Fundamental theorem of curves
Encyclopedia
In differential geometry, the fundamental theorem
of curve
s states that any regular curve
with non-zero curvature has its shape (and size) completely determined by its curvature
and torsion
.
A curve can be described, and thereby defined, by a pair of scalar field
s: curvature
and torsion 
, both of which depend on some parameter which parametrizes
the curve but which can ideally be the arc length
of the curve. From just the curvature and torsion, the vector field
s for the tangent, normal, and binormal vectors can be derived using the Frenet-Serret formulas
. Then, integration
of the tangent field (done numerically, if not analytically) yields the curve.
If a pair of curves are in different positions but have the same curvature and torsion, then they are congruent
to each other.
Fundamental theorem
The fundamental theorem of a field of mathematics is the theorem considered central to that field. The naming of such a theorem is not necessarily based on how often it is used or the difficulty of its proofs....
of curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
s states that any regular curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
with non-zero curvature has its shape (and size) completely determined by its curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
and torsion
Torsion of curves
In the elementary differential geometry of curves in three dimensions, the torsion of a curve measures how sharply it is twisting. Taken together,the curvature and the torsion of a space curve are analogous to the curvature of a plane curve...
.
A curve can be described, and thereby defined, by a pair of scalar field
Scalar field
In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...
s: curvature
and torsion , both of which depend on some parameter which parametrizesParametric equation
In mathematics, parametric equation is a method of defining a relation using parameters. A simple kinematic example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion....
the curve but which can ideally be the arc length
Arc length
Determining the length of an irregular arc segment is also called rectification of a curve. Historically, many methods were used for specific curves...
of the curve. From just the curvature and torsion, the vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
s for the tangent, normal, and binormal vectors can be derived using the Frenet-Serret formulas
Frenet-Serret formulas
In vector calculus, the Frenet–Serret formulas describe the kinematic properties of a particle which moves along a continuous, differentiable curve in three-dimensional Euclidean space R3...
. Then, integration
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
of the tangent field (done numerically, if not analytically) yields the curve.
If a pair of curves are in different positions but have the same curvature and torsion, then they are congruent
Congruence (geometry)
In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...
to each other.