The term curve is associated with two closely related notions. The first notion is kinematic: a parameterized curve is a function of one real variable taking values in some ambient geometric setting. This variable can be interpreted as time, in which case the function describes the evolution of a moving particle. The second notion is geometric; in this sense a curve is an arc, a 1-dimensional subset of an ambient space. The two notions are related: the image of a parameterized curve describes the trajectory of a moving particle. Conversely, a given arc admits multiple parameterizations. A trajectory can be traversed by moving particles at different speeds.
In algebraic geometry, the term curve is used to describe a 1-dimensional variety relative to the complex numbers or some other ground field. This can be potentially confusing, because a curve over the complex numbers refers to an object which, in conventional geometry, one would refer to as a surface. In particular, a Riemann surface can be regarded as as complex curve.
Let be an interval (http://planetmath.org/Interval) of the real line. A parameterized curve is a continuous mapping taking values in a topological space . We say that is a simple curve if it has no self-intersections, that is if the mapping is injective.
We say that is a closed curve, or a loop (http://planetmath.org/loop) whenever is a closed interval, and the endpoints are mapped to the same value; Equivalently, a loop may be defined to be a continuous mapping whose domain is the unit circle. A simple closed curve is often called a Jordan curve.
If then is called a plane curve or planar curve.
A smooth closed curve in is locally if the local multiplicity of intersection of with each hyperplane at of each of the intersection points does not exceed . The global multiplicity is the sum of the local multiplicities. A simple smooth curve in is called (or globally ) if the global multiplicity of its intersection with any affine hyperplane is less than or equal to . An example of a closed convex curve in is the normalized generalized ellipse:
In odd dimension there are no closed convex curves.
In many instances the ambient space is a differential manifold, in which case we can speak of differentiable curves. Let be an open interval, and let be a differentiable curve. For every can regard the derivative (http://planetmath.org/RelatedRates), , as the velocity (http://planetmath.org/RelatedRates) of a moving particle, at time . The velocity is a tangent vector (http://planetmath.org/TangentSpace), which belongs to , the tangent space of the manifold at the point . We say that a differentiable curve is regular, if its velocity, , is non-vanishing for all .
It is also quite common to consider curves that take values in . In this case, a parameterized curve can be regarded as a vector-valued function , that is an -tuple of functions
where , are scalar-valued functions.
A (non-singular) curve , equivalently, an arc, is a connected, 1-dimensional submanifold of a differential manifold . This means that for every point there exists an open neighbourhood of and a chart such that
for some real .
An alternative, but equivalent definition, describes an arc as the image of a regular parameterized curve. To accomplish this, we need to define the notion of reparameterization. Let be intervals. A reparameterization is a continuously differentiable function
are said to be related by a reparameterization if there exists a reparameterization such that
The inverse of a reparameterization function is also a reparameterization. Likewise, the composition of two parameterizations is again a reparameterization. Thus the reparameterization relation between curves, is in fact an equivalence relation. An arc can now be defined as an equivalence class of regular, simple curves related by reparameterizations. In order to exclude pathological embeddings with wild endpoints we also impose the condition that the arc, as a subset of , be homeomorphic to an open interval.
|Date of creation||2013-03-22 12:54:17|
|Last modified on||2013-03-22 12:54:17|
|Last modified by||rmilson (146)|
|Defines||simple closed curve|
|Defines||locally convex curve|