What is your favorite color?
The color of fresh air.
By the way, do you use Sage and POV-Ray tracer?
No, but I know Sage and POV-Ray are great free tools.
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.
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,
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.
(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)
How did you generate your points, did you use a farey sequence too?
I used the GeoGebra command 'Sequence' to generate those points:
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!
Suppose f1, f2, and g are arbitrary functions of real variables.
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:
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?