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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.[1] (Note that some authors define the angle as the deviation from the direction of the curve at some fixed starting point. This is equivalent to the definition given here by the addition of a constant to the angle or by rotating the curve.[2])

Equations

If a curve is given parametrically by (x(t), y(t)), then the tangential angle φ at t is defined (up to a multiple of ) by[3]

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

Here, the prime symbol denotes the derivative with respect to t. Thus, the tangential angle specifies the direction of the velocity vector (x(t), y(t)), while the speed specifies its magnitude. The vector

${\displaystyle {\frac {{\big (}x'(t),\ y'(t){\big )}}{{\big |}x'(t),\ y'(t){\big |}}}}$