= Geodesic endpoints in Minkowski conformal compactification
A complete inertial <timelike geodesic> in <Minkowski spacetime> has $\mathbf x=\mathbf b+\mathbf vt$ with $|\mathbf v|<1$. Under $p=\arctan(t-r)$, $q=\arctan(t+r)$, both $p,q$ tend to the same endpoint, giving <past timelike infinity> and <future timelike infinity>. A complete <null geodesic> has one finite limiting null coordinate at each end and reaches <past null infinity> and <future null infinity>. Complete <spacelike geodesics> have $r>|t|$ asymptotically and both ends approach <spacelike infinity>. The endpoints have finite compactified coordinates but occur at infinite physical <affine parameter>. A <conformal transformation> preserves null paths as unparametrized geodesics; it does not generally preserve timelike or spacelike geodesic paths.
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