how to generate output primitives in computer graphics

1. Computer Graphics
Output Primitives
Marwa K. U. Al-Rikaby
Babylon University
IT College

2. This Presentation has no copyrights.
Contents are taken from:

3. Outlines:
 Introduction.
 Point Drawing.
 Line Drawing Algorithms.
 Circle Generation Algorithms.
 Ellipse Generation Algorithms.
 Other Curves.
 Filled Area Primitives.
 Character Generation.

4. Introduction:
 Picture descriptions:
 Raster:
 Completely specified by the set of intensities for the pixels positions in the
display.
 Shapes and colors are described with pixel arrays.
 Scene displayed by loading pixels array into the frame buffer.
 Vector:
 Set of complex objects positioned at specified coordinates locations within
the scene.
 Shapes and colors are described with sets of basic geometric structures.
 Scene is displayed by scan converting the geometric-structure specifications
into pixel patterns.

5. Introduction:
 Output Primitives:
 Basic geometric structures used to describe scenes.
 Can be grouped into more complex structures.
 Each one is specified with input coordinate data and
other information about the way that object is to be
displayed.
 Examples: point, line and circle each one with
specified coordinates.
 Construct the vector picture.

6. Introduction:
In digital representation:
 Display screen is divided into scan lines and columns.
 Pixels positions are referenced according to scan line number
and column number (columns across scan lines).
 Scan lines start from 0 at screen bottom, and columns start
from 0 at the screen left side.
 Screen locations (or pixels) are referenced with integer values.
 The frame buffer stores the intensities temporarily.
 Video controller reads from the frame buffer and plots the
screen pixels.

7. Point Drawing
y
x

8. Point Drawing:
 Converting a single coordinate position furnished by
an application program into appropriate operation for
the output device in use.
 In raster system:
 Black-white: setting the bit value corresponding to a
specified screen position within the frame buffer to 1.
 RGB: loading the frame buffer with the color codes for the
intensities that are to be displayed at the screen pixel
positions.

9. Line Drawing Algorithms
y2
y1
x1 x2

10. Line Drawing:
 A straight line is specified by two endpoint
positions.
 Line drawing is done by:
 Calculating intermediate positions between the
endpoints.
 Directing the output device to fill in the calculated
positions as in the case of plotting single points.

11. Line Drawing:
 Plotted positions may be only approximations to the
actual line positions between endpoints.
 A computed position (10.48, 20.51) is converted to pixel
(10,21).
 This rounding causes the lines to be displayed with a stairstep
appearance.
 Stairsteps are noticeable in low resolution systems, it can be
improved by:
 Displaying lines on high resolution systems.
 Adjusting intensities along line path.

12. Line Drawing:
 The Cartesian intercept equation for a straight line:
y= m. x +b
Where m is the line slop and b is y intercept.
 For line segment starting in (x1,y1) and ending in (x2,y2), the
slop is:
m= (y1-y2)/(x1-x2)
b= y1- m.x1
 For any given x interval x, we can compute the corresponding y
interval y:
y= m .x
 Or x interval x from a given y:
x= y/m

13. Line Drawing:
Sampling along x axis
 On raster systems, lines are plotted
Y2
with pixels, and step sizes in the
horizontal and vertical directions are
constrained by pixel separations. y1
X1 x2
 Scan conversion process samples a
line at discrete positions and determine Sampling along y axis
the nearest pixel to the line at each Y2
sampled position.
y1
X1 x2

14. Line Drawing Algorithms:
 Digital Differential Analyzer (DDA):
Samples the line at unit intervals in one coordinate
and determine corresponding integer values nearest
the line path for the other coordinate.
 Bresenham’s Line Algorithm:
Scan converts lines using only incremental integer
calculations.

15. Line Drawing Algorithms:
 DDA
 A scan-conversion line algorithm based on calculating either
y or x using line points calculating equations.
 In each case, choosing the sample axis depends on the slop
value.
 The unit step for the selected axis is 1.
 The other axis is calculated depending on the first axis and
the slop m.
 The next slides will show the different cases of the slop:

16. Line Drawing Algorithms:
 DDA
 Case 1:the slop is Positive and less than 1
 Sample at unit x interval (x=1) and compute each successive
y value as :
y k+1= yk+ m
 K takes integer values starting from 1,at the first point, and
increasing by 1 on each step until reaching the final endpoint.
 The calculated y must be rounded to the nearest integer.
 Example:
Describe the line segment which starts at (3,3) and ends at
(23,7).

17. Lines Drawing Algorithms: DDA
24
23
22
21
m= (7-3)/(23-3) 20
=4/20 19
18
=0.2 17
16
15
x=1 14
13
y=0.2 12
11
10
9
8
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

18. Lines Drawing Algorithms: DDA
x y actual point pixel position 24
23
3 3 (3,3) (3,3) 22
4 3.2 (4,3.2) (4,3) 21
5 3.4 (5,3.4) (5,3) 20
6 3.6 (6,3.6) (6,4) 19
18
7 3.8 (7,3.8) (7,4) 17
8 4 (8,4) (8,4) 16
9 4.2 (9,4.2) (9,4) 15
10 4.4 (10,4.4) (10,4) 14
13
11 4.6 (11,4.6) (11,5) 12
12 4.8 (12,4.8) (12,5) 11
13 5 (13,5) (13,5) 10
9
14 5.2 (14,5.2) (14,5) 8
15 5.4 (15,5.4) (15,5) 7
16 5.6 (16,5.6) (16,6) 6
17 5.8 (17,5.8) (17,6) 5
4
.. .. .. .. 3
.. .. .. .. 2
22 6.8 (22,6.8) (22,7) 1
23 7 (23,7) (23,7) 0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

19. Line Drawing Algorithms:
 DDA
 Case 2:the slop is Positive and greater than 1
 Sample at unit y interval (y=1) and compute each successive
y value as :
x k+1= xk+ (1/m)
 K takes integer values starting from 1,at the first point, and
increasing by 1 on each step until reaching the final endpoint.
 The calculated x must be rounded to the nearest integer.
 Example:
Describe the line segment which starts at (3,3) and ends at
(7,23).

20. Line Drawing Algorithms: DDA
24
23
22
m= (23-3)/(7-3) 21
20
= 20/4 19
=5 18
17
16
15
x =1/m 14
=1/5 13
12
=0.2 11
10
9
y =1 8
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

21. Line Drawing Algorithms: DDA
y x actual point pixel position 24
23
3 3 (3,3) (3,3) 22
4 3.2 (3.2,4) (3,4) 21
5 3.4 (3.4,5) (3,5) 20
6 3.6 (3.6,6) (4,6) 19
18
7 3.8 (3.8,7) (4,7) 17
8 4 (4,8) (4,8) 16
9 4.2 (4.2,9) (4,9) 15
10 4.4 (4.4,10) (4,10) 14
13
11 4.6 (4.6,11) (5,11) 12
12 4.8 (4.8,12) (5,12) 11
13 5 (5,13) (5,13) 10
9
14 5.2 (5.2,14) (5,14) 8
15 5.4 (5.4,15) (5,15) 7
16 5.6 (5.6,16) (6,16) 6
17 5.8 (5.8,17) (6,17) 5
4
.. .. .. .. 3
.. .. .. .. 2
22 6.8 (6.8,22) (7,22) 1
23 7 (7,23) (7,23) 0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

22. Line Drawing Algorithms:
 DDA
 Case 3:the slop is negative and its absolute value is less than 1
 Follow the same way in case 1.
 Case 4:the slop is negative and its absolute value is greater than 1
 Follow the same way in case 2.
 In the previous 4 cases, we start from the left to the right. If
the state is reversed then:
If the absolute slop is less than 1, set x=-1 and
y k+1= yk - m
If the absolute slop is greater than 1, set y=-1 and
x k+1= xk - (1/m)

23. Line Drawing Algorithms:
 DDA properties
 A faster method for calculating pixel positions than the direct
use: y= m . x +b.
 Uses x or y to eliminate the multiplication in the above
equation.
 Rounding successive additions of the floating-point
increment can cause the calculated pixel positions to drift
away from the true line path for long line segment.
 Rounding and floating-point arithmetic are time-consuming
operations.

24. Line Drawing Algorithms:
 Bresenham’s Line Algorithm:
 Scan converts lines using only incremental integer
calculations.
 Can be adapted to display circles and other
curves.
 When sampling at unit x intervals, we need to
decide which of two possible pixel positions is
closer to the line path at each sample step by using
a decision parameter.

25. Line Drawing Algorithms:
 Bresenham’s Line Algorithm:
The decision parameter (pk) is an integer number
that is proportional to the difference between the
separations of the two pixel positions from the
actual line path.
Depending on the slop sign and value the decision
parameter determine which pixel coordinates would
be taken in the next step.

26. Line Drawing Algorithms:
 Bresenham’s Line Algorithm:
 Pk for line with positive slop less than 1:
 Sampling at unit x intervals.
 At pixel (xk,yk) we need to decide whether (xk+1,yk) or
(xk+1,yk+1) would be plotted in column xk+1.
 d1 and d2 are the vertical pixel separations from the
mathematical line path.
 y coordinate on the mathematical line at pixel xk+1 is
calculated as:
y= (x+1.m)+b
d1= y-yk
d2= yk+1-y
Pk =x(d1-d2)

27. Line Drawing Algorithms:
 Bresenham’s Line Algorithm:
Properties of Pk for line with positive slop less than 1:
 If the pixel at yk is closer to the line path than the
pixel at yk+l (that is, d1 < d2), then decision
parameter pk is negative and we plot the lower
pixel.
 On reverse, pk is positive and we plot the upper
pixel.
If pk is zero ,then choose either yk+l for all such
cases, or yk.

28. Line Drawing Algorithms:
 Bresenham’s Line Algorithm:
Properties of Pk for line with positive slop less than 1:
 Incremental integer calculations can be used to
obtain values of successive decision
parameters, because change occur on either x
or y coordinates:
pk+1=pk+2y-2x(yk+1-yk)
and p0=2y-x

29. Line Drawing Algorithms: Bresenham’s
24
23
Drawing the line between 22
21
the endpoints (3,3) and 20
(23,13) 19
18
17
m= (13-3)/(23-3)= 0.5 16
15
14
13
x= 20 12
y=10 11
10
9
8
P0= 2(10)-20=0 7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

30. Line Drawing Algorithms: Bresenham’s
24
K P x+1 y+1 23
22
0 0 4 4 21
1 -20 5 4 20
19
2 0 6 5 18
17
3 -20 7 5 16
4 0 8 6 15
14
5 -20 9 6 13
6 0 10 7 12
11
7 -20 11 7 10
9
8 0 12 8 8
9 -20 13 8 7
6
10 0 14 9 5
4
.. .. .. .. 3
.. .. .. .. 2
1
19 -20 23 13 0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

31. Line Drawing Algorithms:
 Bresenham’s line algorithm properties
 It is generalized to lines with arbitrary slope by considering
the symmetry between the various octants and quadrants of
the xy plane.
 The constants 2y and 2y - 2x are calculated once for
each line to be scan converted, so the arithmetic involves
only integer addition and subtraction of these two constants.

32. … Finally
24
23
Special cases can be 22
21
handled separately: 20
19
18
Horizontal y=0 17
Vertical x=0 16
15
Diagonal |x|=|y| 14
13
12
each can be loaded 11
10
directly into the frame 9
8
buffer without processing 7
them through the line- 6
5
plotting algorithms. 4
3
2
1
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

33. Circle Generating Algorithms

34. Circle Properties:
 A set of points that are all at the same distance (r)
from a center point (xc,yc).
 Expressed by Pythagorean equation in Cartesian
coordinates as:
(x – xc )2 + (y - yc )2 = r2
 Any given circle can be drawn on the origin(0,0)
and then moved to its actual position by:
 Each calculated x on the circumference moved to x+xc.
 Each calculated y on the circumference is moved to y+yc.

35. Circle Generating Algorithms:
 Standard Algorithm
 Polar Algorithm
 Symmetry Algorithm
 Mid-point Algorithm (xc, yc )
r (x, y)

36. Standard Algorithm:
 Uses the Pythagorean equation to find
the points along the circumference.
 Steps along x axis by unit steps from xc-r
to xc+r and calculate y for each step as:
y= yc± √r2-(xc-yc)2
 Itinvolves considerable computations at
each step.

37. Standard Algorithm:
 Spacing between plotted pixels is not uniform.
 Spacing can be adjusted by stepping on y instead of x
when the slop is greater than 1, but such approach
increasing computations.

38. Polar Algorithm:
 Uses polar coordinates r and Ө.
 Calculate points by fixed regular step size equations
as:
x= xc + r * cosӨ
y= yc + r * sinӨ
 Step size that is chosen for Ө depends on the
application.
 Larger separations can be connected by line segments
to approximate the circle path.

39. Polar Algorithm:
 In raster displays, step size is 1/r which plots pixel
positions that are approximately one unit apart.
x,y
r
Ө

40. Symmetry Algorithm:
 Reduces computations by considering the
symmetry of circles in the eight octants in the xy
plane.
 Generates all pixel positions
around a circle by calculating
only the points within the
sector from x = 0 to x = y.

41. Mid-point Algorithm:
 Samples at unit intervals and uses decision parameter to
determine the closest pixel position to the specified circle path at
each step.
 Along the circle section from x = 0 to x = y in the first quadrant, the
slope of the curve varies from 0 to -1. Therefore, we can take unit
steps in the positive x direction over this octant and use a decision
parameter to determine which of the two possible y positions is
closer to the circle path at each step. Positions in the other seven
octants are then obtained by symmetry.

42. Mid-point Algorithm:
To apply mid-point method, we define a circle function:
and determine the nearest y position from pk:

43. Mid-point Algorithm:
 If pk < 0, this midpoint is inside the circle and the pixel
on scan line yk is closer to the circle boundary.
Otherwise, the midposition is outside or on the circle
boundary, and we select the pixel on scan line yk – 1 .
 Successive parameters are calculated using
incremental calculations:

44. Mid-point Algorithm:
 Example:
For a circle radius=10 and center (12,12)
we calculate positions in the first octant only , beginning with x=0
and ending when x=y
The initial decision parameter value:
P0=1 - r = -9
Drawing on the origin (0,0) and the initial point (0,10)
2x0=0 2y0=20

45. Mid-point Algorithm:
24
23
K P (xk+1,yk+1) 2xk+1 2yk+1 22
21
0 -9 (1,10) 2 20 20
1 -6 (2,10) 4 20 19
18
2 -1 (3,10) 6 20 17
3 6 (4,9) 8 18 16
15
4 -3 (5,9) 10 18 14
13
5 8 (6,8) 12 16 12
6 5 (7,7) 14 14 11
10
9
8
The first octant is completed 7
and the remained seven 6
5
octants can be plotted 4
3
according to symmetry. 2
1
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

46. Mid-point Algorithm:
24
23
The circle after plotting all 22
21
octants and moving it to its 20
original center (12,12) 19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

47. Ellipse Generating Algorithms

48. Ellipse Properties:
ry
F2 rx F1
d2 d1
P=(x,y)

49. Ellipse Properties:
 An elongated circle.
 A set of points having an equal sum distances from two
fixed positions (foci F1,F2).
d1+d2=constant
 The major axis is the straight line segment extending from
one side of the ellipse to the other through the foci.
 The minor axis spans the shorter dimension of the
ellipse, bisecting the major axis at the halfway position
(ellipse center) between the two foci.
 The two foci and point coordinates on the circumference
are enough information to plot an ellipse with arbitrary
orientation.

50. Ellipse Properties:
 The equation of the ellipse can be written in terms of the
ellipse center coordinates and parameters rx, and ry, as:
 Using polar coordinates r and Ө. we can also describe the
ellipse in standard position with the parametric equations:
x= xc + rx * cosӨ
y= yc + ry * sinӨ
 Symmetry considerations can be used to reduce
computations, the four quadrants are similar ( not the eight
octants).

51. Midpoint Ellipse Algorithm:
 Similar to displaying a raster circle.
 Given parameters rx, ry, and (xc,yc), we determine points ( x
, y) for an ellipse in standard position centered on the
origin, and then we shift the points so the ellipse is centered
at ( xc,yc).
 To display the ellipse in nonstandard position, we could then
rotate the ellipse about its center coordinates to reorient the
major and minor axes.
 This method is applied throughout the first quadrant in two
parts:
 Stepping on x when the slop is less than 1.
 Stepping on y when the slop is greater than 1.

52. Midpoint Ellipse Algorithm:
 The decision parameter considered by:
It has the following properties:
The next position to be selected depends on the sign of this
parameter.
 The same way followed in circle we can use to choose the start
point and depending on the slop and Pk calculate the next nearest
pixel on the ellipse path.

53. Other Curves

54. Overview:
 Include: conics, trigonometric and exponential functions, probability
distributions, general polynomials, and spline functions.
 Positions along paths can be generated from curves explicit
equations y= f(x) .
 Incremental midpoints can be calculated from implicit function
f(x,y)=0.
 The same idea in the previous primitives, choosing the stepping
axis, can be used to obtain uniform spacing between points.
 Symmetries are also available in some curves, such as
 The normal probability distribution function is symmetric about at center position
(the mean).
 All points along one cycle of a sine curve can be generated from the points in a
90" interval.

55. Conic Sections:
 Described by:
 The parameters A,B,C,D,E,F values determine the
section type:

56. Conic Sections:
Parabolic path Hyperbola Path

57. Polynomial and Splines Curves:
 Polynomial is defined as:
where n is a nonnegative integer and the ak are
constants, with a≠0.
 We get:
 A straight line when n=1.
 A quadratic when n=2.
 A cubic polynomial when n=3.
 A quartic when n=4…

58. Polynomial and Splines Curves:
 Designing object shapes or motion paths is typically done by
specifying a few points to define the general curve contour, then
fitting the selected points with a polynomial.
 One way to accomplish the curve fitting is to construct a cubic
polynomial curve section between each pair of specified points.
 Each curve section is then described in parametric form as:
where parameter u varies over the interval 0 to 1.

59. Polynomial and Splines Curves:
 Values for the coefficients of u in the parametric
equations are determined from boundary conditions on
the curve sections:
 Two adjacent curve sections have the same coordinate position at the
boundary.
 Match the two curve slopes at the boundary so that we obtain one
continuous, smooth individual cubic curve.
 Splines are continuous curves formed with polynomial
pieces.

60. Filled Area Primitives

61. Overview:
 Two basic ideas:
 Determine the overlap intervals for scan lines that
cross the area. Used to fill
polygons, circles, ellipses, and other simple
curves.
 Start from a given interior position and paint
outward from this point until we encounter the
specified boundary conditions. Used with more
complex boundaries and in interactive painting
systems.

62. Filled Area Primitives:
 Scan-Line Polygon Fill Algorithm:
 Locates the intersection points along the scan line with the
polygon edges.
 Sorts the intersections from left to right.
 Set the corresponding frame buffer positions between each
intersection with a specified fill color.
 Special handling:
 Scan line passing through a vertex intersects two edges at that
position: add two points to the list of intersections for the scan line.
 Care should be taken to determine which side of intersection is
belonging to the polygon.

63. Filled Area Primitives:
Such situations need
more processing to
determine:
1. Whether there is a
need to duplicate some
intersections.
2. If the points are inside
or outside the shape.
We should trace around
the shape boundaries to
make a decision.

64. Filled Area Primitives:

65. Filled Area Primitives:
Scan line

66. Filled Area Primitives:
Scan line

67. Filled Area Primitives:
Scan line

68. Inside-Outside Tests:
 Odd-even (odd parity or even odd) rule:
 Drawing a line from any position P to a distant
point outside the coordinate extents of the object.
 Counting the number of edge crossings along the
line.
If the number of polygon edges crossed by this
line is odd, then P is an interior point, otherwise it
is an exterior point.
The line chosen must not intersect the polygon
vertices.

69. Inside-Outside Tests:
 Non-zero winding number rule:
 Counts the number of times the polygon edges wind around
a particular point in the counterclockwise direction, which is
called “winding number”.
 Interior points of a two-dimensional object are defined to be
those that have a nonzero value for the winding number.
 Counting method:
 Start winding number with 0.
 Imagine a line passing across the object and not intersect vertices.
 Add 1 to winding number when the line is crossed by an edge
from right to left.
 Subtract 1 from winding number when the line is crossed by an
edge from left to right.
 To determine the direction use either cross product or dot product.

70. Inside-Outside Tests:

71. Scan-Line Fill of Curved Boundary Areas :
In circles and ellipses, only find the two
scan-line Intersections on opposite sides
of the curve.
In other curves, such as ellipse arc, the
interior region is bounded by the ellipse
section and a straight-line segment that
closes the curve by joining the beginning
and ending positions of the arc.
Symmetry can be exploited to reduce
calculations.

72. Boundary-Fill Algorithm:
 Accepts as input the coordinates of an interior point
(x, y), a fill color, and a boundary color.
 Starting from (x,y), the procedure tests neighboring
positions to determine whether they are of the
boundary color.
 If not, they are painted with the fill color, and their
neighbors are tested.
 This process continues until all pixels up to the
boundary color for the area have been tested.

73. Boundary-Fill Algorithm:

74. Boundary-Fill Algorithm:

75. Flood-Fill Algorithm:
 Replacing a specified interior color instead of searching
for a boundary color value.
 Start from a specified interior point (x, y) and reassign all
pixel values that are currently set to a given interior color
with the desired fill color.
 If the area we want to paint has more than one interior
color, we can first reassign pixel values so that all interior
points have the same color.
 Using either a 4-connected or 8-connected approach, we
then step through pixel positions until all interior points
have been repainted.

76. Character Generation

77. Typefaces (Fonts):
 Serif group:
 Has small lines or accents at the ends of the main
character strokes.
 More readable.
 Sans serif group:
 Does not have accents.
 The individual characters easier to recognize.
 More legible, and hence, good for labeling and
short headings.

78. Character Generation Methods:
 Bitmapped Font:
 Uses rectangular grid patterns.
 The simplest to define and display, the character grid
only needs to be mapped to a frame-buffer position.
 Need more space, each variation (size and format)
must be stored in a font cache.
 Outline Font:
 Uses straight-line and curve sections.
 Require less storage since each variation does not
require a distinct font cache.
 Need more time since to the scan conversion process.

79. Character Generation Methods:
Bitmapped Font Outline Font