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. (Note, 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.)
If a curve is given parametrically by , then the tangential angle at is defined (up to a multiple of ) by
Here, the prime symbol denotes derivative. Thus, the tangential angle specifies the direction of the velocity vector , while the speed specifies its magnitude. The vector is called the unit tangent vector, so an equivalent definition is that the tangential angle at is the angle such that is the unit tangent vector at .
If the curve is parameterized by arc length , so , then the definition simplifies to . In this case, the curvature is given by , where is taken to be positive if the curve bends to the left and negative if the curve bends to the right.
If the curve is given by , then we may take as the parameterization, and we may assume is between and . This produces the explicit expression .
Polar tangential angle 
In polar coordinates, define the polar tangential angle as the angle between the tangent line to the curve at the given point and ray from the origin to the point. If denotes the polar tangential angle, then , where is as above and is, as usual, the polar angle.
If the curve is defined in polar coordinates by , then polar tangential angle at is defined (up to a multiple of ) by
If the curve is parameterized by arc length as , so , then the definition becomes .
See also 
- "Natural Equation" at MathWorld
- For example W. Whewell "Of the Intrinsic Equation of a Curve, and its Application" Cambridge Philosophical Transactions Vol. VIII (1849) pp. 659-671. Google Books uses 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.
- MathWorld "Tangential Angle"
- MathWorld "Natural Equation" differentiating equation 1
- "Logarithmic Spiral" at Planet Math
- Williamson for section unless otherwise noted.
- Weisstein, Eric W., "Tangential Angle", MathWorld.
- Weisstein, Eric W., "Natural Equation", MathWorld.
- "Notations" at Encyclopédie des Formes Mathématiques Remarquables
- R.C. Yates (1952). A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards. pp. 123–126.
- "Angle between Tangent and Radius Vector" in An elementary treatise on the differential calculus By Benjamin Williamson p222 9th ed. (1899) online