I got interested in making nice pictures in Mathematica again, after a hiatus of ~5 years. I suppose I should be writing up on how I generated these samples, but I’ll try to formalize the mathematics behind these first. For now, enjoy these short movies!
\Jan
The Pleasure of Figuring Things Out 
Very pretty! A quick search (http://demonstrations.wolfram.com/search.html?query=cycloid) shows that there is no demonstration of a rolling hyperbola, maybe you should make one?
Hi Simon! It’s actually a rolling parabola. The focus of a rolling parabola traces out a catenary, the curve a chain or string hanging from two points makes. I have working code, but explaining how to do it is a bit difficult. I suppose I should write about it in a future entry.
Respected J.M,
I am Chandru1 from stackexchange. I see that you have drawn wonderful diagrams. I would like to have this diagram worked out for me. I don’t know how to draw pics.
Could you please draw the graph of the curve
when 
is even and tell me what happens when 
.
Hello Chandru, I suggest researching on the so-called Lamé curves; some looking around gives me websites that have pretty pictures of these, probably better than what I can hope to create. :)
Lovely graphics. I particularly liked the first one. Seems that the red trajectory is a parabola as well as the blue one.
The rolling curve is a parabola, while the curve being traced out is a catenary. I have a derivation written, but the words to explain what I did so the rest of you can understand are proving to be hard to write. :)
Ok, no problem. The visual result will do. :-)
Keep up the good work!
Neat! There was an article this year in the College Mathematics Journal called “The locus of the focus of a rolling parabola” that you might find interesting.
http://www.jstor.org/stable/10.4169/07468342.41.2.129
Looks pretty neat!
I have a similar post and I have included code that works for the prolate and curtate cycloids.
here is the link
http://blog.isallaboutmath.com/2008/01/19/producing-animations-with-mathematica-60-is-as-easy-as-pie/
Hello,
Could you please post the script of the animations? I need to do some simila ones, and it would be very useful for me to have them. Thank you so much.
Hi, it has been a while since I looked at this. I will make a new post as soon as I can find the related files.