1. Circle Drawing Asst. Prof. Dr. Ahmet Sayar Kocaeli University
Computer Engineering
Advanced Computer Graphics
Spring 2012

2. Properties of Circle
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

3. 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:

4. Brute Force Method
Computational cost: floating point multiplications, subtraction, square-root, rounding

5. 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

6. Common approach
However, unsurprisingly this is not a brilliant solution!
Firstly, the resulting circle has large gaps where the slope approaches the vertical
Secondly, 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

7. 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

8. 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)
(-x, y)

9. 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

10. Midpoint Circle Algorithm
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 SE
dnew = d + (2xp–2yp+5)
y-1/2 M SE d xp
xp+1
r2 = x2+y2
d= F(x,y) = x2+y2-r2

11. Choosing the Next Pixel
decision variable d M
(x, y)
(x+1, y)
(x+1, y-1) E SE
choose SE
choose E

12. 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

13. Change of d when SE is chosen
(x, y)
(x+1, y) E
Mold SE
Mnew
(x+1, y-2)
(x+2, y-2)

14. Initial value of d (0,R) (1,R) M0 (1,R-1)
Remember f(x+1,y+1/2) when we talked about midpoint line drawing M0
(1,R-1)

15. 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?

16. New Decision Variable
Our circle algorithm requires arithmetic with real numbers.
Let’s create a new decision variable kd
kd=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.

17. 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) }

18. 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 point if ( sum < 0 ) // below the line and choose NE sum += 2d y++ x++ sum += 2c
Compare this with the algorithm presented in the previous slide
EXPLAIN !

19. Initial value of decision param: sum
What about starting value?
(xL+1,yL+1/2)
is on the line!
Sum = 2c + d

20. 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()

21. 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()