Math Is Fun Forum

Hi bobbym;

Thank you for your appreciation.

Six-pointed stars:
Sine Star (3)
Cosine Star (3)

Hi All!

A five-pointed star (☆) can be obtained by drawing five lines.
Based on this, I made an animation by drawing five collections
of parallel lines: Starflakes Animation

Hi bobbym,

Thank you! And you're welcome.

UDAV (universal data array visualization)

It's free!

Hi,

The boundary curve is a superellipse: (x/4)^4 + (y/3)^4 = 1.

Hi gAr,

Thanks for the link! I noticed the Buddha's quote in your signature.
Have you seen the Buddhabrot? It's amazing!

Hi bobbym,

I agree with you. It's difficult to write a contour plotter from scratch.
Many thanks to those kind people who make open/free plotters for us.

Thanks for the link!

-π < atan2(x,y) <= π
5π < atan2(x,y) + 6π <= 7π
The set of polar coordinates A = { (r,t) | f(r,t) = 0 , r∈[0,∞) , t∈(5π,7π] }
is transformed to the set of rectangular coordinates B = { (x,y) | f3(x,y) = 0, x∈R, y∈R }.
That is, A and B represent the same figure (curve) in the plane.

Because the polar coordinate representation of a given point in the plane is not unique:
(r, t) = (r, t ± 2n*pi) = (-r, t ± (2n+1)*pi), where n is any integer.
I define f0, f1, f2, ... for (r, t ± 2n*pi) and g0, g1, g2, ... for (-r, t ± (2n+1)*pi).

My idea came from this page. (See the section on 'Converting between polar and Cartesian coordinates'.)

Hi bobbym,

Here is what I use in the plotting process.

Define the following functions:

f0(x,y) = f( sqrt(x^2 + y^2), atan2(x,y) )
f1(x,y) = f( sqrt(x^2 + y^2), atan2(x,y) + 2*pi )
f2(x,y) = f( sqrt(x^2 + y^2), atan2(x,y) + 4*pi )
f3(x,y) = f( sqrt(x^2 + y^2), atan2(x,y) + 6*pi )
. . . . . .

g0(x,y) = f( -sqrt(x^2 + y^2), atan2(x,y) + pi )
g1(x,y) = f( -sqrt(x^2 + y^2), atan2(x,y) + 3*pi )
g2(x,y) = f( -sqrt(x^2 + y^2), atan2(x,y) + 5*pi )
g3(x,y) = f( -sqrt(x^2 + y^2), atan2(x,y) + 7*pi )
. . . . . .

Examples of plotting f(r,t) = 0:

m=8, n=6, k=5:
Plot f0(x,y) = 0, g0(x,y) = 0.

m=8, n=6, k=2.5:
Plot f0(x,y) = 0, f1(x,y) = 0, g0(x,y) = 0, g1(x,y) = 0.

m=8, n=6, k=1.25:
Plot f0(x,y) = 0, f1(x,y) = 0, f2(x,y) = 0, f3(x,y)=0,
and g0(x,y) = 0, g1(x,y) = 0, g2(x,y) = 0, g3(x,y) = 0.