Output Primitives
Dr. S.M. MalaekAssistant: M. Younesi
Output Primitives Output Primitives: Basic geometric
structures (points, straight line segment, circles and other conic sections, quadric surfaces, spline curve and surfaces, polygon color areas, and character strings)
These picture components are often defined in a continuous space.
Output Primitives In order to draw the primitive objects,
one has to first scan convert the object.
Scan convert: Refers to he operation of finding out the location of pixels to the intensified and then setting the values of corresponding bits, in the graphic memory, to the desired intensity code.
Output Primitives Each pixel on the display surface has
a finite size depending on the screen
resolution and hence a pixel cannot cannot representrepresent a single mathematical point.
Scan Converting A Point
Scan Converting A Point A mathematical point (x,y) needs to
be scan converted to a pixel at location (x´, y´).
Scan Converting A Point x´=Round(x) and y´=Round(y) All points that satisfy:
are mapped to pixel (x´,y´) 1
1
yyy
xxx
Scan Converting A Point x´=Round(x+0.5) and y´=Round(y+o.5) All points that satisfy:
are mapped to pixel (x´,y´) 5.05.0
5.05.0
yyy
xxx
Scan Converting A Line
Scan Converting A Line The Cartesian slope- intercept equation
for a straight line is:
12
12
xxyy
m
bxmy
11xmyb
xmy my
x
Scan Converting A Line
These equation form the basic for determining deflection voltage in analog devices.
xmy my
x
|m|<1 |m|>1
Scan Converting A Line On raster system, lines are plotted
with pixels, and step size (horizontal & vertical direction) are constrained by pixel separation.
Scan Converting A Line We must samplesample a line at discrete
positions and determine the nearest pixel to the line at each sampled position.
Digital Differential Analyzer
(DDA Algorithm)
DDA Algorithm Algorithm is an incremental scan
conversion method. Based on calculating either or If |m|<1,
x y)1( x
myy
xmy
kK
1
DDA Algorithm If |m|>1,
)1( x
)1( y
mxx
my
x
kK
11
If
myy
xmy
kK
1
Bresenham’s Line Algorithm
Bresenham’s Line Algorithm A highly efficient incremental
method for scan converting lines. Using only incremental integer
calculation.By testing the sign of an integer parameter, whose value is proportional to the difference between the separation of two pixel positions from the actual line path.
Bresenham’s Line Algorithm A highly efficient incremental
method for scan converting lines. Using only incremental integer
calculation.By testing the sign of an integer parameter, whose value is proportional to the difference between the separation of two pixel positions from the actual line path.
Bresenham’s Line Algorithm
Bresenham’s Line Algorithm
y
yk
yk+1
xk+1
}
}d2
d1
Bresenham’s Line Algorithm
y
yk
yk+1
xk+1
}
}d2
d1
y
yk
yk+1
xk+1
}
}d2
d1
Bresenham’s Line Algorithmd1 = y - yk = m (xk + 1) + b - yk
d2 = (yk + 1) - y = yk + 1 - m (xk + 1) –b
d1 - d2 = 2 m (xk + 1) - 2 yk + 2b -1
Bresenham’s Line Algorithm
Pk = ΔX ( d1 - d2) = 2 ΔY . xk - 2 ΔX . yk + c
c = constante = 2 ΔY + ΔX (2b -1)
Let's get rid of multiplicationsPk+1 = 2 ΔY . xk+1 - 2 ΔX .y + c
Pk+1 - Pk = 2 ΔY (xk+1 - xk) -2 ΔX (yk+1 - yk) (get rid of the constance)
Pk+1 = Pk + 2 ΔY - 2 ΔX (yk+1 - yk)
Pk+1 = Pk + 2ΔY or = Pk + 2(ΔY – ΔX(
with (yk+1 - yk) = 0 or 1 depending on Pk sign
P0 =ΔX (d1-d2) =ΔX[2m(x0+1)-2y0+2b-1] =ΔX[2(mx0+b-y0)+2m-1]
P0=2ΔY -ΔX0
m=Δy/x
Bresenham’s Line Algorithm
Bresenham’s Line Algorithm Example: Digitize the line with endpoint
(20,10) and (30,18) ΔX=10 ΔY= 8P0=2ΔY –Δx=6 2ΔY =16 2(ΔY – ΔX)=-4
P0=2ΔY –ΔXPk+1 = Pk + 2ΔY or = Pk + 2(ΔY – ΔX(
Circle Generation Algorithms
Circle Generation Algorithms The equation of a circle:
We could solve for y in terms of x
Circle Generation Algorithms Computation can be reduced by
considering the symmetry of circles
8 -Way symmetry
Midpoint Circle Algorithm
Circle Generation Algorithms As in the line algorithm, we sample at
unit intervals and determine the closet pixel position to the circle path at each step.
Circle Generation Algorithms Points are generated from 90º to 45º, moves
will be made only in the +x and –y direction.
positive x direction over this octant and use a decision parameter
Circle Generation Algorithms
We define a circle function:
o
ryxyxfcircle
0
0
),( 222
Circle Generation Algorithms Midpoint
Consider the coordinates of the point halfway between pixel T and pixel S
)21
,1( ii
yx
Midpoint
Circle Generation Algorithms
We use it to define a decision parameter
222 )21
()1()21
,1( ryxyxfpiiii
Circle Generation Algorithms If is negative, the midpoint
is inside the pixel, and we choose pixel T.
If we choose pixel S.0i
p
0i
p
Circle Generation Algorithms Parameter for the next step is:
since
2
1
2
11)
21
()1( ryxpiii
11
ii
xx
Circle Generation Algorithms
If T is chosen ( ) we have:
If pixel S is chosen ( ) we have
0i
p
0i
p
Circle Generation Algorithms
In terms of
Circle Generation Algorithms
Initial value for the decision parameter using the original function of (0,r(
When r is an integer we can simply set
Circle Generation Algorithms Example: A circle radius r=10 x=0 to x=y P0=1-r = -9
ExercisesExercises??
1. Scan Converting Arcs
And Sectors?
Character Generation
Character Generation
Letters, numbers, and other character can be displayed in a variety of size and styles.
Character Generation• Typeface: The overall design
style for a set of characters is call typeface: Zar, nazanin, Titr.
Font: Referred to a set of cast metal character forms in a particular size and forma: 10 point Zar.
Character Generation Two different representation are used
for storing computer fonts:
1. Bitmap font (or bitmapped font)
2. Outline font
Bitmap font Bitmap font (or bitmapped font): A
simple method for representing the character shapes in a particular typeface is to use rectangular grid pattern.
Bitmap font The character grid only need to be
mapped to a frame buffer position. Bitmap fonts required more space,
because each variation (size and format) must be stored in a font cash.
Bold Italic
Outline Font Graphic primitives such as lines and arcs
are used to define the outline of each character.
Require less storage since variation does not require a distinct font cash.
Outline Font We can produce boldface, italic, or
different size by manipulating the curve definition for the character outlines.
It does take more time to process the outline fonts, because they must be scan converted into frame buffer.