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axis or optical center
Yes, the axis of the Earth passes through its centre.
axis - An imaginary line that passes through earths center and its north and south poles.
The transverse axis is a connection on a hyperbola. It connects the focus, or center, of the hyperbola, and can connect two together.
(1/2) mr2, assuming the axis of rotation goes through the center, and along the axis of symmetry.