Curve (math)
In mathematics , a curve (from the Latin curvus “bent, curved”) is a one-dimensional object . In contrast to a straight line , for example , a curve does not have to be straight, but can instead take any shape.
One-dimensional means informally that you can only move in one direction (or the opposite direction) on the curve. Whether the curve lies 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 sub-area of mathematics, there are different specifications for this description.
Parametric representations
A curve can be defined as the image of a path . A path is (in contrast to colloquial language) a continuous mapping from an interval into the considered space, e.g. B. in the Euclidean plane .
Examples:
- The image
- describes the unit circle in the plane.
- The image
- describes a curve with a single colon at , corresponding to the parameter values and .
Occasionally, especially with historical names, no distinction is made between path and curve. So the interesting structure in the Hilbert curve is the way; the image of this path is the unit square, so it no longer has any fractal structure.
The parameter representation gives the curve a sense of direction in the direction of the growing parameter.
Equation representations
A curve can also be described by one or more equations in the coordinates. Examples of this are again the pictures of the two curves given by the parameter representations above:
- the equation
- describes the unit circle in the plane.
- the equation
- describes the curve specified above in the parametric representation with a colon.
If the equation is given by a polynomial , as here , the curve is called algebraic .
Function graph
Function graphs are a special case of both of the above forms: The graph of a function
can either be a parametric representation
or as an equation
can be specified.
Is in school mathematics of Kurvendiskussion spoken, one usually thinks only this special case.
Differentiable curves, curvature
Let be an interval and a regular curve , i.e. H. for everyone . The length of the curve is
The function
is a diffeomorphism , and the concatenation of with the inverse diffeomorphism yields a new curve with for all . One says: is parameterized according to the arc length.
Let be an interval and a curve parameterized according to the arc length. The curvature of at the point is defined as . For plane curves, the curvature can also be given a sign : If the rotation is 90 °, then it is determined by . Positive curvature corresponds to left turns, negative right turns.
Closed curves
Be a flat curve. It is closed when , and simply closed if, in addition to is injective. The Jordanian curve theorem states that a simply closed curve divides the plane into a bounded and an unbounded part. If a closed curve is common to all , the curve can be assigned a number of revolutions which indicates how often the curve runs around the zero point.
Smooth closed curves can be assigned to another number, the tangent revolution number , which for a parameterized by arc Läge curve by
given is. The circulation rate of Heinz Hopf says that a simple closed curve tangent winding number or has.
Generally be a topological space . Instead of closed paths with , one also speaks of loops with a base point . Because the quotient space is homeomorphic to the unit circle , one identifies loops with continuous maps . Two loops with a base point are called homotopic if they can be continuously deformed into one another while maintaining the base point, i.e. H. if there is a continuous mapping with , for all and for all . The equivalence classes homotopic loops form a group , the fundamental group of . If , then the fundamental group is isomorphic to over the number of turns .
Space curves
Let be an interval and a curve parameterized according to the arc length. The following names are standard:
(defined whenever ). is the tangential vector, the normal vector and the binormal vector, the triple is called the accompanying tripod . The curvature is defined by the twist . The Frenet formulas apply :
The main theorem of local curve theory says that a curve can be reconstructed from curvature and winding: If smooth functions are with for all (the value 0 is therefore not allowed for), there is exactly one corresponding curve except for movements .
The three vectors of two , or spanned planes through the curve point wear special name:
- The Oskulations flat or osculating plane is by and spanned.
- The normal plane is spanned by and .
- The rectifying plane or extension plane is spanned by and .
Curves as independent objects
Curves without a surrounding space are relatively uninteresting in differential geometry , because every one-dimensional manifold is diffeomorphic to a real straight line or to the unit circle . Properties such as the curvature of a curve cannot be determined intrinsically either.
In algebraic geometry and, related to it, in complex analysis , “curves” are generally understood to be one-dimensional complex manifolds , often referred to as Riemann surfaces . These curves are independent study objects, the most prominent example being the elliptical curves . See curve (algebraic geometry)
Historical
The first book of the elements of Euclid began with the definition “A point is that which has no parts. A curve is a length without a width. "
This definition can no longer be maintained today, because there are, for example, Peano curves , i.e. H. continuous surjective mappings that fill the entire level. On the other hand, it follows from Sard's lemma that every differentiable curve has an area of zero, that is, in fact, as required by Euclid, has “no width”.
literature
- Ethan D. Bloch: A First Course in Geometric Topology and Differential Geometry. Birkhäuser, Boston 1997.
- Wilhelm Klingenberg: A Course in Differential Geometry. Springer, New York 1978.
Web links
Individual evidence
- ^ H. Neunzert, WG Eschmann, A. Blickensdörfer-Ehlers, K. Schelkes: Analysis 2: With an introduction to vector and matrix calculations . A textbook and workbook. 2nd Edition. Springer, 2013, ISBN 978-3-642-97840-1 , 23.5 ( limited preview in the Google book search).
- ↑ H. Wörle, H.-J. Rumpf, J. Erven: Taschenbuch der Mathematik . 12th edition. Walter de Gruyter, 1994, ISBN 978-3-486-78544-9 ( limited preview in Google book search).
- ^ W. Kühnel: Differentialgeometrie. Vieweg-Verlag, 1999, ISBN 978-3-8348-0411-2 , section 2.9.