1. Andy Johnson's CS 488 Course Notes, Lecture 4
  2. Foley, Van Dam, Feiner, and Hughes, "Computer Graphics - Principles and Practice", Chapter XX

Drawing Circles

The algorithm used to draw circles is very similar to the Midpoint Line algorithm.

8 way-symmetry - for a circle centered at (0,0) and given that point (x,y) is on the circle, the following points are also on the circle:

(-x, y)
( x,-y)
( y, x)
(-y, x)
( y,-x)

So it is only necessary to compute the pixels for 1/8 of the circle and then simply illuminate the appropriate pixels in the other 7/8.

given a circle centered at (0,0) with radius R:

We choose to work in the 1/8 of the circle (45 degrees) from x=0 (y-axis) to x = y = R/sqrt(2) (45 degrees clockwise from the y-axis.)

so for any point (Xi,Yi) we can plug Xi,Yi into the above equation and

Given that we have illuminated the pixel at (Xp,Yp) we will next either illuminate

We again create a decision variable d set equal to the function evaluated at the midpoint (Xp+ 1,Yp - 0.5)

d = F(Xp + 1,Yp - 0.5)
We plug the midpoint into the above F() for the circle and see where the midpoint falls in relation to the circle.

dcurrent = F(Xp + 1, Yp - 0.5)
dcurrent = (Xp + 1)^2 + (Yp - 0.5)^2 - R^2
dcurrent = Xp^2 + 2Xp + 1 + Yp^2 - Yp + 0.25 - R^2

if the EAST pixel is chosen then:

if the SOUTHEAST pixel is chosen then:

Unlike the algorithm for lines, deltaE and deltaSE are not constant.

initial point (Xo,Yo) is known to be (0,R)
so initial M is at (1, R - 0.5)
so initial d = F(1, R - 0.5)

unfortunately while deltaSE and deltaE are integral, d is still a real variable, not an integer so:

but since h starts off as an integer (assuming an integral R) and h is only incremented by integral amounts (deltaE and deltaSE) we can ignore the 0.25 and compare h < 0.

The full algorithm ( was ? ) given (in C) in the red book ( version ??? ) as program 3.4 on p.83.
The full algorithm is given (in C) in the white book as figure 3.16 on p.86.

This is still somewhat bad in that there is a multiplication to compute the new value of the decision variable. The book shows a more complicated algorithm which does this multiplication only once.

ellipses F(x,y) = b^2 X^2 + a^2 Y^2 - a^2 b^2 = 0 are handled in a similar manner, except that 1/4 of the ellipse must be dealt with at a time and that 1/4 must be broken into 2 parts based on where the slope of the tangent to the ellipse is -1 (in first quadrant.)

Filling Circles

Circles have the advantage of being convex polygons, so each scan line will have a single span.

The span extrema can be computed using the same method as was used to draw the pixels on the boundary of the circle, computing them for 1/8 of the circle and using 8-way symetry to get the other 7/8. These extrema can then be stored in an array of Xmin and Ymin indexed by Y