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Second region condition, ellipse midpoint

Short answer

The second region is identified when 2·ry²·x ≥ 2·rx²·y, meaning the slope magnitude is at least 1. Region 1 continues while 2ry²x < 2rx²y; otherwise switch to region 2 where y is stepped.

State the condition to identify that you are in the second region of the ellipse using the midpoint algorithm.

In the midpoint ellipse algorithm, the slope of the ellipse at (x, y) is dy/dx = −(ry²x)/(rx²y). We are in the second region when the slope magnitude becomes 1 or more, i.e. when

2·ry²·x ≥ 2·rx²·y

The loop for region 1 runs while 2ry²x < 2rx²y; once this is false, the algorithm switches to region 2 (y is then the unit-step direction).

✓ Answer checked by Claude Opus 5.5 (high effort)

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