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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.

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  • 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.

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