In the
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 ...
in
three dimensions
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
, the torsion of a
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
measures how sharply it is twisting out of the
osculating plane {{Unreferenced, date=May 2019, bot=noref (GreenC bot)
In mathematics, particularly in differential geometry, an osculating plane is a plane in a Euclidean space or affine space which meets a submanifold at a point in such a way as to have a second ...
. Taken together, the
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
and the torsion of a
space 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 ...
are analogous to the curvature of a
plane curve
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic pla ...
. For example, they are coefficients in the system of
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s for the
Frenet frame given by the
Frenet–Serret formulas
In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space \mathbb^, or the geometric properties of the curve itself irrespective ...
.
Definition
![Torus-Knot uebereinander animated](https://upload.wikimedia.org/wikipedia/commons/6/68/Torus-Knot_uebereinander_animated.gif)
Let be a
space 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 ...
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
* ...
and with 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 ...
. If the
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
of at a certain point is not zero then the
principal normal vector
Principal may refer to:
Title or rank
* Principal (academia), the chief executive of a university
** Principal (education), the office holder/ or boss in any school
* Principal (civil service) or principal officer, the senior management level ...
and the
binormal vector at that point are the unit vectors
:
respectively, where the prime denotes the derivative of the vector with respect to the parameter . The torsion measures the speed of rotation of the binormal vector at the given point. It is found from the equation
:
which means
:
As
, this is equivalent to
.
''Remark'': The derivative of the binormal vector is perpendicular to both the binormal and the tangent, hence it has to be proportional to the principal normal vector. The negative sign is simply a matter of convention: it is a byproduct of the historical development of the subject.
Geometric relevance: The torsion measures the turnaround of the binormal vector. The larger the torsion is, the faster the binormal vector rotates around the axis given by the tangent vector (see
graphical illustrations). In the animated figure the rotation of the binormal vector is clearly visible at the peaks of the torsion function.
Properties
* A plane curve with non-vanishing curvature has zero torsion at all points. Conversely, if the torsion of a regular curve with non-vanishing curvature is identically zero, then this curve belongs to a fixed plane.
* The curvature and the torsion of a
helix
A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, ...
are constant. Conversely, any space curve whose curvature and torsion are both constant and non-zero is a helix. The torsion is positive for a right-handed
helix and is negative for a left-handed one.
Alternative description
Let be the
parametric equation of a space curve. Assume that this is a regular parametrization and that the
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
of the curve does not vanish. Analytically, is a three times differentiable
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
of with values in and the vectors
:
are
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
.
Then the torsion can be computed from the following formula:
:
Here the primes denote 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. F ...
s with respect to and the cross denotes the
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
. For , the formula in components is
:
Notes
References
*
{{curvature
Differential geometry
Curves
Curvature (mathematics)
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