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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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    $\begingroup$ 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? $\endgroup$ Jun 27, 2013 at 10:15
  • $\begingroup$ For example, the normalization of the moment curve $(1, t,\cdots, t^{n-1})$. $\endgroup$
    – Jiange Li
    Jul 12, 2013 at 16:55

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As stated in the comments, there is no chance to get a reasonable answer to the general question. However, since you are interested in the moment curve, there is the following relation.

Suppose $p$ is the portion of the area of $\mathbb{S}^{n-1}$ swept by equatorial spheres. Then $p$ is the probability that a random polynomial of degree $n-1$ has a real root. This statement follows from the argument in "How many zeros of a random polynomial are real?" by Alan Edelman and Eric Kostlan.

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