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.