** Using WeBWorK **

A matching problem could be a bit more challenging than a plain multiple choice task --- for unit circle, offer variants of
a)  [ sin(1 - 2 t) , cos(1 - 2 t) ]
b)  [ cos(pi sin(t)) , sin(pi sin(t)) ]
c)  [ sin(t^2) , cos(t^2) ]  or  [ sin(e^t) , cos(e^t) ]
d)  [ (1-t^2)/(1+t^2) , 2t/(1+t^2) ]  for punctured circle
e)  convert r = 2 sin(theta) to rectangular, then shift

Severe ad hoc constraints seem necessary to enable Webwork to handle this as a free response task (even if adaptive parameters are used in the .PG template).  E.g., requiring the parameterization to be directly proportional to arc length would be hard to assess even with access to a CAS.

I have used a similar task in Calc II as a mini-project:  find several non-equivalent parameterizations for a [simple curve], show each "works", and discuss ways in which they differ.