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[[File:Intrinsic coordinates (Whewell equation).png|thumb|The tangential angle {{mvar|φ}} for an arbitrary curve {{mvar|P}}.]]
In [[Geometry|geometry]], the '''tangential angle''' of a [[Curve|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 {{mvar|x}}-axis.<ref name="mwne">{{MathWorld|NaturalEquation|Natural Equation}}</ref> (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.<ref>For example: {{cite journal|first=W. |last=Whewell |title=Of the Intrinsic Equation of a Curve, and Its Application |journal=Cambridge Philosophical Transactions |volume=8 |date=1849 |pages=659–671 |url=https://books.google.com/books?id=2vsIAAAAIAAJ&pg=PA659}} This paper uses {{mvar|φ}} to mean the angle between the tangent and tangent at the origin. This is the paper introducing the Whewell equation, an application of the tangential angle.</ref>)

== Equations ==
If a curve is given [[Parametric equation|parametrically]] by {{math|(''x''(''t''), ''y''(''t''))}}, then the tangential angle {{mvar|φ}} at {{mvar|t}} is defined (up to a multiple of {{math|2π}}) by<ref>{{MathWorld|TangentialAngle|Tangential Angle}}</ref>

: <math>\frac{\big(x'(t),\ y'(t)\big)}{\big|x'(t),\ y'(t)\big|} = (\cos \varphi,\ \sin \varphi).</math>

Here, the [[Prime (symbol)|prime symbol]] denotes the [[Derivative|derivative]] with respect to {{mvar|t}}. Thus, the tangential angle specifies the direction of the [[Physics:Velocity|velocity]] vector {{math|(''x''(''t''), ''y''(''t''))}}, while the [[Physics:Speed|speed]] specifies its magnitude. The vector

:<math>\frac{\big(x'(t),\ y'(t)\big)}{\big|x'(t),\ y'(t)\big|}</math>

is called the unit tangent vector, so an equivalent definition is that the tangential angle at {{mvar|t}} is the angle {{mvar|φ}} such that {{math|(cos ''φ'', sin ''φ'')}} is the unit tangent vector at {{mvar|t}}.

If the curve is parametrized by [[Arc length|arc length]] {{mvar|s}}, so {{math|{{abs|''x''′(''s''), ''y''′(''s'')}} {{=}} 1}}, then the definition simplifies to

:<math>\big(x'(s),\ y'(s)\big) = (\cos \varphi,\ \sin \varphi).</math>

In this case, the [[Curvature|curvature]] {{mvar|κ}} is given by {{math|''φ''′(''s'')}}, where {{mvar|κ}} is taken to be positive if the curve bends to the left and negative if the curve bends to the right.<ref name="mwne" />
Conversely, the tangent angle at a given point equals the definite integral of curvature up to that point:<ref name="Surazhsky2004">{{cite conference | last1=Surazhsky | first1=Tatiana | last2=Surazhsky | first2=Vitaly | title=Sampling planar curves using curvature-based shape analysis |conference= Mathematical methods for curves and surfaces | year = 2004 |place=Tromsø | isbn=978-0-9728482-4-4 | citeseerx=10.1.1.125.2191 }}</ref><ref name="mwne" />
:<math>\varphi(s) = \int_{0}^s \kappa(s) ds + \varphi_0</math>
:<math>\varphi(t) = \int_{0}^t \kappa(t) s'(t) dt + \varphi_0</math>

If the curve is given by the [[Graph of a function|graph of a function]] {{math|''y'' {{=}} ''f''(''x'')}}, then we may take {{math|(''x'', ''f''(''x''))}} as the parametrization, and we may assume {{mvar|φ}} is between {{math|−{{sfrac|π|2}}}} and {{math|{{sfrac|π|2}}}}. This produces the explicit expression

:<math>\varphi = \arctan f'(x).</math>

== {{anchor|Polar}}Polar tangential angle<ref name="williamson" /> ==
In [[Polar coordinate system|polar coordinates]], the '''polar tangential angle''' is defined as the angle between the tangent line to the curve at the given point and ray from the origin to the point.<ref name="Planet">{{PlanetMath|urlname=LogarithmicSpiral|title=Logarithmic Spiral}}</ref> If {{mvar|ψ}} denotes the polar tangential angle, then {{math|''ψ'' {{=}} ''φ'' − ''θ''}}, where {{mvar|φ}} is as above and {{mvar|θ}} is, as usual, the polar angle.

If the curve is defined in polar coordinates by {{math|''r'' {{=}} ''f''(''θ'')}}, then the polar tangential angle {{mvar|ψ}} at {{mvar|θ}} is defined (up to a multiple of {{math|2π}}) by

: <math>\frac{\big(f'(\theta),\ f(\theta)\big)}{\big|f'(\theta),\ f(\theta)\big|} = (\cos \psi,\ \sin \psi)</math>.

If the curve is parametrized by arc length {{mvar|s}} as {{math|''r'' {{=}} ''r''(''s'')}}, {{math|''θ'' {{=}} ''θ''(''s'')}}, so {{math|{{abs|(''r''′(''s''), ''rθ''′(''s''))}} {{=}} 1}}, then the definition becomes

:<math>\big(r'(s),\ r\theta'(s)\big) = (\cos \psi,\ \sin \psi)</math>.

The [[Logarithmic spiral|logarithmic spiral]] can be defined a curve whose polar tangential angle is constant.<ref name="williamson">{{cite book|chapter=Angle between Tangent and Radius Vector |title=An Elementary Treatise on the Differential Calculus |first=Benjamin |last=Williamson |page=222 |edition=9th |date=1899 |chapter-url=https://books.google.com/books?id=vYkAAAAAMAAJ&pg=PA222}}</ref><ref name="Planet" />

== See also ==
* [[Differential geometry of curves]]
* [[Whewell equation]]
* [[Subtangent]]

== References ==
{{reflist}}

=== Further reading ===
* {{cite web|language=French|url=http://mathcurve.com/courbes2d/courbes2d.shtml#notations |title=Notations |website=Encyclopédie des Formes Mathématiques Remarquables}}
* {{cite book | first=R. C. |last=Yates | title=A Handbook on Curves and Their Properties | location=Ann Arbor, MI | publisher=J. W. Edwards | pages=123–126 | year=1952 }}

[[Category:Analytic geometry]]
[[Category:Differential geometry]]

{{Sourceattribution|Tangential angle}}