Process of drawing an ellipse

An ellipse with centre (xc, yc) and semi-axes rx and ry has the equation (x−xc)²/rx² + (y−yc)²/ry² = 1. In a raster system, it is drawn by scan conversion, using the midpoint ellipse algorithm (the direct equation and polar methods need square roots or trigonometry and are slower).

Steps

1. Use symmetry. Shift the ellipse to the origin and compute only the first quadrant; each point (x, y) gives (±x, ±y), then add the centre (xc, yc).
2. Split the quadrant into two regions where the tangent slope is −1: region 1 where 2ry²x < 2rx²y, region 2 elsewhere.
3. Region 1: start at (0, ry). Step x by 1, and use the decision parameter p1 = ry² − rx²ry + ¼rx² to choose between y and y − 1.
4. Region 2: start from the last point of region 1. Step y by −1 and use p2 = ry²(x+½)² + rx²(y−1)² − rx²ry² to choose between x and x + 1, until y = 0.
5. Plot the four symmetric points at each step.

        y
        |   region 1  (x steps)
   ry --*--*--*
        |       *
        |         *   <- slope = −1
        |           *
        |  region 2  *  (y steps)
        +-------------*---- x
                      rx