When a definition applies to a much larger class of spaces $\ X\subseteq \mathbb R^3,\ $ then such definition rather is not directly related to any parametrization for the smaller class of the parametrized curves (which can be rectified). Several of variants of Hausdorff dimension for dimension 1 would be a possible answer. Another could be

$$\lim_{r\rightarrow +0}\ \frac {V(X+B(r))}{\pi\cdot r^2}$$

where $\ X+B(r)\ =\ \{y\in\mathbb R^3:\ \exists_{x\in X}\ d(x\ y)\le r\},\ \ d\ $ is the euclidean distance, and $\ V\ $ stands for volume. This will certainly work properly for the piecewise $\ C^1$-curves, but most likely for all rectifiable curve; and of course--by definition--for many other spaces in the Euclidean space. Again, variations of this definition are possible, e.g. using cubes with edges $\ \left[\frac k{2^n};\frac{k+1}{2^n}\right],\ $ etc.

When a definition applies to a much larger class of spaces $\ X\subseteq \mathbb R^3,\ $ then such definition rather is not directly related to any parametrization for the smaller class of the parametrized curves (which can be rectified). Several of variants of Hausdorff dimension for dimension 1 would be a possible answer. Another could be

$$\lim_{r\rightarrow +0}\ \frac {V(X+B(r))}{\pi\cdot r^2}$$

where $\ X+B(r)\ =\ \{y\in\mathbb R^3:\ \exists_{x\in X}\ d(x\ y)\le r\},\ \ d\ $ is the euclidean distance, and $\ V\ $ stands for volume (see also, optionally, the detailed terminology below). This will certainly work properly for the piecewise $\ C^1$-curves, but most likely for all rectifiable curve (see the justification below); and of course--by definition--for many other spaces in the Euclidean space. Again, variations of this definition are possible, e.g. using cubes with edges $\ \left[\frac k{2^n};\frac{k+1}{2^n}\right],\ $ etc.

TERMINOLOGY

• $\ B(r)\ :=\ \{x\in\mathbb R^3: |x|\le r\}\ $ is the closed ball of radius $\ r,\ $ centered at the origin of $\ \mathbb R^3$;
• $\ X+Y\ :=\, \ \{x+y:\ x\in X\ \ \&\ \ y\in Y\}\quad $ for $\ X\ Y\subseteq\mathbb R^3$.

JUSTIFICATION of the volume formula for the length

In the case of a finitely piece-wise linear curve, the above volume is a sum of the respective cylinders around the intervals plus/minus a negligible error when the radius approaches $\ 0.\ $ The general case of rectifiable curves is obtained by $\ \epsilon/\delta\ $ (:-) which I am ready to provide if asked to.

When a definition applies to a much larger class of spaces $\ X\subseteq \mathbb R^3,\ $ then such definition rather is not directly related to any parametrization for the smaller class of the parametrized curves (which can be rectified). Several of variants of Hausdorff dimension for dimension 1 would be a possible answer. Another could be

$$\lim_{r\rightarrow +0}\ \frac {V(X+B(r))}{\pi\cdot r^2}$$

where $\ X+B(r)\ =\ \{y\in\mathbb R^3:\ \exists_{x\in X}\ d(x\ y)\le r\},\ \ d\ $ is the euclidean distance, and $\ V\ $ stands for volume. This will certainly work properly for the piecewise $\ C^1\ $ curve, but most liely for all rectifiable curve; and of course--by definition--for many other spaces in the Euclidean space. Again, variations of this definition are possible, e.g. using cubes with edges $\ \left[\frac kn;\frac{k+1}n\right],\ $ etc.

When a definition applies to a much larger class of spaces $\ X\subseteq \mathbb R^3,\ $ then such definition rather is not directly related to any parametrization for the smaller class of the parametrized curves (which can be rectified). Several of variants of Hausdorff dimension for dimension 1 would be a possible answer. Another could be

$$\lim_{r\rightarrow +0}\ \frac {V(X+B(r))}{\pi\cdot r^2}$$

where $\ X+B(r)\ =\ \{y\in\mathbb R^3:\ \exists_{x\in X}\ d(x\ y)\le r\},\ \ d\ $ is the euclidean distance, and $\ V\ $ stands for volume. This will certainly work properly for the piecewise $\ C^1$-curves, but most likely for all rectifiable curve; and of course--by definition--for many other spaces in the Euclidean space. Again, variations of this definition are possible, e.g. using cubes with edges $\ \left[\frac k{2^n};\frac{k+1}{2^n}\right],\ $ etc.