1 csce 441 lecture 2: scan conversion of lines jinxiang chai
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CSCE 441 Lecture 2:Scan Conversion of Lines
Jinxiang Chai
2/110
Displays – Cathode Ray Tube
3/110
Displays – Liquid Crystal Displays
4/110
Displays – Plasma
5/110
Displays – Pixels
Pixel: the smallest element of picture
- Integer position (i,j)
- Color information (r,g,b)
x
y
(0,0)
6/110
Digital Scan Conversion
Convert graphics output primitives into a set of pixel values for storage in the frame buffer
7/110
OpenGL Geometric Primitives
All geometric primitives are specified by vertices
GL_QUAD_STRIPGL_QUAD_STRIP
GL_POLYGONGL_POLYGON
GL_TRIANGLE_STRIPGL_TRIANGLE_STRIP
GL_TRIANGLE_FANGL_TRIANGLE_FAN
GL_POINTSGL_POINTSGL_LINESGL_LINES
GL_LINE_LOOPGL_LINE_LOOPGL_LINE_STRIPGL_LINE_STRIP
GL_TRIANGLESGL_TRIANGLES
GL_QUADSGL_QUADS
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OpenGL Geometric Primitives
All geometric primitives are specified by vertices
GL_QUAD_STRIPGL_QUAD_STRIP
GL_POLYGONGL_POLYGON
GL_TRIANGLE_STRIPGL_TRIANGLE_STRIP
GL_TRIANGLE_FANGL_TRIANGLE_FAN
GL_POINTSGL_POINTSGL_LINESGL_LINES
GL_LINE_LOOPGL_LINE_LOOPGL_LINE_STRIPGL_LINE_STRIP
GL_TRIANGLESGL_TRIANGLES
GL_QUADSGL_QUADS
9/110
Outline
How to draw a line?
- Digital Differential Analyzer (DDA)
- Midpoint algorithm
Required readings
- HB 6.1 to 6.8
10/110
Problem
Given two points (P, Q) on the screen (with integer coordinates) determine which pixels should be drawn to display a unit width line
P
Q
11/110
Problem
Given two points (P, Q) on the screen (with integer coordinates) determine which pixels should be drawn to display a unit width line
P
Q
12/110
Problem
Given two points (P, Q) on the screen (with integer coordinates) determine which pixels should be drawn to display a unit width line
P
Q
Line-drawing using “Paint”
14/110
Special Lines - Horizontal
15/110
Special Lines - Horizontal
Increment x by 1, keep y constant
16/110
Special Lines - Vertical
17/110
Special Lines - Vertical
Keep x constant, increment y by 1
18/110
Special Lines - Diagonal
19/110
Special Lines - Diagonal
Increment x by 1, increment y by 1
20/110
How about Arbitrary Lines?
How to draw an arbitrary line?
P
Q
21/110
Slope-intercept equation for a line:
Arbitrary Lines
bmxy m: slope
b: y-intercept
22/110
Slope-intercept equation for a line:
How can we compute the equation from (xL,yL), (xH,yH)?
Arbitrary Lines
bmxy m: slope
)( , HH yx
)( , LL yx
b: y-intercept
23/110
Slope-intercept equation for a line:
How can we compute the equation from (xL,yL), (xH,yH)?
Arbitrary Lines
LL
LH
LH
mxyb
xx
yym
bmxy m: slope
)( , HH yx
)( , LL yx
b: y-intercept
24/110
Arbitrary Lines
Assume
bmxy
10 m
10 m
25/110
Arbitrary Lines
Assume Other lines by swapping x, y or negating
- e.g., reverse the role of x and y if m>1
bmxy
10 m
10 m
26/110
Arbitrary Lines
Assume Other lines by swapping x, y or negating
- e.g., reverse the role of x and y if m>1
- e.g., negate y if
bmxy
10 m
10 m
01 m
27/110
Arbitrary Lines
Assume Other lines by swapping x, y or negating
Take steps in x and determine where to fill y
xmy
bmxy
10 m
10 m
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Digital Differential Analyzer
Start from (xL, yL) and draw to (xH, yH)
where xL< xH
( x0, y0 ) = ( xL, yL )
For ( i = 0; i <= xH-xL; i++ )
DrawPixel ( xi, Round ( yi ) )
xi+1 = xi + 1
yi+1 = m xi+1 + b
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Digital Differential Analyzer
Simple algorithm:
- sample unit intervals in one coordinate
- find the nearest pixel for each sampled point.
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Digital Differential Analyzer (DDA)
Start from (xL, yL) and draw to (xH, yH)
where xL< xH
( x0, y0 ) = ( xL, yL )For ( i = 0; i <= xH-xL; i++ ) //sample unit intervals in x coordinate
DrawPixel ( xi, Round ( yi ) ) //find the nearest pixel for each sampled point.
xi+1 = xi + 1
yi+1 = m ( xi + 1 ) + b
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DDA
Start from (xL, yL) and draw to (xH, yH)
where xL< xH
( x0, y0 ) = ( xL, yL )
For ( i = 0; i <= xH-xL; i++ )
DrawPixel ( xi, Round ( yi ) )
xi+1 = xi + 1
yi+1 = m xi + m + b
32/110
DDA
Start from (xL, yL) and draw to (xH, yH)
where xL< xH
( x0, y0 ) = ( xL, yL )
For ( i = 0; i <= xH-xL; i++ )
DrawPixel ( xi, Round ( yi ) )
xi+1 = xi + 1
yi+1 = m xi + b + m
33/110
DDA
Start from (xL, yL) and draw to (xH, yH)
where xL< xH
( x0, y0 ) = ( xL, yL )
For ( i = 0; i <= xH-xL; i++ )
DrawPixel ( xi, Round ( yi ) )
xi+1 = xi + 1
yi+1 = m xi + b + m
34/110
DDA
Start from (xL, yL) and draw to (xH, yH)
where xL< xH
( x0, y0 ) = ( xL, yL )
For ( i = 0; i <= xH-xL; i++ )
DrawPixel ( xi, Round ( yi ) )
xi+1 = xi + 1
yi+1 = yi + m
35/110
DDA - Example
)2,0(
)4,6(
bmxy
36/110
DDA - Example
)2,0(
)4,6(
bmxy
31
0624
LH
LH
xxyym
37/110
DDA - Example
)2,0(
)4,6(
bmxy
2
0
y
x
31
0624
LH
LH
xxyym
38/110
DDA - Example
)2,0(
)4,6(
bmxy
37
1
y
x
31
0624
LH
LH
xxyym
39/110
DDA - Example
)2,0(
)4,6(
bmxy
37
1
y
x
31
0624
LH
LH
xxyym
40/110
DDA - Example
)2,0(
)4,6(
bmxy
38
2
y
x
31
0624
LH
LH
xxyym
41/110
DDA - Example
)2,0(
)4,6(
bmxy
38
2
y
x
31
0624
LH
LH
xxyym
42/110
DDA - Example
)2,0(
)4,6(
bmxy
3
3
y
x
31
0624
LH
LH
xxyym
43/110
DDA - Example
)2,0(
)4,6(
bmxy
3
3
y
x
31
0624
LH
LH
xxyym
44/110
DDA - Example
)2,0(
)4,6(
bmxy
310
4
y
x
31
0624
LH
LH
xxyym
45/110
DDA - Example
)2,0(
)4,6(
bmxy
310
4
y
x
31
0624
LH
LH
xxyym
46/110
DDA - Example
)2,0(
)4,6(
bmxy
311
5
y
x
31
0624
LH
LH
xxyym
47/110
DDA - Example
)2,0(
)4,6(
bmxy
311
5
y
x
31
0624
LH
LH
xxyym
48/110
DDA - Example
)2,0(
)4,6(
bmxy
4
6
y
x
31
0624
LH
LH
xxyym
49/110
DDA - Example
)2,0(
)4,6(
bmxy
4
6
y
x
31
0624
LH
LH
xxyym
50/110
DDA - Problems
Floating point operations Rounding Can we do better?
( x0, y0 ) = ( xL, yL )
For ( i = 0; i <= xH-xL; i++ )
DrawPixel ( xi, Round ( yi ) )
xi+1 = xi + 1
yi+1 = yi + m
51/110
Outline
How to draw a line?
- Digital Differential Analyzer (DDA)
- Midpoint algorithm
52/110
How to Improve It?
)4,6(
bmxy
31
0624
LH
LH
xxyym
Question: if we draw the current pixel, where is the next pixel?
)2,0(
(xk,yk) (xk+1,yk+1)
53/110
How to Improve It?
)4,6(
bmxy
31
0624
LH
LH
xxyym
Question: if we draw the current pixel, where is the next pixel?
)2,0(
(xk,yk) (xk+1,yk+1)
54/110
How to Improve It?
)4,6(
bmxy
31
0624
LH
LH
xxyym
Question: if we draw the current pixel, where is the next pixel?
)2,0(
(xk,yk) (xk+1,yk+1)
55/110
How to Improve It?
)2,0(
)4,6(
bmxy
31
0624
LH
LH
xxyym
Question: if we draw the current pixel, where is the next pixel?
Next column: E or NE direction; why?
xk+1 = xk+1
yk+1 = yk or yk+1
56/110
Midpoint Algorithm
Given a point just drawn, determine whether we move E or NE on next step
xk+1 = xk+1
yk+1 = yk or yk+1
57/110
Midpoint Algorithm
Given a point just drawn, determine whether we move E or NE on next step
Is the line above or below midpoint ?
Below: move E
Above: move NE
),1( 21 yx
58/110
Above/Below the Line?
)2,0(
)4,6(
59/110
Two Issues
How to evaluate if the midpoint is above or below the line?
How to incrementally evaluate this?
How to avoid floating point operations?
60/110
Midpoint Algorithm – Implicit Forms
How do we tell if a point is above or below the line? bmxy
x
y
)2,0(
)5,6(
)7,3(
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Midpoint Algorithm – Implicit Forms
How do we tell if a point is above or below the line? bmxy
x
y
)2,0(
)5,6(
)7,3(Properties
y=mx+b (x,y) on the line
y<mx+b (x,y) below the line
y>mx+b (x,y) above the line
62/110
Midpoint Algorithm – Implicit Forms
How do we tell if a point is above or below the line? bmxy
x
y
)2,0(
)5,6(
)7,3(Properties
y=mx+b (x,y) on the line
y<mx+b (x,y) below the line
y>mx+b (x,y) above the line
But this still requires floating point operations!
63/110
Midpoint Algorithm – Implicit Forms
How do we tell if a point is above or below the line? bmxy
bmxyyxf ),(
Properties
f(x,y)=0 (x,y) on the line
f(x,y)<0 (x,y) below the line
f(x,y)>0 (x,y) above the line
64/110
Midpoint Algorithm – Implicit Forms
How do we tell if a point is above or below the line?
bxyLH
LH
xxyy
65/110
Midpoint Algorithm – Implicit Forms
How do we tell if a point is above or below the line?
))(()( LHxxyy
LH xxbxyxxLH
LH
66/110
Midpoint Algorithm – Implicit Forms
How do we tell if a point is above or below the line?
bxxxyyyxx LHLHLH )()()(
67/110
Midpoint Algorithm – Implicit Forms
How do we tell if a point is above or below the line?
0)()()( bxxxyyyxx HLHLLH
68/110
Midpoint Algorithm – Implicit Forms
How do we tell if a point is above or below the line?
edycxyxf ),(HL yyc LH xxd )( HL xxbe
0)()()( bxxxyyyxx HLHLLH
69/110
Midpoint Algorithm – Implicit Forms
How do we tell if a point is above or below the line?
edycxyxf ),(HL yyc LH xxd )( HL xxbe
Properties
f(x,y)=0 (x,y) on the line
f(x,y)<0 (x,y) below the line
f(x,y)>0 (x,y) above the line
0)()()( bxxxyyyxx HLHLLH
70/110
Midpoint Algorithm – Implicit Forms
How do we tell if a point is above or below the line?
edycxyxf ),(HL yyc LH xxd )( HL xxbe
Properties
f(x,y)=0 (x,y) on the line
f(x,y)<0 (x,y) below the line
f(x,y)>0 (x,y) above the line
0)()()( bxxxyyyxx HLHLLH
LL
LH
LH
mxyb
xx
yym
Note that e is an integer because
71/110
Midpoint Algorithm – Implicit Forms
x
y
)2,0(
)5,6(
263 xy
72/110
Midpoint Algorithm – Implicit Forms
x
y
)2,0(
263 xy
1236),( xyyxf)5,6(
73/110
Midpoint Algorithm – Implicit Forms
x
y
)2,0(
263 xy
1236),( xyyxf 18)5,0( f)5,6(
74/110
Midpoint Algorithm – Implicit Forms
x
y
)2,0(
263 xy
1236),( xyyxf
18)1,4( f
)5,6(
75/110
Midpoint Algorithm – Implicit Forms
x
y
)2,0(
263 xy
1236),( xyyxf
0)4,4( f)5,6(
76/110
Midpoint Algorithm – Implicit Forms
x
y
)2,0(
263 xy
1236),( xyyxf)5,6(
77/110
Two Issues
How to evaluate if the midpoint is above or below the line?
How to incrementally evaluate this?
78/110
Above/Below the Line?
)2,0(
)4,6(
79/110
Midpoint Algorithm
What about starting value?
What is the middle point?
(xL,yL)
80/110
Midpoint Algorithm
What about starting value?
What is the middle point? (xL+1,yL+1/2)
(xL,yL)
81/110
Midpoint Algorithm
What about starting value?
What is the middle point?
eydxcyxf LLLL )()1(),1( 21
21
(xL+1,yL+1/2)
(xL,yL)
82/110
Midpoint Algorithm
What about starting value?
eydxcyxf LLLL )()1(),1( 21
21
dcyxfyxf LLLL 21
21 ),(),1(
83/110
Midpoint Algorithm
What about starting value?
eydxcyxf LLLL )()1(),1( 21
21
dcyxfyxf LLLL 21
21 ),(),1(
is on the line!),( LL yx
84/110
Midpoint Algorithm
What about starting value?
eydxcyxf LLLL )()1(),1( 21
21
dcyxf LL 21
21 ),1(
85/110
Midpoint Algorithm
What about starting value?
eydxcyxf LLLL )()1(),1( 21
21
dcyxf LL 2),1( 21
Multiplying coefficients by 2 doesn’t change sign of f
86/110
Midpoint Algorithm
Need value of to determine E or NE
Build incremental algorithm Assume we have value of
Find value of if E chosenFind value of if NE chosen
),1( 21 yxf
),1( 21 yxf
),2( 21 yxf
),2( 23 yxf
87/110
Midpoint Algorithm
Need value of to determine E or NE
Build incremental algorithm Assume we have value of
Find value of if E chosen
),1( 21 yxf
),1( 21 yxf
),2( 21 yxf
88/110
Midpoint Algorithm
Need value of to determine E or NE
Build incremental algorithm Assume we have value of
Find value of if E chosenFind value of if NE chosen
),1( 21 yxf
),1( 21 yxf
),2( 21 yxf
),2( 23 yxf
89/110
Midpoint Algorithm
If E was chosen, find ),2( 21 yxf
eydxcyxf )()2(),2( 21
21
90/110
Midpoint Algorithm
If E was chosen, find ),2( 21 yxf
eydxcyxf )()2(),2( 21
21
),1(),2( 21
21 yxfcyxf
91/110
Midpoint Algorithm
If NE was chosen, find ),2( 23 yxf
eydxcyxf )()2(),2( 23
23
92/110
Midpoint Algorithm
If NE was chosen, find ),2( 23 yxf
eydxcyxf )()2(),2( 23
23
),1(),2( 21
23 yxfdcyxf
93/110
Midpoint Algorithm – Summary
x=xL y=yL
d=xH - xL c=yL - yH
sum=2c+d draw(x,y)
while ( x < xH)if ( sum < 0 )
sum += 2dy++
x++sum += 2cdraw(x,y)
94/110
Midpoint Algorithm – Summary
x=xL y=yL
d=xH - xL c=yL - yH
sum=2c+d //initial valuedraw(x,y)
while ( x < xH) if ( sum < 0 )
sum += 2dy++
x++sum += 2cdraw(x,y)
95/110
Midpoint Algorithm – Summary
x=xL y=yL
d=xH - xL c=yL - yH
sum=2c+d //initial valuedraw(x,y)
while ( x < xH) // iterate until reaching the end pointif ( sum < 0 )
sum += 2dy++
x++sum += 2cdraw(x,y)
96/110
Midpoint Algorithm – Summary
x=xL y=yL
d=xH - xL c=yL - yH
sum=2c+d //initial valuedraw(x,y)
while ( x < xH) // iterate until reaching the end pointif ( sum < 0 ) // below the line and choose NE
sum += 2dy++
x++sum += 2cdraw(x,y)
97/110
Midpoint Algorithm – Summary
x=xL y=yL
d=xH - xL c=yL - yH
sum=2c+d //initial valuedraw(x,y)
while ( x < xH) // iterate until reaching the end pointif ( sum < 0 ) // below the line and choose NE
sum += 2dy++
x++sum += 2cdraw(x,y)
- Update the current pixel
- Update the function value of midpoint
98/110
Midpoint Algorithm – Summary
x=xL y=yL
d=xH - xL c=yL - yH
sum=2c+d //initial valuedraw(x,y)
while ( x < xH) // iterate until reaching the end pointif ( sum < 0 ) // below the line and choose NE
sum += 2dy++
x++sum += 2cdraw(x,y)
- Draw the pixel based on the function value
of midpoint
99/110
x=xL y=yL
d=xH - xL c=yL - yH
sum=2c+ddraw(x,y)
while ( x < xH)if ( sum < 0 )
sum += 2dy++
x++sum += 2cdraw(x,y)
Midpoint Algorithm – Example
)2,0(
)4,6(
0x 2y 2sum
d =6 c = -2
100/110
Midpoint Algorithm – Example
)2,0(
)4,6(
0x 2y 2sumx=xL y=yL
d=xH - xL c=yL - yH
sum=2c+ddraw(x,y)
while ( x < xH)if ( sum < 0 )
sum += 2dy++
x++sum += 2cdraw(x,y)
d =6 c = -2
101/110
Midpoint Algorithm – Example
)2,0(
)4,6(
1x 2y 2sumx=xL y=yL
d=xH - xL c=yL - yH
sum=2c+ddraw(x,y)
while ( x < xH)if ( sum < 0 )
sum += 2dy++
x++sum += 2cdraw(x,y)
d =6 c = -2
102/110
x=xL y=yL
d=xH - xL c=yL - yH
sum=2c+ddraw(x,y)
while ( x < xH)if ( sum < 0 )
sum += 2dy++
x++sum += 2cdraw(x,y)
Midpoint Algorithm – Example
)2,0(
)4,6(
2x 3y 6sum
d =6 c = -2
103/110
Midpoint Algorithm – Example
)2,0(
)4,6(
3x 3y 2sumx=xL y=yL
d=xH - xL c=yL - yH
sum=2c+ddraw(x,y)
while ( x < xH)if ( sum < 0 )
sum += 2dy++
x++sum += 2cdraw(x,y)
d =6 c = -2
104/110
Midpoint Algorithm – Example
)2,0(
)4,6(
4x 3y 2sumx=xL y=yL
d=xH - xL c=yL - yH
sum=2c+ddraw(x,y)
while ( x < xH)if ( sum < 0 )
sum += 2dy++
x++sum += 2cdraw(x,y)
d =6 c = -2
105/110
x=xL y=yL
d=xH - xL c=yL - yH
sum=2c+ddraw(x,y)
while ( x < xH)if ( sum < 0 )
sum += 2dy++
x++sum += 2cdraw(x,y)
Midpoint Algorithm – Example
)2,0(
)4,6(
5x 4y 6sum
d =6 c = -2
106/110
x=xL y=yL
d=xH - xL c=yL - yH
sum=2c+ddraw(x,y)
while ( x < xH)if ( sum < 0 )
sum += 2dy++
x++sum += 2cdraw(x,y)
Midpoint Algorithm – Example
)2,0(
)4,6(
6x 4y 2sum
d =6 c = -2
107/110
Midpoint Algorithm
Only integer operations Exactly the same as the more commonly
found Bresenham’s line drawing algorithm Extends to other types of shapes (circles)
108/110
Midpoint Algorithm: Circle
0)()( 220
20 ryyxx
109/110
Midpoint Algorithm: Circle
0)()( 220
20 ryyxx
110/110
Midpoint Algorithm: Circle
0)()( 220
20 ryyxx
111/110
Midpoint Algorithm: Circle
0)()( 220
20 ryyxx
Two issues:- What are two possible solutions?
- What is the evaluation function?
112/110
Midpoint Algorithm: Circle
113/110
Midpoint Algorithm: Circle
(xk+1,yk)(xk+1,yk-1)
Two issues:- What are two possible solutions?
- What is the evaluation function?
114/110
Midpoint Algorithm: Circle
(xk+1,yk) Midpoint: (xk+1,yk-1/2)(xk+1,yk-1)
Two issues:- What are two possible solutions?
- What is the evaluation function?
115/110
Midpoint Algorithm: Circle
(xk+1,yk) Midpoint: (xk+1,yk-1/2)(xk+1,yk-1)
Two issues:- What are two possible solutions?
- What is the evaluation function?
220
20 )()(),( ryyxxyxf
116/110
Circle, Ellipse, etc
Need implicit forms
- circle:
- ellipse:
Equations for incremental calculations Initial positions More details in section 6.4-6.6 of the
textbook
022 FEyDxCxyByAx
0)()( 220
20 ryyxx
117/110
Next Lecture
How to draw a polygon?
- section 6.10- 6.13
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