Tangential angle

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 $2pi$) by

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

Thus the tangential angle specifies the direction of the velocity vector $(x\text{'}(t),\; y\text{'}(t))$ while the speed specifies its magnitude. The vector $frac\{(x\text{'}(t),\; y\text{'}(t))\}$ 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 $|x\text{'}(s),\; y\text{'}(s)|\; =\; 1$, then the definition simplifies to $(x\text{'}(t),\; y\text{'}(t))\; =\; (cos\; varphi,\; sin\; varphi)$. In this case the curvature $kappa$ is given by $varphi\text{'}(s)$ where $kappa$ 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 $-pi/2$ and $pi/2$. This produces the explicit expression $varphi\; =\; arctan\; f\text{'}(x)$.

ee also

* Differential geometry of curves

References

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