Editing Differentiable curve (section)

=== Normal vector or curvature vector ===

A curve ''[[normal vector]]'', sometimes called the '''curvature vector''', indicates the deviance of the curve from being a straight line.
It is defined as the [[vector rejection]] of the particle's [[acceleration]] from the tangent direction:
<math display="block">\overline{\mathbf{e}_2}(t) = \boldsymbol{\gamma}''(t) - \bigl\langle \boldsymbol{\gamma}''(t), \mathbf{e}_1(t) \bigr\rangle \, \mathbf{e}_1(t),</math>
where the acceleration is defined as the second derivative of position with respect to time:
<math display="block"> \boldsymbol{\gamma}''(t_0) = \left.\frac{\mathrm{d}^2}{\mathrm{d}t^2}\boldsymbol{\gamma}(t)\right|_{t=t_0} </math>

Its normalized form, the unit normal vector, is the second Frenet vector {{math|'''e'''<sub>2</sub>(''t'')}} and is defined as
<math display="block">\mathbf{e}_2(t) = \frac{\overline{\mathbf{e}_2}(t)} {\left\| \overline{\mathbf{e}_2}(t) \right\|}.</math>

The tangent and the normal vector at point {{math|''t''}} define the [[osculating plane]] at point {{math|''t''}}.

It can be shown that {{math|'''ē'''<sub>2</sub>(''t'') ∝ '''e'''{{prime}}<sub>1</sub>(''t'')}}. Therefore,
<math display="block">\mathbf{e}_2(t) = \frac{\mathbf{e}_1'(t)}{\left\| \mathbf{e}_1'(t) \right\|}.</math>