parametric equations: both x and y are functions of another variable (the parameter) t:
As t varies over its domain, x and y wander about in the plane, creating a parametric curve. As t ranges from a to b, the curve moves from initial point (f(a),g(a)) to terminal point (f(b),g(b)).
cycloid: the curve traced out by a point on the perimeter of a circle as the circle rolls along a straight line. The cycloid figures heavily in the brachistochrone problem (shortest time) and the tautochrone problem (equal time).
conchoids of Nicomedes: the family of functions given by
named ``conchoids'' because of their shell-like forms.
None.
We now relax, and allow curves that fail the vertical line test, thinking of them as trajectories or pathways of a particle as it moves in time. Time is hence a ``parameter'' which locates the particle at a particular point in the plane.
We don't have to think of t as a time. For example, we might consider the following problem:
where t is a parameter. Where does the quadratic have its critical point?
For what value 
is 
?
What is the corresponding minimum?
As t varies the critical point varies, following the parametric curve
Parametric curves are terribly important for computer graphics, as computer gamers can well imagine. As mentioned in the text, the letters in a laser printer may well be drawn as parametric curves (Bezier curves).
The cycloid is an important example of a parametric curve, which proves to be the solution to both the brachistochrone and the tautochrone problems.