Do ants know the direct (approximately shortest) path to their nest from the food morsel?
Terms that may help you research are dead reckoning or path integration. From Wikipedia:

In navigation, dead reckoning is the process of calculating one's current position by using a previously determined position, or fix, by using estimations of speed and course over elapsed time. The corresponding term in biology, used to describe the processes by which animals update their estimates of position or heading, is path integration.

There are many animals and certainly some species of ants that use dead reckoning/path integration for navigation. Here are some papers about the Cataglyphis genus by one scientist and some colleagues:
Wehner, R., &amp; Räber, F. (1979). Visual spatial memory in desert ants, Cataglyphis bicolor (Hymenoptera: Formicidae). Experientia, 35(12), 1569-1571.
Wehner, R., &amp; Wehner, S. (1986). Path integration in desert ants. Approaching a long-standing puzzle in insect navigation. Monitore Zoologico Italiano-Italian Journal of Zoology, 20(3), 309-331.
Müller, M., &amp; Wehner, R. (1994). The hidden spiral: systematic search and path integration in desert ants, Cataglyphis fortis. Journal of Comparative Physiology A, 175(5), 525-530.
Collett, M., Collett, T. S., Chameron, S., &amp; Wehner, R. (2003). Do familiar landmarks reset the global path integration system of desert ants?. Journal of experimental biology, 206(5), 877-882.
The last one on the list is particularly relevant because it seems to show that the ants are not using landmarks as markers of position, because moving landmarks doesn't alter their paths.
My intuition is that the extent of path integration will depend on the environment. Species living in a fairly &quot;flat&quot; terrain can gain a lot from knowing the direction towards their home. In a tree canopy, however, the direction &quot;home&quot; doesn't matter a whole lot if your path is constrained by paths down foliage, etc.
Therefore it's unlikely to be able to make any general statement about ants as a whole group, and better to investigate individual genera/species.
However, the Cataglyphis genus from the papers above live in a desert, and their dead reckoning abilities seem to be the ones most studied (there are many many more papers than the ones I pasted here, try searching Google Scholar for &quot;path integration&quot; ant or &quot;path integration&quot; Cataglyphis).