Your chart and hat look lovely!
Heres a link to surface crochet which i posted some time back, it’s a nice example of curves. The same effect can be acheived with chain stitch which @Remi posted. Both methods can create elongated stitches too which can look very nice.
You can also embroider flowers, leaves, swirls, little knots or dots and so on. Methods of producing flower petals and little french knots could perhaps be modified away from the flower look and more like the pulli kolam designs.
There’s a picture and text tutorial here for dots, french knots, in case you want some for the design