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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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I think it will be very difficult to answer this problem in general because it depends so heavily on the curve. Is there a specific class of curves that you are interested in? –  Maarten Derickx Jun 27 '13 at 10:15
    
For example, the normalization of the moment curve $(1, t,\cdots, t^{n-1})$. –  Jiange Li Jul 12 '13 at 16:55

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