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In
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, the tangential angle of a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
in the Cartesian plane, at a specific point, is the angle between the
tangent line In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
to the curve at the given point and the -axis. (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.)


Equations


Parametric

If a curve is given parametrically by , then the tangential angle at is defined (up to a multiple of ) by : \frac = (\cos \varphi,\ \sin \varphi). Here, the prime symbol denotes the
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
with respect to . Thus, the tangential angle specifies the direction of the
velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
vector , while the
speed In kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a non-negative scalar quantity. Intro ...
specifies its magnitude. The vector :\frac 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 parametrized by
arc length Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a problem in vector calculus and in differential geometry. In the ...
, so , then the definition simplifies to :\big(x'(s),\ y'(s)\big) = (\cos \varphi,\ \sin \varphi). In this case, the
curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
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. Conversely, the tangent angle at a given point equals the definite integral of curvature up to that point: :\varphi(s) = \int_^s \kappa(s) ds + \varphi_0 :\varphi(t) = \int_^t \kappa(t) s'(t) dt + \varphi_0


Explicit

If the curve is given by the
graph of a function In mathematics, the graph of a function f is the set of ordered pairs (x, y), where f(x) = y. In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in a plane (geometry), plane and often form a P ...
, then we may take as the parametrization, and we may assume is between and . This produces the explicit expression :\varphi = \arctan f'(x).


Polar tangential angle

In
polar coordinates In mathematics, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are *the point's distance from a reference ...
, 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. 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 the polar tangential angle at is defined (up to a multiple of ) by : \frac = (\cos \psi,\ \sin \psi). If the curve is parametrized by arc length as , , so , then the definition becomes :\big(r'(s),\ r\theta'(s)\big) = (\cos \psi,\ \sin \psi). The
logarithmic spiral A logarithmic spiral, equiangular spiral, or growth spiral is a self-similarity, self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewi ...
can be defined a curve whose polar tangential angle is constant.


See also

*
Differential geometry of curves Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have been thoroughly investigated using the ...
* Whewell equation * Subtangent


References


Further reading

* * {{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 Analytic geometry Differential geometry