Comment by oasisaimlessly

G^n curvature solely depends on the geometry of the curve, while C^n continuity also depends on how you parameterize the curve. So, G^n is what you want if you're talking about a purely geometric shape rather than an (x(t), y(t)) trajectory.

* reference: section 2.1 of https://graphics.stanford.edu/courses/cs348a-21-winter/Reade...