# 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]→R$^n${\textstyle f:[a, b]\to \mathbb {R} ^{n}} is a curve in the R$^n${\textstyle \mathbb {R} ^{n}}.
* Complex Analysis: Continuous mapping f:[a, b]→C{\textstyle f:[a, b]\to \mathbb {C} } is a path in C{\textstyle \mathbb {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$^n${\textstyle \mathbb {R} ^{n}} or C{\textstyle \mathbb {C} }).

### Example 1 - Plot

γ$_1$:R→R$^2$, {\textstyle \gamma _{1}\colon \mathbb {R} \to \mathbb {R} ^{2}, } t↦γ$_1$(t)=(t$^2$−1, t(t$^2$−1)){\textstyle t\mapsto \gamma _{1}(t)={\bigt^{2}-1, t(t^{2}-1){\big )}}

### Example 1 Curve as a solution of an algebraic equation

* $\gamma _{1}\colon \mathbb {R} \to \mathbb {R} ^{2},$ $t\mapsto \gamma _{1}(t)={\bigt^{2}-1,t(t^{2}-1){\big )}$ resp. $y^{2}=x^{2}(x+1)$.
* Determine for the curve all (t$_1$, t$_2$)∈R$^2${\textstyle (t_{1}, t_{2})\in \mathbb {R} ^{2}} with γ$_1$(t$_1$)=γ$_1$(t$_2$)∈R$^2${\textstyle \gamma _{1}(t_{1})=\gamma _{1}(t_{2})\in \mathbb {R} ^{2}}

### Examples 2

The mapping
* γ$_2$$^~$:[0, 2π)→R$^2$, t↦γ$_2$$^~$(t)=(cos⁡t, sin⁡t){\textstyle {\widetilde {\gamma _{2}}}\colon [0, 2\pi )\to \mathbb {R} ^{2}, \quad t\mapsto {\widetilde {\gamma _{2}}}(t)=(\cos t, \sin t)}
describes the Unit circle in the plane R$^2${\textstyle \mathbb {R} ^{2}}.
* γ$_2$:[0, 2π)→C, t↦γ$_2$(t)=cos⁡(t)+isin⁡(t){\textstyle \gamma _{2}\colon [0, 2\pi )\to \mathbb {C} , \quad t\mapsto \gamma _{2}(t)=\cos(t)+i\sin(t)}
describes the Unit circle in the Gaussian number level C{\textstyle \mathbb {C} }.

### Examples 3

The mapping
    γ$_3$:R→R$^2$, t↦γ$_3$(t)=(t$^2$−1, t(t$^2$−1)){\textstyle \gamma _{3}\colon \mathbb {R} \to \mathbb {R} ^{2}, \quad t\mapsto \gamma _{3}(t)={\bigt^{2}-1, t(t^{2}-1){\big )}}
describes a curve with a simple double point at (0, 0){\textstyle (0, 0)}, corresponding to the parameter values t=1{\textstyle t=1} and t=−1{\textstyle 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 γ:[a, b]→C{\textstyle \gamma :[a, b]\to \mathbb {C} } or γ:[a, b]→R$^n${\textstyle \gamma :[a, b]\to \mathbb {R} ^{n}} be a path. is the image of a path
    Trace(γ):={γ(t) | a≤t≤b}{\textstyle Trace(\gamma ):=\left\{\gamma (t)\ |\ a\leq t\leq b\right\}}.

### Difference - Graph und Curve

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

### Task - Plot Graph und Curve

use CAS4Wiki :
    ${\begin{array}{rrcl}\gamma :&[0,6\pi ]&\rightarrow &\mathbb {R} ^{2}\\&t&\mapsto &\gamma (t)=\left(3\cdot \cos(t),\sin(t)\right)\end{array}}$

### Animation of the track
## Curves in Geogebra

First create a slider for the variable t∈[0, 2π]{\textstyle t\in [0, 2\pi ]} and two points K$_1$=(2cos⁡(t), 2sin⁡(t))∈R$^2${\textstyle K_{1}=(2\cos(t), 2\sin(t))\in \mathbb {R} ^{2}} or K$_2$=(cos⁡(3t), sin⁡(3t))∈R$^2${\textstyle K_{2}=(\cos(3t), \sin(3t))\in \mathbb {R} ^{2}} and generate with K:=K$_1$+K$_2${\textstyle K:=K_{1}+K_{2}} the sum of both location vectors of K$_1${\textstyle K_{1}} and K$_2${\textstyle 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.
The construction about will create an interactive representation of the the follow path γ$_4${\textstyle \gamma _{4}}. Observe the point $K$ in Geogebra.
    $\gamma _{4}(t):=(2\cos(t),2\sin(t))+(\cos(3t),\sin(3t))\in \mathbb {R} ^{2}$
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{\textstyle x^{2}+y^{2}=1} describes the unit circle in the plane.
* The equation y$^2$=x$^2$(x+1){\textstyle 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→R, x↦f(x){\textstyle f\colon D\to \mathbb {R} , \quad x\mapsto f(x)}
can be either as a parameter representation γ:D→R$^2$, t↦(t, f(t))}{\textstyle \gamma \colon D\to \mathbb {R} ^{2}, \quad t\mapsto (t, f(t))\}} or as equation y=f(x){\textstyle y=f(x)}, wherein the solution quantity of the equation represents the curve by {(x, y)∈R$^2$∣y=f(x)}{\textstyle \{(x, y)\in \mathbb {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 γ:[a, b]→C{\textstyle \gamma \colon [a, b]\to \mathbb {C} } are continuous mappings with γ(a)=γ(b){\textstyle \gamma (a)=\gamma (b)}. In the function theory, we need curves γ:[a, b]→C{\textstyle \gamma \colon [a, b]\to \mathbb {C} } in C{\textstyle \mathbb {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 γ:[a, b]→C{\textstyle \gamma \colon [a, b]\to \mathbb {C} } by
    $\mu (\gamma ,z):={\frac {1}{2\pi i}}\int _{\gamma }{\frac {1}{\xi -z}}\,d\xi :=\int _{a}^{b}{\frac {1}{\gamma (t)-z}}\cdot \gamma '(t)\,dt$
is given. The circulation theorem analogously to a curve in R$^2${\textstyle R^{2}}, states that a simple closed curve has the number of revolutions 1{\textstyle 1} or −1{\textstyle -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 $\mathbb {R}$ 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 $f\colon \mathbb {R} \to \mathbb {R} ^{2}$ that fill the entire plane $\mathbb {R} ^{2}$. 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 $\mathbb {R} ^{2}$ for a curve $\gamma :[a,b]\to \mathbb {R} ^{2}$ with tangent vector $\gamma ':[a,b]\to \mathbb {R} ^{2}$
* Rolling curves Bicycle reflectors as an example of curves - Cycloid
* Rolling curve for Example 2