I know that in $\mathbb{R}^2$ we can define the curvature of a parametrized curve $\textbf{x}(t)=\bigl( x(t), y(t)\bigr)$ as $$ \kappa(t) = \dfrac{\text{det}(\textbf{x}',\textbf{x}'')}{||\textbf{x}'||^3}.$$ If I use a change of parameter $\overline{t}=\phi(t)$ we can reparametrize the curvature $$ \boxed{\overline{\kappa}(\overline{t}) =\kappa(t)}\tag{1}.$$ Equation (1) is the definition of a scalar. The curvature is a scalar magnitude.
My question is: Why is the meaning of the next equation, very similar to equation (1)? In particular, what functions verify the next equations? $$ \overline{\kappa}(t)=\kappa(t) .\tag{2}$$ Perhaps it's related to a geodesic. My teacher says that the solution is in a General Relativity book but it can be answered without any physic concept.