Tangential angle

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In geometry, the tangential angle of a curve in the Cartesian plane, at a specific point, is the angle between the tangent line to the curve at the given point and the x-axis. More specifically, if a curve is given parametrically by $(x (t), y (t))$ then the tangential angle $\varphi$ at $t$ is defined (up to a multiple of $2π$ ) by

$\frac{(x'(t), y'(t))}{|x'(t), y'(t)|} = (\cos \varphi, \sin \varphi).$

Thus the tangential angle specifies the direction of the velocity vector $(x'(t), y'(t))$ while the speed specifies its magnitude. The vector $\frac{(x'(t), y'(t))}{|x'(t), y'(t)|}$ is called the unit tangent vector, so an equivalent definition is that the tangential angle at $t$ is the angle $\varphi$ such that $(\cos \varphi, \sin \varphi)$ is the unit tangent vector at $t$ .

If the curve is parameterized by arc length $s$ , so $| x'(s), y'(s) | = 1$ , then the definition simplifies to $(x'(t), y'(t)) = (\cos \varphi, \sin \varphi)$ . In this case the curvature $κ$ is given by $\varphi'(s)$ where $κ$ is taken to be positive is the curve bends to the left and negative if the curve bends to the right.

If the curve is given by $y = f (x)$ then we may take $(x, f (x))$ as the parameterization and we may assume $\varphi$ is between $- π / 2$ and $π / 2$ . This produces the explicit expression $\varphi = \arctan f'(x)$ .