What is a simple closed curve

Simply closed curves

What is a simple closed curve?

......In contrast to the open curve, the simply closed curve has no beginning and no end.

It creates a coherent area in the Euclidean plane.
 

Flat curves of this kind are more precisely called closed Jordan curves, especially if they are smooth. They divide the Euclidean plane into an inner and an outer area according to the Jordanian curve theorem.

Well-known curves of this type follow.
On this page I present a number of simple methods to create closed curves.

Freehand drawings (1st method)   Top

......A simple way to draw any closed curve is to do a freehand sketch.
In this way you can ensure that it does not have any special properties.
The curve was drawn with the MS-Paint program, which can draw Bézier curves.

......Closed curves are sometimes not that easy to recognize as such.

I found this hint in the booklet (2).


Compound curves (2nd method) great
Curve pieces are put together in such a way that a closed curve is created.


Oval made of four arcs of a circle

Egg line made up of two ellipses

Heart curve
from two lines and two semicircles

Over the circle Top
The circle is a simply closed curve. It leads to different methods of drawing curves.

The circle is described in different ways (1), (2) and (3) in coordinate systems.

......Let P be an arbitrary point of the circle with the radius R, which is given in the center position.
Three representations of the circle follow.
(1) Coordinate equation: x² + y² = R² or [y = sqr (R²-x²) and y = -sqr (R²-x²)],
(2) Parameter equations: x (t) = R * cos (t), y (t) = R * sin (t),
(3) Polar equation: r (t) = R.
In the (simple) polar equation (3), a point is given by the pair (radius OP, angle t). The radius is the distance of the point from the zero point (0 | 0). The angle lies between the radius and the positive x-axis, its apex in the zero point.

Variations of the center equation of the circle (3rd method)    Top
Closed curves can arise if the sum of two positive terms in x and y is constant.


Curves in parametric representations (4th method)   Top

......As mentioned, the circle has the parametric representation
x (t) = R * cos (t) and
y (t) = R * sin (t).

These equations are modified.



Curves with polar coordinates (5th method)    Top
When looking for three-pointed figures, I came across closed curves in polar coordinates.




The equations have the same structure. They contain a constant number and a term that contains the sine or cosine and thus ensures a periodic course. The triple beam is ensured by a factor of 3 in 3t.


The way is illustrated for the example r (t) = 2 + cos (3t) + sin (3t). The exit is the circle with R = 2. A rosette is widened by a circle. - The definition range is 0 <= t <= 2 * pi in each case.

Here is a wide field to experiment with.

Curves with one parameter (6th method)      Top
The method is explained using a third-degree curve with the equation y² = -x³ + 2x + b.
For b = 10 there is a loop, which constricts more and more with decreasing parameter b and leads to a closed curve.


y² = -x³ + 2x + 10

y² = -x³ + 2x + 2

y² = -x³ + 2x + 1
If you discover loops, you can often change a parameter so that closed curves are created.

Curves with a glitch (7th method)      Top
Closed curves also arise if equations describing loops are perturbed with a summand. The summand is small compared to 1.


Reflection on a straight line (8th method)   Top

......If you do not reflect straight lines on a straight line, symmetrical figures can arise, the edge curves of which are closed.
Circular arcs serve as an example.

This method can be transferred to functions by mirroring the graph in the area between two zeros on the x-axis.

f (x) = sin (x)

f (x) = - sin (x) / \ f * (x) = - sin (x)

D = {x | 0 <= x <= pi}


In the next chapter, the mirroring of a straight line is applied to special functions.

Nth order curves (9th method)    Top
First of all, it is about third-order curves with the equation y² = ax³ + bx² + cx + d.

If the function with f (x) = ax³ + bx² + cx + d has three zeros, a maximum and a minimum position arise. If one goes over to the root function g (x) = sqrt (ax³ + bx² + cx + d), this is only defined if the graph of f lies above the x-axis. If you mirror the graph of g on the x-axis, a closed curve is created. This is illustrated by the following example.


f (x) = x (x-1) (x + 1) = x³-x

g (x) = sqrt [x³-x]

y² = x³-x
The relation y² = x³-x belongs to the elliptic curves.

The method also works with algebraic equations of the form y² = ax4+ bx3+ cx2+ dx + e. The associated fourth order curves can be closed. Here too you can get to them via zeros.

y = (x²-4) (x²-1)

y = sqrt [(x²-4) (x²-1)]

y = sqrt [(x²-4) (x²-1)] / \ y = -sqrt [(x²-4) (x²-1)]
y²- (x²-4) (x²-1) = 0

These considerations go back to a letter from Ionel Dancescu from Hunedoara / Romania.
It creates numerous curves using the equation y4-2u (u + 1) y2+ u2(u-1)2= 0, whereby for u certain functions like u (x) = a²-x² have to be inserted. The simple case (y-u) ²-u = 0 with u (x) = 1-x² is shown here.
...(y + x²-1) ² + x² - 1 = 0 .................................... ....

Mechanically generated curves (10th method)   Top
The drawing of a circle with a compass and the gardener's construction of the ellipse are known.



You can find more about the drawings on my pages ellipse and Express lines.

classic Top
There are a number of closed curves that have played and do play a role in mathematics. In addition to the curves of circle, ellipse, egg line, heart curve, strophoide, trifolium, deltoid, quadrifolium and astroide mentioned above, the following pictures fit into this collection.


Cardioids

Arch triangle

Bean curve
The list will be continued e.g. at MathWorld (URL below).

Pathological curves Top
Closed curves seem simple and graphic.
However, the following two fractals show peculiarities that one would not have expected in them.


Koch curve
If A and U are the area and the circumference of the starting triangle, then the area of ​​the Koch curve for n tends towards infinity towards (8/5) A. The scope is growing across all borders.

Sierpinski curve (variation)
If A and U are the area and the circumference of the starting square, then the area of ​​the Sierpinski curve for n tends towards infinity towards (5/12) A. (found at http://mathworld.wolfram.com/SierpinskiCurve.html, derivation?).


A confusing curve is shown above. Here is evidence that the curve is closed.

Closed curve

Inner area

Outer area

credentials Top
(1) I.N.Bronstein, K.A.Semendjajew: Taschenbuch der Mathematik, Leipzig 1987
(2) F.L. Bauer, invitation to mathematics, Deutsches Museum, Munich (ISBN 3-924-18349-X]



Several drawings originate from my pages three-rayed figures, oval lines, ellipses, straight strophoids, equidistant, heart curves, circles, curves in the polar coordinate system, square and four-rayed figures.

Closed curves on the internet Top

German

Wikipedia
Jordan's Curve Theorem, Curve (Mathematics), Elliptic Curve

English

Eric W. Weisstein (MathWorld)
Jordan Curve, Jordan Curve Theorem, Algebraic Curve
Closed curves:

Richard Parris (freeware program WINPLOT)
The official website is closed. Download the German program e.g. from heise

Wikipedia
Curve, Mandelbrot set, Jordan curve theorem, Algebraic equation, Elliptic curve, Bicorn, Edwards curve

The Wolfram Demonstrations Project
Plotting Closed Curves with a Four Bar Linkage


Feedback: Email address on my main page

URL of my homepage:
http://www.mathematische-basteleien.de/

© 2010 Jürgen Köller

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