Curve

In mathematics, a curve ( " bent, curved " from the Latin curvus ) a one-dimensional object.

One-dimensional here means informal, can move that one on the curve only in one direction (or the opposite direction). Whether the curve in the two-dimensional plane ( " plane curve " ) or in a higher-dimensional space (see space curve ), is irrelevant in this conceptual context.

Depending on the area of ​​mathematics, there are different clarifications of this description.

Parametric equations

A curve can be defined as the image of a path. One way is (different from the colloquial language ) is a continuous mapping of an interval in the space considered, eg in the Euclidean plane.

Examples:

• The figure

• The figure

Occasionally, especially in historical terms, no distinction is made between the path and the curve. So the interesting structure in the Hilbert curve of the road; the image of this path is the unit square, ie has no fractal structure more.

Equation representations

A curve can be described by one or more equations in the coordinates. Examples of this are again the images of the two given by the parametric equations above curves:

• The equation

• The equation

The equation is as given here by a polynomial, is called the curve algebraically.

Function graph

Function graphs are a special case of both the abovementioned forms: The graph of a function

Can be either a parametric representation

Or as an equation

Be specified.

Spoken in school mathematics of curve sketching, one usually thinks only this special case.

Differentiable curves, curvature

Be an interval and is a regular curve, ie for everyone. Is the length of the curve

The function

Is a diffeomorphism, and the concatenation of the inverse diffeomorphism provides a new curve for all. It is said that is parameterized by arc length.

Be an interval and a parameterized by arc length curve. The curvature of at the point is defined as. For plane curves can see the curvature yet signed off: Is the rotation by 90 °, then it is determined by. Positive curvature corresponds to left turns, right turns negative.

Closed curves

Be a plane curve. It is called closed if, and simply closed if, in addition to injective. The Jordan curve theorem states that a simple closed curve divides the plane into a bounded and an unbounded part. If a closed curve for all, you can assign a number of turns of the curve that indicates how many times the curve passes around the zero point.

Smooth closed curves can be assigned to another number, the circulation number represented by a parameterized by arc Läge curve

Is given. The circulation rate of Heinz Hopf states that a simple closed curve winding number or has.

Be generally a topological space. Instead of closed paths with one speaks also of loops with base point. Because the quotient space is homeomorphic to the unit circle, one identifies loops with continuous maps. Two loops with base point called homotopic if one can be continuously deformed into each other while maintaining the base point, ie if it is a continuous mapping with, for all and for all. The equivalence classes homotoper loops form a group, the fundamental group of. Is, then the fundamental group on the number of turns is isomorphic to.

Space curves

Be an interval and a parameterized by arc length curve. The following notations are standard:

(defined whenever ). is the tangent vector, the normal vector and the Binormalenvektor, the triple is called accompanying tripod, the plane spanned by and with base Oskulationsebene. The curvature is defined by the convolution. Apply the Frenet formulas:

The main theorem of the space curve theory states that one can reconstruct a curve of curvature and torsion: are smooth functions for all (the value 0 is not allowed for so ), so there are up to movements exactly one corresponding curve.

Curves as separate objects

Curves without surrounding space are relatively uninteresting in differential geometry, because each one-dimensional manifold diffeomorphic to the real line or the unit circle is. Also, features such as the curvature of a curve are intrinsically not detectable.

In algebraic geometry and related information in complex analysis is meant by " curves " usually 1-dimensional complex manifolds, often referred to as Riemann surfaces. These curves are independent objects of study, the most prominent example are the elliptic curves. See curve ( algebraic geometry)

Historical

The first book of Euclid began with the definition " A point is that which has no part. A curve is a length without breadth. "

This definition can no longer be maintained, because there is, for example, Peano curves, ie continuous surjective pictures that fill the entire plane. On the other hand, it follows from the lemma of Sard that every differentiable curve area of ​​zero, so in fact, as required by Euclid " no width " has.