#jsDisabledContent { display:none; } My Account | Register | Help

# Tangential angle

Article Id: WHEBN0015785733
Reproduction Date:

 Title: Tangential angle Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Tangential angle

The tangential angle \varphi for an arbitrary curve P

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, 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 \varphi at t is defined (up to a multiple of 2\pi) by[3]

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

Here, the prime symbol denotes derivative. 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))}{|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'(s),\ y'(s)) = (\cos \varphi,\ \sin \varphi). In this case, the curvature \kappa is given by \varphi'(s), where \kappa is taken to be positive if the curve bends to the left and negative if the curve bends to the right.[4]

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).

## 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.[5] If \psi denotes the polar tangential angle, then \psi = \varphi - \theta, where \varphi is as above and \theta is, as usual, the polar angle.

If the curve is defined in polar coordinates by r = f(\theta), then polar tangential angle \psi at \theta is defined (up to a multiple of 2\pi) by

\frac{(f'(\theta),\ f(\theta))}{|f'(\theta),\ f(\theta)|} = (\cos \psi,\ \sin \psi).

If the curve is parameterized by arc length s as r = r(s),\ \theta = \theta(s), so |r'(s),\ r\theta'(s)| = 1, then the definition becomes (r'(s),\ r\theta'(s)) = (\cos \psi,\ \sin \psi).

The logarithmic spiral can be defined a curve whose polar tangential angle is constant.[5][6]

## References

1. ^ "Natural Equation" at MathWorld
2. ^ 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.
3. ^ MathWorld "Tangential Angle"
4. ^ MathWorld "Natural Equation" differentiating equation 1
5. ^ a b Planet Math"Logarithmic Spiral" at
6. ^ Williamson for section unless otherwise noted.
• Weisstein, Eric W., "Tangential Angle", MathWorld.
• Weisstein, Eric W., "Natural Equation", MathWorld.
• Encyclopédie des Formes Mathématiques Remarquables"Notations" at
• "Angle between Tangent and Radius Vector" in An elementary treatise on the differential calculus By Benjamin Williamson p222 9th ed. (1899) online
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.

Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.