EMILY: Lorelai, your daughter’s being impossible. She won’t pose in an appropriate manner.
RORY: I’m trying to, Grandma. It’s just awkward.
Rory goes to her grandmother’s house after school to pose for her portrait, but it turns out this means sitting on a “throne” (a needlework armchair) in a red velvet dress with a hungry honking swan, and her arm raised in an odd pose. The living room portrait is quite normal, so it’s strange that Emily suddenly wants this bizarre “swan princess” picture of Rory. It might explain why Lorelai’s own portrait was never completed. Rory is already looking quite grumpy, just as Lorelai got fed up with being painted.
Lorelai is able to quickly convince Emily that a picture of Rory sitting and reading would be natural and appropriate, saving the day.