Symmetry property of a circle

A circle centred at the origin is symmetric about both axes and about the lines y = x and y = −x. So if (x, y) is a point on the circle, the following 8 points also lie on it:

(x, y)  (y, x)  (−x, y)  (−y, x)
(x, −y) (y, −x) (−x, −y) (−y, −x)

For a circle centred at (xc, yc), the points are (xc ± x, yc ± y) and (xc ± y, yc ± x).

Use in computation

• Only the points of one octant (from x = 0 to x = y, i.e. a 45° arc) need to be calculated; the other 7 octants are obtained by changing signs and swapping x and y.
• The expensive computation (decision parameter, square root or trigonometric functions) is therefore done for only 1/8 of the circle, reducing the work by about 8 times.
• Each point calculated in this one octant gives 8 pixels.

Example: for r = 5 only the 4 octant points (0,5), (1,5), (2,5), (3,4) are calculated; symmetry then gives the whole circle of 28 distinct pixels (the 4 calculated points plus 24 more).