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CPCS 391 Computer Graphics 1
Instructor: Dr. Sahar ShabanahLecture 3
+Scan conversion Algorithms
Primitives and Attributes
Why Scan Conversion?
Algorithms for Scan Conversion:
Lines
Circles
Ellipses
Filling
Polygons
2
+Scan Conversion Problem
To represent a perfect image as a bitmapped image.
3
+Line Drawing Algorithms
Lines are used a lot - want to get them right.
Lines should appear straight, not jagged.
Horizontal, vertical and diagonal easy, others difficult
Lines should terminate accurately.
Lines should have constant density.
Line density should be independent of line length or angle.
Lines should be drawn rapidly.
Efficient algorithms.
4
+DDA: Digital Differential Analyzer
y i1 mx i1 B
m(x i x) B
mx i mx B
(mx i B) mx
y i mx
y i1 y i m x 1
(xi,yi)
(xi,Round(yi))
(xi,Round(yi+m))
(xi,yi +m)
Desired Line
Line: Left to Right:1- Slope m>0: sample at unit x intervals ( Δx= 1), calculate each succeeding y value as
5
+DDA
x i1 x i 1
my 1
2- Slope m<0: sample at unit y
intervals ( Δy= 1), calculate each
succeeding x value as
Line: from Right to Left
3- Slope m> 0:
y i1 y i m x 1
4- Slope m< 0:
x i1 x i 1
my 1
6
+DDA
Faster than brute force.
Based on Calculating either ∆x or ∆y.
Mathematically well defined
Floating point
Round off error.
Time consuming arithmetic
Advantages Disadvantages
7
+Bresenhams Line Algorithm
Accurate
Efficient
Integer Calculations
Uses Symmetry for other lines
Adapted to display circles, ellipses and curves
It has been proven that the algorithm gives an optimal fit for lines
8
+Bresenhams Line Algorithm
d2
d1
Xk+1
yk+1
y
yk
9
+Bresenhams Line Algorithm
y m(x i 1) b
d1 y y
m(xk1) b yk
d2 (yk1) y
yk1 m(xk1) b
d1 d2 2m(xk1) 2y 2b 1 (3 11)
decision parameter, use m y
xpk x(d1 d2)
2yxk 2xy c (3 12)
10
+Bresenhams Line Algorithm
The sign of pk is the same as the sign of d1 – d2,
since Δx> 0 for our example. Parameter c is independent and will be eliminated in the recursive calculations for pk.
If the pixel at yk is closer to the line path than the pixel at yk+l (that is, d1 < d2), then decision parameter pk is negative. In that case, we plot the lower pixel; otherwise, we plot the upper pixel.
pk1 2yxk1 2xyk c
11
+ Bresenhams Line Algorithm
This recursive calculation of decision parameters is performed at each integer x position, starting at the left coordinate endpoint of the line. The first parameter, po is evaluated from Eq. 3-12 at the starting pixel position (xo, yo) and with m evaluated as Δy/Δx:
pk1 pk 2y(xk1 xk ) 2x(yk1 yk )
but xk1 xk 1
pk1 pk 2y 2x(yk1 yk )
p0 2y 2x
yk1 yk 0or1
12
+Bresenhams Line Drawing Algorithm1. Input the two line endpoints, store the left endpoint
(x0,y0).
2. Plot the first point (x0,y0).
3. Calculate constants ∆x, ∆y, and 2∆y - 2∆x and 2∆y, get starting values for decision parameter pk, p0=2∆y-∆x
4. At each xk along the line, starting at k = 0, do the following test: if pk < 0, the next point to plot is(xk+1, yk)
pk+1 = pk + 2∆y
else, the next point to plot is(xk+1, yk+1)
pk+1=pk +2∆y-2∆x
5. Repeat step 4. ∆x times.
13
+Bresenhams Line Algorithm
14
+Midpoint Line Algorithm
If (BlueLine < Midpoint) Plot_East_Pixel();
Else Plot_Northeast_Pixel();
15
+
Find an equation, given a line and a point, that will tell us if the point is above or below that line?
Midpoint Line Algorithm
y y
xx B
xy yx Bx
yx xy Bx
F(x,y) ax by c 0
F(x,y) yx xy Bx
now,d F(M) M midpoint
F(x p 1, y p 12)
y(x p 1) x(y p 12) Bx
y d y d
If F(x,y) ==0 (x,y) on the line <0 for points below the
line >0 for points above the
line d=F(M)
16
+Midpoint Line Algorithm P=(xp, yp) is pixel chosen by the algorithm in previous step
To calculate d incrementally we require dnew
If d > 0 then choose NE
d y(x p 1) x(y p 12) Bx
dnew F(x p 2,y p 32)
y(x p 2) x(y p 32) Bx
dnew d y xNE
1 2 4 3 4
dnew d NE
NE y x
P=(xp, yp)
Yp+2
M
E
NE
xp+1xp xp+2
Pre
vio
us
Curr
en
t
Next
),1( 21 yxp
),2( 23 yxp
yp
Yp+1
MNE
17
+Midpoint Line Algorithm If d < 0 then choose E
d y(x p 1) x(y p 12) Bx
dnew F(x p 2,y p 12)
y(x p 2) x(y p 12) Bx
dnew d yE{
dnew d E
E y
P=(xp, yp)
M
E
NE
xp+1xp xp+2
Pre
vio
us
Curr
en
t
Next
),1( 21 pp yx
),2( 21 pp yx
yp
Yp+1
Yp+2
ME
18
+Midpoint Line Algorithm To find Initial value of d
d0 F(x0 1, y0 12)
y(x p 1) x(y p 12) Bx
yx p xy p Bx y 12x
F(x0, y0) y 12x
d0 y 12x
[as (x0, y0) is on the line]
P=(x0, y0)
M
E
NE
x0+1x0
Sta
rt
Init
ial d
o
),1( 21
00 yx
Only fractional value
d0 2y x
NE 2(y x)
E 2y
Multiply by 2 to avoid fractions. Redefine d0, E, NE
19
+Midpoint Line Algorithm
Midpoint: Looks at which side of the line the mid point falls on.
Bresenham: Looks at sign of scaled difference in errors.
It has been proven that Midpoint is equivalent to Bresenhams for lines.
20
+void MidpointLine(int x0, int y0, int x1, int y1, int color){
int dx = x1 – x0, dy = y1 – y0;
int d = 2*dy – dx;
int dE = 2*dy, dNE = 2*(dy – dx);
int x = x0, y = y0;
WritePixels(x, y, color);
while (x < x1) {
if (d <= 0) { // Current d
d += dE; // Next d at E
x++;
} else {
d += dNE; // Next d at NE
x++;
y++} Write8Pixels(x, y, color);}}
21
+Midpoint Circle Algorithm
Implicit of equation of circle is: x2 + y2 - R2
= 0, at origin
Eight way symmetry require to calculate one octant
For each pixel (x,y),
there are 8 symmetric pixels
In each iteration
only calculate one pixel,
but plot 8 pixels
22
23+Midpoint Circle Algorithm
Define decision variable d as:
SE Choose
Circle outside s
E Choose
Circle inside s
iM
d
iM
d
Ryx
yxFMFd
pp
pp
0
0
1
),1()(
22
212
21
P=(xp, yp)
M
E
xp+1xp xp+2
Pre
vio
us
Curr
ent
Next
),1( 21 pp yx ),2( 2
1 pp yx
ME
yp
yp – 1
yp – 2
SE MSE
E choose we
either Choose
0d
€
(x p + 2, y p − 32)
24+Midpoint Circle Algorithm
If d <= 0 then midpoint m is inside circle we choose E Increment x y remains unchanged
Edd
xdd
Ryx
yxFd
Ryxd
new
E
pnew
pp
ppnew
pp
32
2
),2(
1
22
212
21
22
212
P=(xp, yp)
M
E
xp+1xp xp+2
Pre
vio
us
Curr
ent
Next
),1( 21 pp yx ),2( 2
1 pp yx
ME
yp
yp – 1
yp – 2
d < 0
25+Midpoint Circle Algorithm
If d > 0 then midpoint m is outside circle we choose E Increment x Decrement y
€
d = x p +1( )2
+ y p − 12( )
2
− R2
dnew = F(x p + 2,y p − 32)
= x p + 2( )2
+ y p − 32( )
2
− R2
dnew − d = 2x p − 2y p + 5ΔSE
1 2 4 4 3 4 4
dnew = d + ΔSE
Pre
vio
us
P=(xp, yp)
M
SE
xp+1
xp xp+2
Curr
ent
Next
),1( 21 pp yx
€
(x p + 2,y p − 32)
MSE
yp
yp – 1
yp – 2
d > 0
26+Midpoint Circle Algorithm
Initial condition
Starting pixel (0, R)
Next Midpoint lies at (1, R – ½)
d0 = F(1, R – ½) = 1 + (R2 – R + ¼) – R2 = 5/4 – R
To remove the fractional value 5/4 : Consider a new decision variable h as, h = d – ¼ Substituting d for h + ¼,
d0=5/4 – R h = 1 – R
d < 0 h < – ¼ h < 0 Since h starts out with an integer value and is incremented by
integer value (E or SE), e can change the comparison to just h < 0
27+Midpoint Circle Algorithmvoid MidpointCircle(int radius, int value) {
int x = 0;int y = radius ;int d = 1 – radius ;CirclePoints(x, y, value);while (y > x) {
if (d < 0) { /* Select E */d += 2 * x + 3;
} else { /* Select SE */
d += 2 * ( x – y ) + 5;y – –;
}x++;CirclePoints(x, y, value);
}}
28+Midpoint Circle Algorithm
Void CirclePoints(int x, int y, float value){
SetPixel(x,y);SetPixel(x,-y); SetPixel(-x,y); SetPixel(-x,-y); SetPixel(y,x);
SetPixel(y,-x); SetPixel(-y,x);
SetPixel(-y,-x);}
29+ Midpoint Circle Algorithm Second-order differences can be used to enhance
performance.
522
32),(
pp
ppp yxSE
xEyx
2
2
52)1(2
3)1(2),1(
SESE
EE
yxSE
xEyx
new
new
ppnew
pnewpp
4
2
5)1(2)1(2
3)1(2)1,1(
SESE
EE
yxSE
xEyx
new
new
ppnew
pnewpp
E is chosen
SE is chosen 52
3),0(
RSE
ER
:value Initial
M
SE
MSE
E
30+Midpoint Circle Algorithmvoid MidpointCircle(int radius, int value) {
int x = 0;int y = radius ;int d = 1 – radius ;int dE = 3;int dSE = -2*radius +5;CirclePoints(x, y, value);while (y > x) {
if (d < 0) { /* Select E */d += dE;dE += 2;dSE += 2;
} else { /* Select SE */d += dSE;dE += 2;dSE += 4;y – –;}
x++;CirclePoints(x, y, value);}
}
31+Midpoint Ellipse Algorithm
Implicit equation is: F(x,y) = b2x2 + a2y2 – a2b2 = 0
We have only 4-way symmetry
There exists two regions In Region 1 dx > dy
Increase x at each step y may decrease
In Region 2 dx < dy Decrease y at each step x may increase
(x1,y1)(-x1,y1)
(x1,-y1)(-x1,-y1)
(-x2,y2)
(-x2,-y2)
(x2,y2)
(x2,-y2)
32+Midpoint Ellipse Algorithm
Region 1
Region 2S SE
E
SE
Gradient Vector
TangentSlope = -1
ya
xb
dx
dy
dx
dyaybx
2
2
22 022
yaxb
dx
dy
22
1 1 Region In
33+Midpoint Ellipse Algorithm
In region 1
SE
ppnew
pp
ppnew
E
pnew
pp
ppnew
pp
pp
yaxbdd
bayaxb
yxFd
SEto move then 0 d if
xbdd
bayaxb
yxFd
E to move then 0 d if
bayaxb
yxFd
)22()32(
)()2(
),2(
)32(
)()2(
),2(
)()1(
),1(
22
222322
23
2
222122
21
222122
21
P=(xp, yp)
M
E
xp+1
xp
xp+2
Pre
vio
us C
urr
ent
Nex
t
),1( 21 pp yx ),2( 2
1 pp yx
ME
yp
yp – 1
yp – 2
SE
MSE
),2( 23 pp yx
34+Midpoint Ellipse Algorithm
In region 2
SE
ppnew
pp
ppnew
S
pnew
pp
ppnew
pp
pp
yaxbdd
bayaxb
yxFd
SEto move then 0 d if
yadd
bayaxb
yxFd
Sto move then 0 d if
bayaxb
yxFd
)32()22(
)2()(
)2,(
)32(
)2()(
)2,(
)1()(
)1,(
22
222232
23
2
222212
21
222212
21
P=(xp, yp)
MS
xp+1
xp
xp+2
Pre
vio
us Curr
ent
Nex
t
)1,( 21 pp yx
)2,( 21 pp yx
MS
yp
yp – 1
yp – 2
SE
MSE
)2,( 23 pp yx
35+Midpoint Ellipse AlgorithmDPx=2*ry*ry; Dpy =2*rx*rx; x=0; y=ry; Px=0; Py =2*rx*rx*ry; f =ry*ry +rx*rx(0.25-ry ); ry2=ry *ry; Set4Pixel(x,y); while (px<py ) //Region I{ x=x+1; Px=Px+DPx; if (f>0) // Bottom case{y=y -1; Py =Py -Dpy ; f=f+ry2+Px-Py;} else // Top casef=f+ry2+Px; Set4Pixel(x,y);
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