In
geometry, 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 the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
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
If a curve is given
parametrically
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
by , then the tangential angle at is defined (up to a multiple of ) by
:
Here, the
prime symbol
The prime symbol , double prime symbol , triple prime symbol , and quadruple prime symbol are used to designate units and for other purposes in mathematics, science, linguistics and music.
Although the characters differ little in appearance fr ...
denotes the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
with respect to . Thus, the tangential angle specifies the direction of the
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
vector , while the
speed
In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (ma ...
specifies its magnitude. The vector
:
is called the
unit tangent vector
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (al ...
, 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 may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
, 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.
Conversely, the tangent angle at a given point equals the definite integral of curvature up to that point:
:
:
If the curve is given by the
graph of a function , then we may take as the parametrization, and we may assume is between and . This produces the explicit expression
:
Polar tangential angle
In
polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
, 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
:
.
If the curve is parametrized by arc length as , , so , then the definition becomes
:
.
The
logarithmic spiral
A logarithmic spiral, equiangular spiral, or growth spiral is a 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" ("ewige Linie"). More ...
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