carceral (adj.)youdaoicibaDictYouDict
"pertaining to prisons or a prison," 1570s, from Latin carceralis, from carcer "prison, jail; starting place in a race course" (see incarceration).
incarcerate (v.)youdaoicibaDictYouDict
1550s, a back-formation from incarceration, or else from Medieval Latin incarceratus, past participle of incarcerare "to imprison" (see incarceration). Related: Incarcerated; incarcerating.
incarceration (n.)youdaoicibaDictYouDict
early 15c., "retention of pus," from Medieval Latin incarcerationem (nominative incarceratio), noun of action from past participle stem of incarcerare "to imprison," from in- "in" (see in- (2)) + carcer "prison, an enclosed space," from Proto-Italic *kar-kr(o)-, of uncertain origin.
It seems best to connect carcer with other IE words for 'circle, round object', such as Latin. curvus, Gr. κιρκος 'ring', OIc. hringr, although not all of these have a good IE etymology. The reduplication in Latin carcer could be iconic; thus, the original meaning would have been 'enclosure'. [de Vaan]