# 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.

## 1 Answer

• 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.