What you are looking at

The circangle is the quarter ellipse with legs a and b and the arc as its hypotenuse. With a = 1, its arc length is exactly E(k), always longer than the straight chord and shorter than the two legs together.

The yellow line is the book’s tangential definition of the ellipse. For every point P on the arc there is a line from the origin O that meets the tangent at P at a right angle, at the foot F. Its length is , the support function. On the circle F and P coincide and p is the radius at every point; for any other k the right angle moves off the curve, and keeping it there draws the ellipse exactly.

The lower chart follows the two complete elliptic integrals as k rises. Both start at K(0), the circangle quarter turn, so the circle’s π is just . After that they part: E(k) falls to 1 as the arc flattens onto the leg, and K(k) climbs without limit.