>will the ray intersect the curve 10% along it's length? It can't. The line and the curve are the same percentage of the way through at three points - the midp

>will the ray intersect the curve 10% along it's length? It can't. The line and the curve are the same percentage of the way through at three points - the midpoint and both ends. Near the midpoint, the line is shorter than the curve per unit of arc, since it's basically parallel to the curve and closer to the center. Near the end, the line is longer per unit arc, since they're the same distance from the center but the line is slanted at an angle. So if you plot out length travelled versus angle, you get two plots. The curve goes straight up-and-to-the-right, while the line will race above to start, peter out, cross over at the midpoint, and then catch back up at the end.