# 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 . ${\ displaystyle \ mathbb {R} ^ {2}}$ Examples:

• The image
${\ displaystyle {[0,2 \ pi [} \ to \ mathbb {R} ^ {2}, \ quad t \ mapsto (\ cos t, \ sin t)}$ describes the unit circle in the plane.
• The image
${\ displaystyle \ mathbb {R} \ to \ mathbb {R} ^ {2}, \ quad t \ mapsto {\ big (} t ^ {2} -1, t (t ^ {2} -1) {\ big)}}$ describes a curve with a single colon at , corresponding to the parameter values and .${\ displaystyle (0,0)}$ ${\ displaystyle t = 1}$ ${\ displaystyle t = -1}$ 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
${\ displaystyle x ^ {2} + y ^ {2} = 1}$ describes the unit circle in the plane.
• the equation
${\ displaystyle y ^ {2} = x ^ {2} (x + 1)}$ 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

${\ displaystyle f \ colon D \ to \ mathbb {R}, \ quad x \ mapsto f (x)}$ can either be a parametric representation

${\ displaystyle D \ to \ mathbb {R} ^ {2}, \ quad t \ mapsto (t, f (t))}$ or as an equation

${\ displaystyle \ Gamma _ {f} = \ {(x, y) \ in \ mathbb {R} ^ {2} \ mid y = f (x) \}}$ 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 ${\ displaystyle [a, b] \ subset \ mathbb {R}}$ ${\ displaystyle c \ colon [a, b] \ to \ mathbb {R} ^ {n}}$ ${\ displaystyle | c '(x) | \ neq 0}$ ${\ displaystyle x \ in (a, b)}$ ${\ displaystyle l = \ int _ {a} ^ {b} | c '(t) | \, dt}$ The function

${\ displaystyle x \ mapsto \ int _ {a} ^ {x} | c '(t) | \, dt}$ 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. ${\ displaystyle [a, b] \ to [0, l]}$ ${\ displaystyle c}$ ${\ displaystyle {\ tilde {c}} \ colon [0, l] \ to \ mathbb {R} ^ {n}}$ ${\ displaystyle | {\ tilde {c}} '(x) | = 1}$ ${\ displaystyle x \ in (0, l)}$ ${\ displaystyle {\ tilde {c}}}$ 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. ${\ displaystyle [a, b] \ subset \ mathbb {R}}$ ${\ displaystyle c \ colon [a, b] \ to \ mathbb {R} ^ {n}}$ ${\ displaystyle c}$ ${\ displaystyle s}$ ${\ displaystyle \ kappa (s) = | c '' (s) |}$ ${\ displaystyle J}$ ${\ displaystyle \ kappa (s)}$ ${\ displaystyle c '' (s) = \ kappa (s) \ cdot Jc '(s)}$ ## 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. ${\ displaystyle c \ colon [0,1] \ to \ mathbb {R} ^ {2}}$ ${\ displaystyle c (0) = c (1)}$ ${\ displaystyle c}$ ${\ displaystyle [0,1)}$ ${\ displaystyle c}$ ${\ displaystyle c (t) \ neq (0,0)}$ ${\ displaystyle t \ in [0,1]}$ Smooth closed curves can be assigned to another number, the tangent revolution number , which for a parameterized by arc Läge curve by ${\ displaystyle c \ colon [0, l] \ to \ mathbb {R} ^ {2}}$ ${\ displaystyle {\ frac {1} {2 \ pi}} \ int _ {0} ^ {l} \ kappa (t) \, dt}$ given is. The circulation rate of Heinz Hopf says that a simple closed curve tangent winding number or has. ${\ displaystyle 1}$ ${\ displaystyle -1}$ 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 . ${\ displaystyle X}$ ${\ displaystyle c \ colon [0,1] \ to X}$ ${\ displaystyle c (0) = c (1)}$ ${\ displaystyle c (0)}$ ${\ displaystyle [0.1] / \ {0.1 \}}$ ${\ displaystyle S ^ {1}}$ ${\ displaystyle S ^ {1} \ to X}$ ${\ displaystyle c_ {1}, c_ {2}}$ ${\ displaystyle x}$ ${\ displaystyle H \ colon [0,1] ^ {2} \ to X}$ ${\ displaystyle H (s, 0) = c_ {1} (s)}$ ${\ displaystyle H (s, 1) = c_ {2} (s)}$ ${\ displaystyle s}$ ${\ displaystyle H (0, t) = H (1, t) = x}$ ${\ displaystyle t}$ ${\ displaystyle X}$ ${\ displaystyle X = \ mathbb {R} ^ {2} - \ {0 \}}$ ${\ displaystyle \ mathbb {Z}}$ ## Space curves

Let be an interval and a curve parameterized according to the arc length. The following names are standard: ${\ displaystyle [a, b] \ subset \ mathbb {R}}$ ${\ displaystyle c \ colon [a, b] \ to \ mathbb {R} ^ {3}}$ {\ displaystyle {\ begin {aligned} t (s) & = c '(s) \\ n (s) & = {\ frac {t' (s)} {| t '(s) |}} \\ b (s) & = t (s) \ times n (s) \ end {aligned}}} (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 : ${\ displaystyle t '(s) \ neq 0}$ ${\ displaystyle t (s)}$ ${\ displaystyle n (s)}$ ${\ displaystyle b (s)}$ ${\ displaystyle (t, n, b)}$ ${\ displaystyle \ kappa (s) = | t '(s) | = | c' '(s) |}$ ${\ displaystyle \ tau (s)}$ ${\ displaystyle b '(s) = - \ tau (s) n (s)}$ ${\ displaystyle {\ begin {matrix} t '& = && \ kappa n \\ n' & = & - \ kappa t & + & \ tau b \\ b '& = && - \ tau n \ end {matrix}} }$ 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 . ${\ displaystyle \ kappa, \ tau \ colon [0, l] \ to \ mathbb {R}}$ ${\ displaystyle \ kappa (s)> 0}$ ${\ displaystyle s \ in [0, l]}$ ${\ displaystyle \ kappa}$ The three vectors of two , or spanned planes through the curve point wear special name: ${\ displaystyle t}$ ${\ displaystyle n}$ ${\ displaystyle b}$ • The Oskulations flat or osculating plane is by and spanned.${\ displaystyle t}$ ${\ displaystyle n}$ • The normal plane is spanned by and .${\ displaystyle n}$ ${\ displaystyle b}$ • The rectifying plane or extension plane is spanned by and .${\ displaystyle t}$ ${\ displaystyle b}$ ## 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. ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle S ^ {1}}$ 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”. ${\ displaystyle f \ colon \ mathbb {R} \ to \ mathbb {R} ^ {2}}$ ## 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.