Output Primitives in Computer Graphics and Multimedia

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COMPUTER
GRAPHICS
CHAPTER 3
2D GRAPHICS ALGORITHMS

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2D Graphics Algorithms
 Output Primitives
 Line Drawing Algorithms
 DDA Algorithm
 Midpoint Algorithm
 Bersenhem’s Algorithm
 Circle Drawing Algorithms
 Midpoint Circle Algorithm
 Antialising
 Fill-Area Algorithms

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Output Primitives
 The basic objects out of which a
graphics display is created are
called.
 Describes the geometry of objects
and – typically referred to as
geometric primitives.
 Examples: point, line, text, filled
region, images, quadric surfaces,
spline curves
 Each of the output primitives has its
own set of attributes.

Output Primitives
 Points
 Attributes: Size, Color.
glPointSize(p);
glBegin(GL_POINTS);
glVertex2d(x1, y1);
glVertex2d(x2, y2);
glVertex2d(x3, y3);
glEnd()

Output Primitives
 Lines
 Attributes: Color, Thickness, Type
glLineWidth(p);
glBegin(GL_LINES);
glVertex2d(x1, y1);
glVertex2d(x2, y2);
glVertex2d(x3, y3);
glVertex2d(x4, y4);
glEnd()

Output Primitives
 Polylines (open)
 A set of line segments joined end to end.
glLineWidth(p);
glBegin(GL_LINE_STRIP);
glVertex2d(x1, y1);
glVertex2d(x2, y2);
glVertex2d(x3, y3);
glVertex2d(x4, y4);
glEnd()

Output Primitives
 Polylines (closed)
 A polyline with the last point connected to
the first point .
Note: A closed polyline cannot be filled.
glBegin(GL_LINE_LOOP);
glVertex2d(x1, y1);
glVertex2d(x2, y2);
glVertex2d(x3, y3);
glVertex2d(x4, y4);
glEnd()

Output Primitives
 Polygons
 A set of line segments joined end to end.
 Attributes: Fill color, Thickness, Fill
pattern
Note: Polygons can be filled.
glBegin(GL_POLYGON);
glVertex2d(x1, y1);
glVertex2d(x2, y2);
glVertex2d(x3, y3);
glVertex2d(x4, y4);
glEnd()

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Output Primitives
 Text
 Attributes: Font, Color, Size,
Spacing, Orientation.
 Font:
 Type (Helvetica, Times, Courier etc.)
 Size (10 pt, 14 pt etc.)
 Style (Bold, Italic, Underlined)

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Output Primitives
 Images
 Attributes: Image Size, Image
Type, Color Depth.
 Image Type:
 Binary (only two levels)
 Monochrome
 Color.
 Color Depth:
Number of bits used to represent
color.

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Output Primitives
Output Primitive Attributes
Point Size
Color
Line Thickness (1pt, 2pt …)
Type (Dashed, Dotted, Solid)
Color
Text Font (Arial, Courier, Times Roman…)
Size (12pt, 16pt ..)
Spacing
Orientation (Slant angle)
Style (Bold, Underlined, Double lined)
Color
Filled Region Fill Pattern
Fill Type (Solid Fill, Gradient Fill)
Fill Color
Images Color Depth (Number of bits/pixel)

Line Drawing
• Line drawing is fundamental to computer graphics.
• We must have fast and efficient line drawing functions.
Rasterization Problem: Given only the two end points, how
to compute the intermediate pixels, so that the set of pixels
closely approximate the ideal line.

Line Drawing - Analytical Method
y y
x x
y x
b a
m
b a
c a ma



 
y
x
y=mx+c
ax bx
A(ax,ay)
B(bx,by)

• Directly based on the analytical equation of a line.
• Involves floating point multiplication and addition
• Requires round-off function.
double m = (double)(by-ay)/(bx-ax);
double c = ay - m*ax;
double y;
int iy;
for (int x=ax ; x <=bx ; x++) {
y = m*x + c;
iy = round(y);
setPixel (x, iy);
}
Line Drawing - Analytical Method

Compute one point based on the previous point:
(x0, y0)…….…………..(xk, yk) (xk+1, yk+1) …….
I have got a pixel on the line (Current Pixel).
How do I get the next pixel on the line?
Next pixel on next column
(when slope is small)
Next pixel on next row
(when slope is large)
Incremental Algorithms

Current
Pixel
(xk, yk)
To find (xk+1, yk+!):
xk+1 = xk+1
yk+1 = ?
(5,2)
(6,1)
(6,2)
(6,3)
• Assumes that the next pixel to be set is on the next column of
pixels (Incrementing the value of x !)
• Not valid if slope of the line is large.
Incrementing along x

Digital Differential Analyzer Algorithm is an incremental
algorithm.
Assumption: Slope is less than 1 (Increment along x).
Current Pixel = (xk, yk).
(xk, yk) lies on the given line. yk = m.xk + c
Next pixel is on next column. xk+1 = xk+1
Next point (xk+1, yk+1) on the line yk+1 = m.xk+1 + c
= m (xk+1) +c
= yk + m
Given a point (xk, yk) on a line, the next point is given by
xk+1 = xk+1
yk+1 = yk + m
Line Drawing - DDA

• Does not involve any floating point multiplication.
• Involves floating point addition.
• Requires round-off function
Line Drawing - DDA
double m = (double) (by-ay)/(bx-ax);
double y = ay;
int iy;
for (int x=ax ; x <=bx ; x++) {
iy = round(y);
setPixel (x, iy);
y+ = m;
}

xk+1 = xk+1
yk+1 = Either yk or yk+1
Midpoint algorithm is an incremental algorithm
Midpoint Algorithm
Assumption:
Slope < 1
Current
Pixel

Candidate Pixels
Current Pixel
( xk, yk)
Midpoint
Line
Coordinates of Midpoint = ( xk+1, yk+(1/2) )
( xk+1, yk)
( xk+1, yk+1)
Midpoint Algorithm - Notations

Midpoint Below Line Midpoint Above Line
Midpoint Algorithm:
Choice of the next pixel
•If the midpoint is below the line, then the next pixel is (xk+1, yk+1).
•If the midpoint is above the line, then the next pixel is (xk+1, yk).

A(ax,ay)
B(bx,by)
Equation of a line revisited.
Let w = bx  ax, and h = by  ay.
Then, h (x  ax)  w (y  ay) = 0.
(h, w , ax , ay are all integers).
In other words, every point (x, y) on the line
satisfies the equation F(x, y) =0, where
F(x, y) = h (x  ax)  w (y  ay).
Equation of the line:
y x
y y x x
y a x a
b a b a
 

 

Midpoint Algorithm:
Regions below and above the line.
F (x,y) > 0
(for any point below line)
F(x,y) < 0
(for any point above line)
F(x,y) = 0

F(MP) > 0
0
)
,
( 
y
x
f
Midpoint below line
F(MP) < 0
Midpoint above line
Midpoint Algorithm
Decision Criteria

Midpoint Algorithm
Decision Criteria
F(MP) = F(xk+1, yk+ ½) = Fk (Notation)
If Fk < 0 : The midpoint is above the line. So the next
pixel is (xk+1, yk).
If Fk  0 : The midpoint is below or on the line. So the
next pixel is (xk+1, yk+1).
Decision Parameter

Midpoint Algorithm – Story so far.
Midpoint Below Line
Next pixel = (xk+1, yk+1)
Fk > 0
yk+1 = yk+1
Midpoint Above Line
Next pixel = (xk+1, yk)
Fk < 0
yk+1 = yk

Midpoint Algorithm
Update Equation
Fk = F(xk+1, yk+ ½) = h (xk+1  ax)  w (yk+½  ay)
But, Fk+1 = Fk + h  w (yk+1 yk). (Refer notes)
So,
Fk< 0 : yk+1 = yk. Hence, Fk+1 = Fk + h .
Fk  0 : yk+1 = yk+1. Hence, Fk+1 = Fk + h  w.
F0 = h  w/2.
Update Equation

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Midpoint Algorithm
int h = by-ay;
int w = bx-ax;
float F=h-w/2;
int x=ax, y=ay;
for (x=ax; x<=bx; x++){
setPixel(x, y);
if(F < 0)
F+ = h;
else{
F+ = h-w;
y++;
}
}

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Bresenham’s Algorithm
int h = by-ay;
int w = bx-ax;
int F=2*h-w;
int x=ax, y=ay;
for (x=ax; x<=bx; x++){
setPixel(x, y);
if(F < 0)
F+ = 2*h;
else{
F+ = 2*(h-w);
y++;
}
}

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 To determine the closest pixel position to
the specified circle path at each step.
 For given radius r and screen center
position (xc, yc), calculate pixel positions
around a circle path centered at the
coodinate origin (0,0).
 Then, move each calculated position (x, y)
to its proper screen position by adding xc to
x and yc to y.
 Along the circle section from x=0 to x=y in
the first quadrant, the gradient varies from
0 to -1.
Midpoint Circle Drawing Algorithm

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 8 segments of octants for a circle:

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 Circle function: fcircle (x,y) = x2 + y2 –r2
> 0, (x,y) outside the circle
< 0, (x,y) inside the circle
= 0, (x,y) is on the circle
boundary
{
fcircle (x,y) =

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yk
yk-1
midpoint
Next pixel = (xk+1, yk)
Fk < 0
yk+1 = yk
yk
yk-1
midpoint
Next pixel = (xk+1, yk-1)
Fk >= 0
yk+1 = yk - 1

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We know xk+1 = xk+1,
Fk = F(xk+1, yk- ½)
Fk = (xk +1)2 + (yk - ½)2 - r2 -------- (1)
Fk+1 = F(xk+1, yk- ½)
Fk+1 = (xk +2)2 + (yk+1 - ½)2 - r2 -------- (2)
(2) – (1)
Fk+1 = Fk + 2(xk+1) + (y2
k+1 – y2
k) - (yk+1 – yk) + 1
If Fk < 0, Fk+1 = Fk + 2xk+1+1
If Fk >= 0, Fk+1 = Fk + 2xk+1+1 – 2yk+1

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For the initial point, (x0 , y0) = (0, r)
f0 = fcircle (1, r-½ )
= 1 + (r-½ )2 – r2
= 5 – r
4
≈ 1 – r

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Example:
Given a circle radius = 10, determine the circle octant
in the first quadrant from x=0 to x=y.
Solution:
f0 = 5 – r
4
= 5 – 10
4
= -8.75
≈ –9

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Initial (x0, y0) = (1,10)
Decision parameters are: 2x0 = 2, 2y0 = 20
k Fk x y 2xk+1 2yk+1
0 -9 1 10 2 20
1 -9+2+1=-6 2 10 4 20
2 -6+4+1=-1 3 10 6 20
3 -1+6+1=6 4 9 8 18
4 6+8+1-18=-3 5 9 10 18
5 -3+10+1=8 6 8 12 16
6 8+12+1-16=5 7 7 14 14

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void circleMidpoint (int xCenter, int yCenter, int radius)
{
int x = 0;
Int y = radius;
int f = 1 – radius;
circlePlotPoints(xCenter, yCenter, x, y);
while (x < y) {
x++;
if (f < 0)
f += 2*x+1;
else {
y--;
f += 2*(x-y)+1; }
}
circlePlotPoints(xCenter, yCenter, x, y);
}

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void circlePlotPoints (int xCenter, int yCenter,
int x, int y)
{
setPixel (xCenter + x, yCenter + y);
setPixel (xCenter – x, yCenter + y);
setPixel (xCenter + x, yCenter – y);
setPixel (xCenter – x, yCenter – y);
setPixel (xCenter + y, yCenter + x);
setPixel (xCenter – y, yCenter + x);
setPixel (xCenter + y, yCenter – x);
setPixel (xCenter – y, yCenter – x);
}

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Antialiasing
Antialiasing is a technique used to diminish
the jagged edges of an image or a line, so
that the line appears to be smoother; by
changing the pixels around the edges to
intermediate colors or gray scales.
Eg. Antialiasing disabled:
Eg. Antialiasing enabled:

Antialiasing (OpenGL)
Antialiasing disabled Antialiasing enabled
Setting antialiasing option for lines:
glEnable (GL_LINE_SMOOTH);

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Fill Area Algorithms
 Fill-Area algorithms are used to
fill the interior of a polygonal
shape.
 Many algorithms perform fill
operations by first identifying
the interior points, given the
polygon boundary.

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The basic filling algorithm is commonly used
in interactive graphics packages, where the
user specifies an interior point of the region to
be filled.
Basic Filling Algorithm
4-connected pixels

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[1] Set the user specified point.
[2] Store the four neighboring pixels in
a stack.
[3] Remove a pixel from the stack.
[4] If the pixel is not set,
Set the pixel
Push its four neighboring pixels
into the stack
[5] Go to step 3
[6] Repeat till the stack is empty.

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void fill(int x, int y) {
if(getPixel(x,y)==0){
setPixel(x,y);
fill(x+1,y);
fill(x-1,y);
fill(x,y+1);
fill(x,y-1);
}
}
Basic Filling Algorithm (Code)

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 Requires an interior point.
 Involves considerable amount of
stack operations.
 The boundary has to be closed.
 Not suitable for self-intersecting
polygons

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 Boundary Fill Algorithm
 For filling a region with a single
boundary color.
 Condition for setting pixels:
 Color is not the same as border color
 Color is not the same as fill color
 Flood Fill Algorithm
 For filling a region with multiple
boundary colors.
 Condition for setting pixels:
 Color is same as the old interior color
Types of Basic Filling Algorithms

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void boundaryFill(int x, int y,
int fillColor, int borderColor)
{
getPixel(x, y, color);
if ((color != borderColor)
&& (color != fillColor)) {
setPixel(x,y);
boundaryFill(x+1,y,fillColor,borderColor);
boundaryFill(x-1,y,fillColor,borderColor);
boundaryFill(x,y+1,fillColor,borderColor);
boundaryFill(x,y-1,fillColor,borderColor);
}
}
Boundary Fill Algorithm (Code)

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void floodFill(int x, int y,
int fillColor, int oldColor)
{
getPixel(x, y, color);
if (color != oldColor)
{
setPixel(x,y);
floodFill(x+1, y, fillColor, oldColor);
floodFill(x-1, y, fillColor, oldColor);
floodFill(x, y+1, fillColor, oldColor);
floodFill(x, y-1, fillColor, oldColor);
}
}
Flood Fill Algorithm (Code)

Filling Polygons (OpenGL)
Enabling polygon fill (Default):
glPolygonMode(GL_FRONT_AND_BACK, GL_FILL);
Disabling polygon fill:
glPolygonMode(GL_FRONT_AND_BACK, GL_LINE);

Output Primitives in Computer Graphics and Multimedia

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