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Circle Drawing
Asst. Prof. Dr. Ahmet SayarKocaeli University
Computer EngineeringAdvanced Computer Graphics
Spring 2012
A circle is a set of points that are all at a given distance r from a center position ( xc , yc ).
For any circle point ( x , y ), this distance relationship is expressed by the Pythagorean theorem in Cartesian coordinates as
We could use this equation to calculate the position of points on a circle circumference by stepping along the x axis in unit steps from xc - r to xc + r and calculating the corresponding y values at each position as
Properties of Circle
A Simple Circle Drawing Algorithm
The equation for a circle is:
where r is the radius of the circle
So, we can write a simple circle drawing algorithm by solving the equation for y at unit x intervals using:
222 ryx
22 xry
Brute Force Method
20020 220 y
20120 221 y
20220 222 y
61920 2219 y
02020 2220 y
Computational cost: floating point multiplications, subtraction, square-root, rounding
Issues on Brute Force Method
Problems in the brute force method– Involves considerable computations at each steps. Lots of float-
point calculations– Spacing between plotted pixel positions is not uniform
Eliminating unequal spacing– Expressing the circle equation in parametric polar form
x= xc+ r cos(alpha), y= yc+ r sin(alpha) for 0o < alpha < 360o
– For a more continuous boundary, set the step size at 1/r
Reducing computation cost– Considering the symmetry of circles– Still require a good deal of computation time for a octant
Common approach
However, unsurprisingly this is not a brilliant solution!Firstly, the resulting circle has large gaps where the slope approaches the verticalSecondly, the calculations are not very efficient
– The square (multiply) operations– The square root operation – try really hard to
avoid these!
We need a more efficient, more accurate solution
Midpoint Circle Algorithm (1/4)
Only need to draw the arc between (0, r) and (r/sqrt(2),r/sqrt(2))
The coordinates of the other pixels can be implied from the 8-way symmetry rule.
45o
Eight-Way Symmetry
The first thing we can notice to make our circle drawing algorithm more efficient is that circles centred at (0, 0) have eight-way symmetry
(x, y)
(y, x)
(y, -x)
(x, -y)(-x, -y)
(-y, -x)
(y, -x)
(-x, y)
2
R
Mid-Point Circle Algorithm
Similarly to the case with lines, there is an incremental algorithm for drawing circles – the mid-point circle algorithm
In the mid-point circle algorithm we use eight-way symmetry so only ever calculate the points for the top right eighth of a circle, and then use symmetry to get the rest of the points
Midpoint Circle Algorithm),( pp yxP E
SE
M
Choose E as the next pixel if M lies inside the circle, and SE otherwise.
d =d<0: Select E
dnew = d + (2xp+3)
d>0: Select SEdnew = d + (2xp–2yp+5)
22
2
2
1)1(
2
1,1 RyxyxF
xp xp+1
y-1/2
d
r2 = x2+y2
d= F(x,y) = x2+y2-r2
Choosing the Next Pixel
M
(x, y) (x+1, y)
(x+1, y-1)
E
SE
0)2/1,1( yxF
0)2/1,1( yxF
choose SE
choose E
)2/1,1()( yxFMFddecision variable d
Change of d when E is chosen
Mold
(x, y) (x+1, y)
(x+1, y-1)
E
SE
(x+2, y)
(x+2, y-1)
Mnew
32
)2/1()1(
)2/1()2(222
222
xddd
Ryxd
Ryxd
oldnew
old
new
Change of d when SE is chosen
Mold
(x, y) (x+1, y)E
SE
(x+1, y-2) (x+2, y-2)
Mnew
522
)2/1()1(
)2/3()2(222
222
yxddd
Ryxd
Ryxd
oldnew
old
new
Initial value of d
Rd
RRd
RFd
MFd
4/5
)2/1()1(
)2/1,1(
)(
0
2220
0
00
(0,R)
M0
(1,R)
(1,R-1)
Remember f(x+1,y+1/2) when we talked about midpoint line drawing
Midpoint Circle Algorithm (3/4)
Start with P (x = 0, y = R).
x = 0;y = R;d = 5/4 – R; /* real */While (x < y) {
If (d >= 0) // SE is chosen y = y – 1 d = d + 2 * (x – y) + 5
else // E is chosen d = d + 2 * x + 3
x = x+1; WritePixel(x, y)
}
Where is the arch drawn?
New Decision Variable
Our circle algorithm requires arithmetic with real numbers.
Let’s create a new decision variable kdkd=d-1/4
Substitute kd+1/4 for d in the code.
Note kd > -1/4 can be replaced with h > 0 since kd will always have an integer value.
Midpoint Circle Algorithm
Start with P (x = 0, y = R).
x = 0;y = R;kd = 1 – R;While (x < y) {
If (kd >= 0) // SE is chosen y = y – 1 kd = kd + 2 * (x – y) + 5
else // E is chosen kd = kd + 2 * x + 3
x = x+1; WritePixel(x, y)
}
LINE DRAWING with Midpoint Algorithm
x=xL y=yL
d=xH - xL c=yL - yH
sum=2c+d //initial value // decision param sum is multiplied with 2, act val (c+d/2)draw(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)
Compare this with the algorithm presented in the previous slide
EXPLAIN !
Initial value of decision param: sum
What about starting value?
eydxcyxf LLLL )()1(),1( 21
21
dcyxfyxf LLLL 21
21 ),(),1(
(xL+1,yL+1/2)
dcyxfyxf LLLL 21
21 ),(),1(
is on the line!),( LL yx
dcyxf LL 21
21 ),1(
Sum = 2c + d
Drawing circle in OpenGL
OpenGL does not have any primitives for drawing curves or circles. However we can approximate a circle using a triangle fan like this:
glBegin(GL_TRIANGLE_FAN)
glVertex2f(x1, y1)
for angle# = 0 to 360 step 5
glVertex2f(x1 + sind(angle#) * radius#, y1 + cosd(angle#) * radius#)
next
glEnd()
If you want to use mid-point to draw circle
If you calculate the points on circle through the given mid-point algorithm, then use
glBegin(GL_POINTS)
glVertex2f(x1, y1)
glEnd()
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