Correct me if I am wrong but assume I have a point in 3D which I would like to rotate around all arbitrary axes fixed at common origin. Then this is true that all orbits circled by rotated point will form an empty sphere of radius equal to the distance between point and origin? If I am right could you give me some references. Just in case to explore why my intuition is correct.
2026-03-30 10:38:39.1774867119
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Do rotations of one point around all arbitrary axes form a sphere?
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Consider the sphere centered at the origin and extending to the point. Consider any other point on that sphere. Consider the great circle connecting the two points. Make believe that great circle is the Equator. Then a rotation around the axis through the North and South Poles wil take the one point to the other.
This is quite obvious, for any two points $x,y$ equidistant from a common origin, you can always construct a pencil of rotation axes: they lie on the plane passing through the origin and is perpendicular to $xy$.
This is some basic euclidean geometry stuff, since no affine or projective stuff is involved. If you ask for a reference, I dont know, Felix Klein for sure knew them quite well. Or you could try this simple book "Geometric Concepts for Geometric Design".