I am trying to get the generalized form of the cissoid traced at the vertex by the roulette curve of two identical parabolas, starting with the general form y= ax^2+bx+c.
(note identical, so a,b,c is the same for both parabolas).

I have the generalized roulette of the fixed function f(t) and rolled function r(t) as:
t -> f(t) + (p - r(t))(f'(t)/r'(t)
[ Roulette (curve) - Wikipedia, the free encyclopedia ]

(side questions:
1. What does the t->g(t) form represent? Is this the same as graphing y=g(x) ?
2. Wikipedia says 'p' is the generator, and I presume the generator must be specially chosen for the vertex. How do I choose p?)

Continuing further:

For my purpose, f' and r' are equal, so f'(t)/r'(t) = 1 and I ignore the term.
t -> f(t) + ( p - r(t) )
and the generalized roulette transform above becomes:
t -> f(t) - r(t) + p
t -> p

Did I do something wrong? How can t -> p, wouldn't that be a flat line?

(another off-topic question: does this forum editor support latex-style formatting?)

Other references that may be useful:
Cissoid of Diocles - Wikipedia, the free encyclopedia
Parabola Pedal Curve -- from Wolfram MathWorld

Thank you for your help!