dr. scott schaefer scan conversion of lines. 2/78 displays – cathode ray tube
TRANSCRIPT
Dr. Scott Schaefer
Scan Conversion of Lines
2/78
Displays – Cathode Ray Tube
3/78
Displays – Liquid Crystal Displays
4/78
Displays – Plasma
5/78
Displays – Pixels
Pixel: the smallest element of picture
- Integer position (i,j)
- Color information (r,g,b)
x
y
(0,0)
6/78
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
7/78
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
8/78
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
9/78
Special Lines - Horizontal
10/78
Special Lines - Horizontal
Increment x by 1, keep y constant
11/78
Special Lines - Vertical
12/78
Special Lines - Vertical
Keep x constant, increment y by 1
13/78
Special Lines - Diagonal
14/78
Special Lines - Diagonal
Increment x by 1, increment y by 1
15/78
How about Arbitrary Lines?
16/78
Slope-intercept equation for a line:
Arbitrary Lines
bmxy m: slope
b: y-intercept
17/78
Slope-intercept equation for a line:
How can we compute the equation from (xL,yL), (xH,yH)?
Arbitrary Lines
bmxy m: slope
b: y-intercept
)( , HH yx
)( , LL yx
18/78
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
b: y-intercept
)( , HH yx
)( , LL yx
19/78
Arbitrary Lines
Assume Other lines by swapping x, y or negating
Take steps in x and determine where to fill y
bmxy
10 m
20/78
Simple Algorithm
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
21/78
Simple Algorithm
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
22/78
Simple Algorithm
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
23/78
Simple Algorithm
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
24/78
Simple Algorithm
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
25/78
Simple Algorithm
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
26/78
Simple Algorithm - Example
)2,0(
)4,6(
bmxy
27/78
Simple Algorithm - Example
)2,0(
)4,6(
bmxy
31
0624
LH
LH
xxyym
28/78
Simple Algorithm - Example
)2,0(
)4,6(
bmxy
2
0
y
x
31
0624
LH
LH
xxyym
29/78
Simple Algorithm - Example
)2,0(
)4,6(
bmxy
37
1
y
x
31
0624
LH
LH
xxyym
30/78
Simple Algorithm - Example
)2,0(
)4,6(
bmxy
38
2
y
x
31
0624
LH
LH
xxyym
31/78
Simple Algorithm - Example
)2,0(
)4,6(
bmxy
3
3
y
x
31
0624
LH
LH
xxyym
32/78
Simple Algorithm - Example
)2,0(
)4,6(
bmxy
310
4
y
x
31
0624
LH
LH
xxyym
33/78
Simple Algorithm - Example
)2,0(
)4,6(
bmxy
311
5
y
x
31
0624
LH
LH
xxyym
34/78
Simple Algorithm - Example
)2,0(
)4,6(
bmxy
4
6
y
x
31
0624
LH
LH
xxyym
35/78
Simple Algorithm - 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
36/78
Midpoint Algorithm
Given a point just drawn, determine whether we move E or NE on next step
37/78
Midpoint Algorithm
Given a point just drawn, determine whether we move E or NE on next step
Is the line above or below ?
Below: move E
Above: move NE
),1( 21 yx
38/78
Midpoint Algorithm
)2,0(
)4,6(
39/78
Midpoint Algorithm – Implicit Forms
How do we tell if a point is above or below the line?
bmxy
40/78
Midpoint Algorithm – Implicit Forms
How do we tell if a point is above or below the line?
bxyLH
LH
xxyy
41/78
Midpoint Algorithm – Implicit Forms
How do we tell if a point is above or below the line?
))(()( LHxxyy
LH xxbxyxxLH
LH
42/78
Midpoint Algorithm – Implicit Forms
How do we tell if a point is above or below the line?
bxxxyyyxx LHLHLH )()()(
43/78
Midpoint Algorithm – Implicit Forms
How do we tell if a point is above or below the line?
0)()()( bxxxyyyxx HLHLLH
44/78
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
45/78
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
46/78
Midpoint Algorithm – Implicit Forms
x
y
)2,0(
)5,6(
263 xy
47/78
Midpoint Algorithm – Implicit Forms
x
y
)2,0(
263 xy
1236),( xyyxf)5,6(
48/78
Midpoint Algorithm – Implicit Forms
x
y
)2,0(
263 xy
1236),( xyyxf 18)5,0( f)5,6(
49/78
Midpoint Algorithm – Implicit Forms
x
y
)2,0(
263 xy
1236),( xyyxf
18)1,4( f
)5,6(
50/78
Midpoint Algorithm – Implicit Forms
x
y
)2,0(
263 xy
1236),( xyyxf
0)4,4( f)5,6(
51/78
Midpoint Algorithm – Implicit Forms
x
y
)2,0(
263 xy
1236),( xyyxf)5,6(
52/78
Need value of to decide E or NE Build incremental algorithm Assume we have value of
Find value of if E chosenFind value of if NE chosen
Midpoint Algorithm
),1( 21 yxf
),1( 21 yxf
),2( 21 yxf
),2( 23 yxf
53/78
Midpoint Algorithm
Need value of to decide 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
),1( 21 yxf
),2( 21 yxf
),2( 23 yxf
54/78
Midpoint Algorithm
If E was chosen, find ),2( 21 yxf
eydxcyxf )()2(),2( 21
21
55/78
Midpoint Algorithm
If E was chosen, find
),1(),2( 21
21 yxfcyxf
),2( 21 yxf
eydxcyxf )()2(),2( 21
21
56/78
Midpoint Algorithm
If NE was chosen, find ),2( 23 yxf
eydxcyxf )()2(),2( 23
23
57/78
Midpoint Algorithm
If NE was chosen, find
),1(),2( 21
23 yxfdcyxf
),2( 23 yxf
eydxcyxf )()2(),2( 23
23
58/78
Midpoint Algorithm
What about starting value?
eydxcyxf LLLL )()1(),1( 21
21
59/78
Midpoint Algorithm
What about starting value?
eydxcyxf LLLL )()1(),1( 21
21
dcyxfyxf LLLL 21
21 ),(),1(
60/78
Midpoint Algorithm
What about starting value?
eydxcyxf LLLL )()1(),1( 21
21
dcyxfyxf LLLL 21
21 ),(),1(
is on the line!),( LL yx
61/78
Midpoint Algorithm
What about starting value?
eydxcyxf LLLL )()1(),1( 21
21
dcyxf LL 21
21 ),1(
62/78
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
63/78
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)
64/78
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)
65/78
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)
66/78
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)
67/78
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)
68/78
Midpoint Algorithm – Example
)2,0(
)4,6(
0x 2y 2sumx=xL y=yL
d=6 c=-2sum=2c+ddraw(x,y)
while ( x < xH)if ( sum < 0 )
sum += 2dy++
x++sum += 2cdraw(x,y)
69/78
Midpoint Algorithm – Example
)2,0(
)4,6(
0x 2y 2sumx=xL y=yL
d=6 c=-2sum=2c+ddraw(x,y)
while ( x < xH)if ( sum < 0 )
sum += 2dy++
x++sum += 2cdraw(x,y)
70/78
Midpoint Algorithm – Example
)2,0(
)4,6(
1x 2y 2sumx=xL y=yL
d=6 c=-2sum=2c+ddraw(x,y)
while ( x < xH)if ( sum < 0 )
sum += 2dy++
x++sum += 2cdraw(x,y)
71/78
Midpoint Algorithm – Example
)2,0(
)4,6(
2x 3y 6sumx=xL y=yL
d=6 c=-2sum=2c+ddraw(x,y)
while ( x < xH)if ( sum < 0 )
sum += 2dy++
x++sum += 2cdraw(x,y)
72/78
Midpoint Algorithm – Example
)2,0(
)4,6(
3x 3y 2sumx=xL y=yL
d=6 c=-2sum=2c+ddraw(x,y)
while ( x < xH)if ( sum < 0 )
sum += 2dy++
x++sum += 2cdraw(x,y)
73/78
Midpoint Algorithm – Example
)2,0(
)4,6(
4x 3y 2sumx=xL y=yL
d=6 c=-2sum=2c+ddraw(x,y)
while ( x < xH)if ( sum < 0 )
sum += 2dy++
x++sum += 2cdraw(x,y)
74/78
Midpoint Algorithm – Example
)2,0(
)4,6(
5x 4y 6sumx=xL y=yL
d=6 c=-2sum=2c+ddraw(x,y)
while ( x < xH)if ( sum < 0 )
sum += 2dy++
x++sum += 2cdraw(x,y)
75/78
Midpoint Algorithm – Example
)2,0(
)4,6(
6x 4y 2sumx=xL y=yL
d=6 c=-2sum=2c+ddraw(x,y)
while ( x < xH)if ( sum < 0 )
sum += 2dy++
x++sum += 2cdraw(x,y)
76/78
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)
77/78
Circle, ellipse, etc…
Need implicit forms
- circle:
- ellipse:
Equations for incremental calculations Initial positions More details in section 6.4-6.5 of the textbook
022 FEyDxCxyByAx
0)()( 220
20 ryyxx
78/78
OpenGL: Drawing Lines
glBegin (GL_LINES);glVertex2f (p1.x, p1.y);glVertex2f (p2.x, p2.y);glVertex2f (p3.x, p3.y);glVertex2f (p4.x, p4.y);glVertex2f (p5.x, p5.y);glVertex2f (p6.x, p6.y);
glEnd();
P1
P2 P3
P4
P5P6