For a point on the unit sphere, we know the plane perpendicular to the line through the origin and the point cuts the sphere with a big circle. When the point moves along a sphere curve, the corresponding big circle will rotate. What is the area swept out these big circles? (Probably a point in the region swept out will be contained by more than one big circles, but we only count it once when estimate the area. If we count the multiplicity, the area is easy to get, which is promotional to length of the curve.)
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