In fact it's even more complicated if we want to get deeper :
1) first Pocci's completely right about the projection : the ground track of a descending pattern alongside a cone would be a portion of a spiral
BUT... (because there's a but)
2) in fact there are not two but THREE origins : the first documents mentionning the HAC show for some of them the cone... but some of them show a cylinder... and the projection of a cylinder is indeed a circle !
So there are in fact THREE HACs : Heading Alignment Circle/Cone/Cylinder !
(although cylinder is not anymore in the official terminology of NASA)
3) and the HORIZ SIT display that we all know well shows a circle which has a changing radius, which is then a portion of a spiral.
If you have a look at the DPS dictionnary for example (go to page 3-66) they also mention a "Heading Alignment CIRCLE" there.