It was established by TMA, @WillSawin, and @DouglasZare, in their responses to the MO question, "Thales' semicircle theorem in higher dimensions," that the natural generalization of Thales' semicircle theorem to $\mathbb{R}^3$ is false.

Q. Is there any natural measure/quantity that is preserved as a point $p$ varies over a unit hemisphere centered at $o$, and somehow projects a geometric figure on the base unit circle of the hemisphere?

It is a bit surprising to me that the straightforward generalizations fail. One obvious and useless answer to Q is that $||p-o||=1$. Perhaps Q is unanswerable, but I have the sense that something more substantive than the radius should be preserved...