This animation demonstrates the math we will need to keep graphical elements aligned even when the elements are at different insets from the paisley shell.

The premise is that we start with a position on the piecewise curve and then calculate which piece it is on and its position in that piece. We then transpose that information onto the inset curve and evaluate its position.

Unfortunately the result is a failure, because in the spiral segments alignment of θ is different than visual/artistic alignment.