Complex Analysis/Curves

Introduction

In the Mathematics a curve (of lat. curvus for "bent", "curved") is a one dimensionals object in a two-dimensional plane (i.e. a curve in the plane) or in a higher-dimensional space.

Parameter representations

• Multidimensional analysis: A continuous mapping $f:[a,b]\to {ℝ}^n$ is a curve in the ${ℝ}^n$.
• Complex Analysis: Continuous mapping $f:[a,b]\to ℂ$ is a path in $ℂ$ (see also path for integration).

Explanatory notes

A curve/a way is a mapping. It is necessary to distinguish the track of the path or the image of a path from the mapping graph. A path is a steady mapping of a interval in the space considered (e.g. ${ℝ}^n$ or $ℂ$).

Example 1 - Plot

${\gamma }_1:ℝ\to {ℝ}^2,$ $t\mapsto {\gamma }_1(t)=(t^2-1,t(t^2-1))$

Example 1 Curve as a solution of an algebraic equation

${\gamma }_1:ℝ\to {ℝ}^2,$ $t\mapsto {\gamma }_1(t)=(t^2-1,t(t^2-1))$ resp. $y^2=x^2(x+1)$. Determine for the curve all $(t_1,t_2)\in {ℝ}^2$ with ${\gamma }_1(t_1)={\gamma }_1(t_2)\in {ℝ}^2$

Examples 2

The mapping

• $\widetilde{{\gamma }_2}:[0,2\pi )\to {ℝ}^2,t\mapsto \widetilde{{\gamma }_2}(t)=(\cos t,\sin t)$

describes the Unit circle in the plane ${ℝ}^2$.

• ${\gamma }_2:[0,2\pi )\to ℂ,t\mapsto {\gamma }_2(t)=\cos (t)+i\sin (t)$

describes the Unit circle in the Gaussian number level $ℂ$.

Examples 3

The mapping

${\gamma }_3:ℝ\to {ℝ}^2,t\mapsto {\gamma }_3(t)=(t^2-1,t(t^2-1))$

describes a curve with a simple double point at $(0,0)$, corresponding to the parameter values $t=1$ and $t=-1$.

Direction

As a result of the parameter representation, the curve receives a directional direction in the direction of increasing parameter.

Curve as Image of Path

Let $\gamma :[a,b]\to ℂ$ or $\gamma :[a,b]\to {ℝ}^n$ be a path. is the image of a path

$Trace(\gamma ):=\{\gamma (t)|a\le t\le b\}$.

Difference - Graph und Curve

For a curve $\gamma :[a,b]\to {ℝ}^2$ the Supr or curve is a subset of ${ℝ}^2$, while the graph of function $Graph(\gamma )\subset {ℝ}^3$ is.

Task - Plot Graph und Curve

use CAS4Wiki  :

$\begin{array}{cccc}\gamma :& [0,6\pi ]& \to & {ℝ}^2\\ & t& \mapsto & \gamma (t)=(3\cdot \cos (t),\sin (t))\end{array}$

Animation of the track

Animation: Abrollkurve

Curves in Geogebra

First create a slider for the variable $t\in [0,2\pi ]$ and two points $K_1=(2\cos (t),2\sin (t))\in {ℝ}^2$ or $K_2=(\cos (3t),\sin (3t))\in {ℝ}^2$ and generate with $K:=K_1+K_2$ the sum of both location vectors of $K_1$ and $K_2$. Analyze the parameterization of the curves.

Geogebra - Interactive Implementation

Create a value slider in Geogebra with the variable name $t$ and create the following 3 points step by step in the command line of Geogebra and move the value slider for $t$ after that.

  K_1:(2*cos(t),2 * sin(t)) 
  K_2:(cos(3*t),sin(3*t))
  K: K_1+K_2 

The construction about will create an interactive representation of the the follow path ${\gamma }_4$. Observe the point $K$ in Geogebra.

$${\gamma }_4(t):=(2\cos (t),2\sin (t))+(\cos (3t),\sin (3t))\in {ℝ}^2$$

See also interaktive Example in Geogebra

Representations of Image Sets by Equations

A curve can also be described by one or more equations in the coordinates. The solution of the equations represents the curve:

• The equation $x^2+y^2=1$ describes the unit circle in the plane.
• The equation $y^2=x^2(x+1)$ describes the curve indicated above in parameter representation with double point.

If the equation is given by a Polynomial, the curve is called algebraic.

Graph of a function

Functiongraphs are a special case of the two forms indicated above: The graph of a function

$f:D\to ℝ,x\mapsto f(x)$

can be either as a parameter representation $\gamma :D\to {ℝ}^2,t\mapsto (t,f(t))\}$ or as equation $y=f(x)$, wherein the solution quantity of the equation represents the curve by $\{(x,y)\in {ℝ}^2\mid y=f(x)\}$. If theMathematics education of Curve sketching is spoken, this special case is usually only said.

Closed curves

Closed curves $\gamma :[a,b]\to ℂ$ are continuous mappings with $\gamma (a)=\gamma (b)$. In the function theory, we need curves $\gamma :[a,b]\to ℂ$ in $ℂ$, which can be continuously differentiated. These are called integration paths.

Number of circulations in the complex numbers

Smooth closed curves can be assigned a further number, thenumber of revolutions, which curve is parameterized according to the arc curve $\gamma :[a,b]\to ℂ$ by

$$\mu (\gamma ,z):=\frac{1}{2\pi i}\underset{\gamma }{\int }\frac{1}{\xi -z}d\xi :=\underset{a}{\overset{b}{\int }}\frac{1}{\gamma (t)-z}\cdot {\gamma }^{\prime }(t)dt$$

is given. The circulation theorem analogously to a curve in $R^2$, states that a simple closed curve has the number of revolutions $1$ or $-1$.

Curves as Independent Objects

Curves without an ambient space are relatively uninteresting in w:en:Differential Geometry because every one-dimensional manifold is diffeomorphic to the real line $ℝ$ or to the unit circle $S^1$. Also, properties like the curvature of a curve are intrinsically undetectable.

In algebraic geometry and, correspondingly, in complex analysis, "curves" typically refer to one-dimensional complex manifolds, often also called Riemann surfaces. These curves are independent objects of study, with the most prominent example being elliptic curves. See curve (algebraic geometry)

Historical

The first book of Elements by Euclid began with the definition:

"A point is that which has no parts. A curve is a length without breadth."

This definition can no longer be upheld today because, for example, there are Peano curves, i.e., continuous surjective mappings $f:ℝ\to {ℝ}^2$ that fill the entire plane ${ℝ}^2$. On the other hand, the Sard's Lemma implies that every differentiable curve has zero area, i.e., as Euclid demanded, it truly has no breadth.

Interactive Representations of Curves in GeoGebra

• Tangent vector of a curve in ${ℝ}^2$ for a curve $\gamma :[a,b]\to {ℝ}^2$ with tangent vector ${\gamma }^{\prime }:[a,b]\to {ℝ}^2$

• Rolling curves Bicycle reflectors as an example of curves - Cycloid

• Rolling curve for Example 2

See also

• Space curves in ${ℝ}^3$

• Curves in geometry

• Differentiable curves, curvature

• Cycloid

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.

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