E.g., given the endpoint coordinates for a line segment, a display algorithm must calculate the positions for those pixels that lie along the line path between the endpoints.
Since a pixel position occupies a finite area of the screen
the finite size of a pixel must be taken into account by the implementation algorithms.
for the present, we assume that each integer screen position references the centre of a pixel area.

8. 3.1 COORDINATE REFERENCE FRAMES
Once pixel positions have been identified the color values must be stored in the frame buffer
Assume we have available a low-level procedure of the form
setPixel (x, y);
stores the current color setting into the frame buffer at integer position (x, y), relative to the position of the screen-coordinate origin
getPixel (x, y, color);
retrieves the current frame-buffer setting for a pixel location;
parameter color receives an integer value corresponding to the combined RGB bit codes stored for the specified pixel at position (x,y).
additional screen-coordinate information is needed for 3D scenes. For a two-dimensional scene, all depth values are 0.

9. 3.1 Absolute and Relative Coordinate Specifications
So far, the coordinate references discussed are given as absolute coordinate values
values are actual positions wrt origin of current coordinate system
some graphics packages also allow positions to be specified using relative coordinates
as an offset from the last position that was referenced (called the current position)

10. 3-5 LINE-DRAWING ALGORITHMS
A straight-line segment in a scene is defined by the coordinate positions for the endpoints of the segment.
To display the line on a raster monitor, the graphics system must
first project the endpoints to integer screen coordinates and
determine 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.
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.

11. 3-5 LINE-DRAWING ALGORITHMS
On raster systems, lines are plotted with pixels, and step sizes in the horizontal and vertical directions are constrained by pixel separations.
That is, we must "sample" a line at discrete positions and determine the nearest pixel to the line at sampled position.
Sampling is measuring the values of the function at equal intervals
Idea: A line is sampled at unit intervals in one coordinate and the corresponding integer values nearest the line path are determined for the other coordinate.

12. Towards the Ideal Line We can only do a discrete approximation
Illuminate pixels as close to the true path as possible, consider bi-level display only
Pixels are either lit or not lit
In the raster line alg.,
we sample at unit intervals and
determine the closest pixel position to the specified line path at each step

13. What is an ideal line Must appear straight and continuous
Only possible with axis-aligned and 45o lines
Must interpolate both defining end points
Must have uniform density and intensity
Consistent within a line and over all lines
What about anti-aliasing ?
Aliasing is the jagged edges on curves and diagonal lines in a bitmap image.
Anti-aliasing is the process of smoothing out those jaggies.
Graphics software programs have options for anti-aliasing text and graphics.
Enlarging a bitmap image accentuates the effect of aliasing.
Must be efficient, drawn quickly
Lots of them are required

14. Simple Line
The Cartesian slope-intercept equation for a straight line is :
y = mx + b
with m as the slope of the line and b as the y intercept.
Simple approach:
increment x, solve for y

15. Line-Drawing Algorithms: DDA Bresenham’s Midpoint Algorithm

16. Algorithms for displaying lines are based on the Cartesian slope-intercept equation
y = m.x + b
where m and b can be calculated from the line endpoints: m = (y1-y0) / (x1-x0)
b = y0 - m. x0
For any x interval x along a line the corresponding y interval y = m.x y1 y0 x0 x1

17. Simple Line Based on slope-intercept algorithm from algebra:
y = mx + b
Simple approach:
increment x, solve for y
Floating point arithmetic required

18. Does it Work? It works for lines with a slope of 1 or less,
but doesn’t work well for lines with slope greater than 1 – lines become more discontinuous in appearance and we must add more than 1 pixel per column to make it work.
Solution? - use symmetry.

19. Modify algorithm per octant
OR, increment along x-axis if dy<dx else increment along y-axis

20. DDA Algorithm
The digital differential analyser (DDA) is a scan-conversion line algorithm based on using x or y.
A line is sampled at unit intervals in one coordinate and the corresponding integer values nearest the line path are determined for the other coordinate.

21. Line with positive slope
If m<=1,
Sample at unit x intervals (dx=1)
Compute successive y values as
yk+1=yk+m <=k<=xend-x0
Increment k by 1 for each step
Round y to nearest integer value.
If m>1,
Sample at unit y intervals (dy=1)
Compute successive x values as
xk+1=xk+1/m <=k<=yend-y0
Round x to nearest integer value.

22. inline int round (const float a) { return int (a + 0.5); }
void lineDDA (int x0, int y0, int xEnd, int yEnd) {
int dx = xEnd - x0, dy = yEnd - y0, steps, k;
float xIncrement, yIncrement, x = x0, y = y0;
if (fabs (dx) > fabs (dy))
steps = fabs (dx);
else
steps = fabs (dy);
xIncrement = float (dx) / float (steps);
yIncrement = float (dy) / float (steps);
setPixel (round (x), round (y));
for (k = 0; k < steps; k++) {
x += xIncrement;
y += yIncrement; }

23. DDA algorithm Need a lot of floating point arithmetic.
2 ‘round’s and 2 adds per pixel.
Is there a simpler way ?
Can we use only integer arithmetic ?
Easier to implement in hardware.

24. Bresenham's line algorithm
Accurate and efficient
Uses only incremental integer calculations
The method is described for a line segment with a positive slope less than one
The method generalizes to line segments of other slopes by considering the symmetry between the various octants and quadrants of the xy plane

25. Bresenham's line algorithm
Decide what is the next pixel position
(11,11) or (11,12)
Specified
line path 13 12 11 10 10 11 12 13

26. Illustrating Bresenham’s Approach
For the pixel position xk+1=xk+1, which one we should choose:
(xk+1,yk) or (xk+1, yk+1)
yk+3
yk+2
y=mx+b
yk+1 yk xk
xk+1
xk+2
xk+3

27. Bresenham’s Approach y=m(xk + 1)+b dlower=y-yk =m(xk + 1)+b-yk
dupper
dlower
y=m(xk + 1)+b
dlower=y-yk
=m(xk + 1)+b-yk
dupper=(yk+1)-y
= yk+1 -m(xk + 1)-b
dlower- dupper= 2m(xk + 1)+2yk+2b-1
Rearrange it to have integer calculations:
m=Δy/Δx
Decision parameter: pk= Δx(dlower- dupper)=2Δy.xk - 2Δx. yk + c

28. The Decision Parameter
Decision parameter: pk= Δx(dlower- dupper)=2Δy.xk - 2Δx. yk + c
pk has the same sign with dlower- dupper since Δx>0.
c is constant and has the value c= 2Δy + Δx(2b-1)
c is independent of the pixel positions and is eliminated from decision parameter pk.
If dlower< dupper then pk is negative.
Plot the lower pixel (East)
Otherwise
Plot the upper pixel (North East)

29. Succesive decision parameter
At step k+1
pk+1= 2Δy.xk+1 - 2Δx. yk+1 + c
Subtracting two subsequent decision parameters yields:
pk+1-pk= 2Δy.(xk+1-xk) - 2Δx. (yk+1-yk)
xk+1=xk+1 so
pk+1= pk + 2Δy - 2Δx. (yk+1-yk)
yk+1-yk is either 0 or 1 depending on the sign of pk
First parameter p0
p0=2 Δy - Δx

30. Bresenham's Line-Drawing Algorithm for I m I < 1
1. Input the twoline endpoints and store the left endpoint in (x0 ,y0).
2. Load (x0 ,y0) into the frame buffer; that is, plot the first point.
3. Calculate constants Δx, Δy, 2Δy, and 2Δy - 2Δx, and obtain the starting value for the decision parameter as
p0=2 Δy - Δx
4. 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 + 2Δy
Otherwise, the next point to plot is (xk+1, yk+1) and
pk+1=pk + 2Δy - 2Δx
5. Repeat step 4 Δx -1 times.

31. Trivial Situations: Do not need Bresenham

32. Example Draw the line with endpoints (20,10) and (30, 18).
Δx=30-20=10, Δy=18-10=8,
p0 = 2Δy – Δx=16-10=6
2Δy=16, and 2Δy - 2Δx=-4
Plot the initial position at (20,10), then

34. Example
Line end points:
Deltas:
initially

35. Graph 13

36. Continue the process... 13

37. Graph 13

38. Graph 13

39. Graph 13

40. /* Bresenham line-drawing procedure for |m| < 1.0. */
void lineBres (int x0, int y0, int xEnd, int yEnd) {
int dx = fabs (xEnd - x0), dy = fabs(yEnd - y0);
int x, y, p = 2 * dy - dx;
int twoDy = 2 * dy, twoDyMinusDx = 2 * (dy - dx);
/* Determine which endpoint to use as start position. */
if (x0 > xEnd) {
x = xEnd; y = yEnd; xEnd = x0; }
else {
x = x0; y = y0;
setPixel (x, y);
while (x < xEnd) {
x++;
if (p < 0)
p += twoDy;
y++;
p += twoDyMinusDx;

41. Line-drawing algorithm should work in every octant, and special cases

42. Simulating the Bresenham algorithm in drawing 8 radii on a circle of radius 20
Horizontal, vertical and 45 radii handled as special cases

43. Scan Converting Circles
Explicit: y = f(x)
We could draw a quarter circle by incrementing x from 0 to R in unit steps and solving for +y for each step.
Method needs lots of computation, and gives non-uniform pixel spacing

44. Scan Converting Circles
Parametric:
Draw quarter circle by stepping through the angle from 0 to 90
avoids large gaps but still unsatisfactory
How to set angular increment
Computationally expensive trigonometric calculations

45. Scan Converting Circles
Implicit: f(x,y) = x2+y2-R2
If f(x,y) = 0 then it is on the circle.
f(x,y) > 0 then it is outside the circle.
f(x,y) < 0 then it is inside the circle.
Try to adapt the Bresenham midpoint approach
Again, exploit symmetries

47. Generalising the Bresenham midpoint approach
Set up decision parameters for finding the closest pixel to the cırcumference at each sampling step
Avoid square root calculations by considering the squares of the pixel separation distances
Use direct comparison without squaring.
Adapt the midpoint test idea: test the halfway position between pixels to determine if this midpoint is inside or outside the curve
This gives the midpoint algorithm for circles
Can be adapted to other curves: conic sections

48. Eight-way Symmetry - only one octant’s calculation needed

49. The 2nd Octant is a good arc to draw
It is a well-defined function in this domain
single-valued
no vertical tangents: |slope|  1
Lends itself to the midpoint approach
only need consider E or SE
Implicit formulation F(x,y) = x 2 + y 2 – r 2
For (x,y) on the circle, F(x,y) = 0
F(x,y) > 0  (x,y) Outside
F(x,y) < 0  (x,y) Inside

50. Choose E or SE Decision variable d is x 2 + y 2 – r 2
Then d = F(M)  0  SE
Or d = F(M) <  E

51. F (M)  0  SE
current pixel
ideal curve

52. F (M) < 0  E
ideal curve

53. Use the implicit form of the circle equation
Decision Variable p
As in the Bresenham line algorithm we use a decision variable to direct the selection of pixels.
Use the implicit form of the circle equation
p = F (M ) = x 2 + y 2 – r 2

54. Midpoint coordinates are

55. Assuming we have just plotted point at (xk,yk) we determine whether move E or SE by evaluating the circle function at the midpoint between the two candidate pixel positions
pk is the decision variable
if pk <0 the midpoint is inside the circle
Thus the pixel above the midpoint is closer to the ideal circle, and we select pixel on scan line yk. i.e. Go E

56. If pk >0 the midpoint is outside the circle.
Thus the pixel below the midpoint is closer to the ideal circle, and we select pixel on scan line yk-1. i.e. Go SE
Calculate successive decision parameter values p by incremental calculations.

57. recursive definition for successive decision parameter values p
Where yk+1 = yk if p<0 (move E)
yk+1 = yk-1 if p>0 (move SE)
yk+1 and xk+1 can also be defined recursively

58. Initialisation
x0 = 0, y0 = r
Initial decision variable found by evaluating circle function at first midpoint test position
For integer radius r p0 can be rounded to p0 =1-r
since all increments are integer.

60. Midpoint Circle Algorithm (cont.)

61. Example
r=10

62. Another method: Approximate it using a polyline.

63. Fill area : an area that is filled with solid colour or pattern

65. Identifying a concave polygon
has at least one interior angle >180 degrees
extensions of some edges will intersect other edges
One test:
Express each polygon edge as a vector, with a consistent orientation.
Can then calculate cross-products of adjacent edges

66. Identifying a concave polygon
When polygon edges are oriented with an anti-clockwise sense
cross product at convex vertex has positive sign
concave vertex gives negative sign

67. Normals n = u x v (vector cross product)
Every plane has a vector n normal (perpendicular, orthogonal) to it
n = u x v (vector cross product) v u P

68. Vector method for splitting concave polygons
E1=(1,0,0) E2=(1,1,0)
E3=(1,-1,0) E4=(0,3,0)
E5=(-3,0,0) E6=(0,-3,0)
All z components have 0 value.
Cross product of two vectors EjxEk is perpendicular to them with z component
EjxEky-EkxEjy E1 E2 E3 E4 E5 E6 1 2 3

69. Example continued E1xE2= (0,0,1) E2xE3= (0,0,-2) E3xE4= … E4xE5= …
Since E2xE3 has negative sign, split the polygon along the line of vector E2 E1 E2 E3 E4 E5 E6 1 2 3

70. Rotational method
Rotate the polygon so that each vertex in turn is at coordinate origin.
If following vertex is below the x axis, polygon is concave.
Split the polygon by x axis.

71. E5 E6 E4 E2 E3 E1

72. E5 E6 E1 E2 E4 E3

74. Inside-Outside? nonzero winding-number rule
A winding number is an attribute of a point with respect to a polygon that tells us how many times the polygon encloses (or wraps around) the point. It is an integer, greater than or equal to 0. Regions of winding number 0 (unenclosed) are obviously outside the polygon, and regions of winding number 1 (simply enclosed) are obviously inside the polygon.
Initially 0
+1: edge crossing the line from right to left
-1: left to right

75. Winding Number Count clockwise encirclements of point
Alternate definition of inside: inside if winding number  0
winding number = 1
winding number = 2

76. Polygon tables
store descriptions of polygon geometry and topology, and surface parameters: colour, transparency, light-reflection
organise in 2 groups
geometric data
attribute data

77. Polygon Tables:
Geometric data
Data can be used for consistency checking
Additional geometric data stored: slopes, bounding boxes

78. Shared Edges
Vertex lists will draw filled polygons correctly but if we draw the polygon by its edges, shared edges are drawn twice
Can store mesh by edge list

80. Inward and Outward Facing Polygons
The order {v1, v6, v7} and {v6, v7, v1} are equivalent in that the same polygon will be rendered by OpenGL but the order {v1, v7, v6} is different
The first two describe outwardly
facing polygons
Use the right-hand rule =
counter-clockwise encirclement
of outward-pointing normal
OpenGL can treat inward and
outward facing polygons differently