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**benice**- Replies: 2

References:

1.**Nested Ellipses (Ellipse Whirl)**

2.**Whirl**

Definitions of convex and concave functions (Wikipedia):

︶ Convex (= convex down = concave up) function

︵ Concave (= convex up = concave down) function

According to the definitions above, we have:

1. All non-vertical lines (y = ax + b) are both concave up and concave down;

2. A vertical line (x = c) is neither concave up nor concave down.

pione = pione + 1/double(sone);

pitwo = pitwo - 1/double(stwo);

It seems that there are no available scripting commands for axis ticks.

You may change the axes ratio or the zoom-in scale dynamically.

See this page for some examples.

Sometimes cheating is good:

anonimnystefy wrote:

Speaking of which, here is a question.

Does every non-constant periodic function have a smallest positive period?

Let f be the function defined by f(x) = 1 if x is rational and f(x) = 0 if x is irrational.

Then every rational number is a period of f.

Agnishom wrote:

What is your favorite color?

The color of fresh air.

Agnishom wrote:

By the way, do you use Sage and POV-Ray tracer?

No, but I know Sage and POV-Ray are great free tools.

Maburo wrote:

How do you trace the curve in order to plot the distances corresponding to θ?

I don't need to trace the curve in order to plot it. The software do the work for me.

If you use Graph or gnuplot, you can **see this page for examples of parametric plots**.

Graph can help you trace parametric curves. Select the parametric function you want

to evaluate or trace, and then press Ctrl+E. See **this page** for more details.

Maburo wrote:

Did you plot both y with θ and x with θ? For example, in your first plot, the blue curve was the distance along the x axis for corresponding values of θ, and the red curve was for distance along y axis?

Yes, blue for x and red for y.

The function used in the first two plots is y = sin(x).

The equation of the monster in the third image is here.

The ring of green circles in the last image can be expressed as follows:

x = cos(t-mod(t,s)) + r*cos(n*mod(t,s))

y = sin(t-mod(t,s)) + r*sin(n*mod(t,s))

r=0.1, n=32, s=2*pi/n,

t=0...2*pi.

Maburo wrote:

However, they can still be drawn. Are you able to have a computer draw these?

Case 1: **Polar functions r = f(θ)**

Use a function grapher to plot

y = f(θ)*cos(θ) in θ-y plane.

Case 2: **Implicit equations f(x,y) = 0**

Use an implicit equation plotter to plot

f(y*cot(θ), y) = 0 & y/sin(θ) > 0 in θ-y plane.

Case 3: **Parametric equations (x,y) = (f(t),g(t))**

Use a parametric curve plotter to plot

(θ, y) = (atan2(g(t),f(t)), g(t)) in θ-y plane.

Case 4: **Functions y = f(x)**

Let g(x,y) = y - f(x) and see Case 2

or let (x,y) = (t,f(t)) and see Case 3.

All these cases can be done using Graph or gnuplot.

**benice**- Replies: 1

**(A) Concentric Circles**

Regions enclosed by an infinite family of concentric circles.

Regions enclosed by five infinite families of concentric circles. (Animation Pictures)

**(B) Tangent Circles**

Regions enclosed by an infinite family of tangent circles.

Regions enclosed by five infinite families of tangent circles. (Animation Pictures)

**benice**- Replies: 4

**Nested polygons** generate logarithmic spiral curves and are good examples of contraction mappings.

We can use nested polygons to tile the plane and get beautiful tessellations:

These pictures are very easy to make using GeoGebra.

More tessellations and a few GeoGebra examples are **here**.

**benice**- Replies: 6

People kill too many trees!

A **mothertree** is coming, and she will revenge her dead children.

------> **Fractal Spacetree**

Hi All;

Thank you for your appreciation.

n872yt3r wrote:

I am the almighty Pythagoras! Give me back my shoes.

Ha! ha! you can't beat me!

I am a Super Saiyan now!

**benice**- Replies: 51

**benice**- Replies: 4

bobbym wrote:

Hi benice;

Hi bobbym;

You are great! I found 5 faces at (11,10), (17,10), (42.5,10), (4.5, 4), (11,4).

The one at (17,10) is much like a cartoon face. (Here is the enlarged picture.)

**benice**- Replies: 3

**Can you find any faces in the picture below?**

bobbym wrote:

How did you generate your points, did you use a farey sequence too?

Hi,

I used the **GeoGebra** command 'Sequence' to generate those points:

Sequence[Sequence[(p/q,GCD[p,q]/q),p,1,q-1],q,2,300] .

bobbym wrote:

Do you have a rationalize command in the language or grapher you are using? If you do then there is a way

I have downloaded **Maxima** 5.28 (which has a rationalize command), but I have no idea how to do that.

Yesterday, I constructed an approximate method which can plot the graph of y = f(sin(x)) directly.

( See this page for some examples. )

Thank you for your help bobbym!

**benice**- Replies: 3

Hi All,

Suppose f1, f2, and g are arbitrary functions of real variables.

Let

f(x) =

f1(x) if x is rational,

f2(x) if x is irrational.

What software can be used to define f(x) such that we can plot y = f(g(x)) directly?

For example, consider the Thomae function:

f(x) =

1/q if x=p/q is rational, gcd(p,q)=1 and q>0,

0 if x is irrational.

How to define f as a function such that we can plot y = f(sin(x)) directly?

bobbym wrote:

Hi;

It is a question of scaling but this looks like the Empire State Building to me. I used n = 4.

Hi, thanks for the rescaled graph. The Empire State Building reminds me of King Kong.

Here are the graphs (for n=1~8) with equal scaling on the x & y axes.

**benice**- Replies: 3

Do you like church? Here is a church function:

W I L D H O R S E S

**benice**- Replies: 1

The first frame of the above animation is derived from the tenth picture

in this page: **Tessellation using Inequalities (1)**