Ellipse symmetry and drawing algorithm

Symmetry in ellipse drawing

An ellipse centred at the origin is symmetric about the x-axis and the y-axis. So if (x, y) is a point on the ellipse, then (−x, y), (x, −y) and (−x, −y) are also on it. This four-way symmetry means only the first quadrant needs to be calculated, and every calculated point gives 4 pixels, reducing the computation to one fourth. (Unlike a circle, an ellipse has no symmetry about y = x.)

Midpoint ellipse algorithm (rx, ry, centre (xc, yc))

The first quadrant is divided into region 1 (slope magnitude < 1, x is the unit step) and region 2 (slope magnitude ≥ 1, y is the unit step). The boundary is where 2ry²x = 2rx²y. The ellipse function f(x, y) = ry²x² + rx²y² − rx²ry² is tested at the midpoint of the two candidate pixels.

Input rx, ry and centre (xc, yc).
Region 1 (slope magnitude < 1):
  x = 0, y = ry
  p1 = ry² − rx²·ry + rx²/4
  while 2·ry²·x < 2·rx²·y:
      plot the 4 symmetric points of (x, y)
      x = x + 1
      if p1 < 0:
          p1 = p1 + 2·ry²·x + ry²
      else:
          y = y − 1
          p1 = p1 + 2·ry²·x − 2·rx²·y + ry²
Region 2 (slope magnitude ≥ 1), start from last (x, y):
  p2 = ry²(x + ½)² + rx²(y − 1)² − rx²·ry²
  while y ≥ 0:
      plot the 4 symmetric points of (x, y)
      y = y − 1
      if p2 > 0:
          p2 = p2 − 2·rx²·y + rx²
      else:
          x = x + 1
          p2 = p2 + 2·ry²·x − 2·rx²·y + rx²
Each plotted point is shifted by (xc, yc): (xc±x, yc±y).