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In the elementary
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 synthe ...
in
three dimensions Three-dimensional space (also: 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 informal meaning of the t ...
, the torsion 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 app ...
measures how sharply it is twisting out of the plane of curvature. 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 canonica ...
and the torsion of a space curve are analogous to 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 canonica ...
of a plane curve. For example, they are coefficients in the system of
differential equation In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the diffe ...
s for the
Frenet frame
Frenet frame
given by the
Frenet–Serret formulas spanned by T and N In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a continuous, differentiable curve in three-dimensional Euclidean space ℝ3, or the geometric properties of the ...
.


Definition

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 app ...

space curve
parametrized by
arc length Arc length is the distance between two points along a section of a curve. Determining the length of an irregular arc segment is also called of a curve. The advent of infinitesimal calculus led to a general formula that provides closed-form solu ...

arc length
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'' (album ...
. 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 canonica ...
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 head teacher of a primary or secondary school * Principal (civil service) or principal officer, the senior management ...
and the binormal vector at that point are the unit vectors : \mathbf=\frac, \quad \mathbf=\mathbf\times\mathbf, 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 : \mathbf' = -\tau\mathbf. which means : \tau = -\mathbf\cdot\mathbf'. ''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 The right-handed helix (cos ''t'', sin ''t'', ''t'') from ''t'' = 0 to 4π with arrowheads showing direction of increasing ''t'' A helix (), plural helixes or helices (), is a shape like a corkscrew or spiral staircase. It is a type of smooth spac ...

helix
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-handedhttp://mathworld.wolfram.com/Torsion.html helix and is negative for a left-handed one.


Alternative description

Let be the
parametric equation In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric objec ...
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 canonica ...
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-oriented ...
of with values in and the vectors : \mathbf(t), \mathbf(t) are
linearly independent In the theory of vector spaces, a set of vectors is said to be if at least one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be ...
. Then the torsion can be computed from the following formula: :\tau = \frac = \frac . 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. For ...
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 three-dimensional space \mathbb^3, and is denoted by the symbol \times. Given ...

cross product
. For , the formula in components is : \tau = \frac.


Notes


References

* {{curvature Differential geometry Curves Curvature (mathematics) ru:Дифференциальная геометрия кривых#Кручение