Generation of a long process of a CNS neuron, that carries efferent (outgoing) action potentials from the cell body in spinal cord towards target cells in the thalamus. This axonal process is a member of those that make up the trigeminothalamic tract, one of the major routes of nociceptive and temperature signaling from the face. [ GOC:jid https://www.worldcat.org/search?q=bn%3A0878937420 GOC:dph http://purl.obolibrary.org/obo/go/references/0000021 GOC:dgh GOC:cls ]