A is a continous map from γ : [0, 1] → SO(3) ⋊ G where G = ℝ3 or possibly SO(2). In other words, a drawing is a continuous path in a group like the one above. Call the drawing group (the semidirect product) D. Then the complete drawing is the composition [0, 1] → G → ℝ2. Let P be the , i.e. the image in ℝ2 under this map.
The image of γ in D is a 1-dimensional submanifold.
Let γ : [0, 1] → G be a drawing. Let 𝔤 be the Lie algebra of G. To start drawing on the group, choose a direction X ∈ 𝔤 and plot the continuous path t ↦ exp (tX). This is a drawing on G in the direction X.
I do not have anything more to say on this subject at the moment. I need to learn more about Lie theory, geometry and physics to understand systems like this more.