Tangential angle

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'(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))} 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 kappa is given by varphi'(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'(x).

ee also

* Differential geometry of curves

References

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