Problem understanding the indicatrix tangent definition I'm self studying differential geometry of curves and surfaces from a book that gives the following definition of indicatrix tangent: > Given an arc length parametrized curve $\alpha$, we consider the tangent vector as a curve, $T:I \rightarrow \mathbb{R^2}$. > > $\alpha$ is arc length parametrized, so $T$ is in the unit circle and defines an angle $\theta$ for each $s \in I$. > > $\theta:I \rightarrow \mathbb{R}$ is a differentiable function such that: $T(s)=(\cos(\theta(s)),\sin(\theta(s))) $. It gives no further information about the angle $\theta$, and I don't understand it' s geometric meaning.

* Every smooth vector field induces a flow. Think of a piece of hair with a small force of blowing on it.

* Since you have a smooth vector field, you could consider the flow of a section.

* For instance, consider a small interval for the piece of hair above. The flow will perturb the section into a section which is diffeomorphic. You can call this section the flow section or vector flow curve (since it is induced by the overarching vector field).

* For example, small wind forces shouldn't tear a string of hair

* So this indicatrix thing is just the tangent flow curve.