A Curvilinear Slide is used to make shapes that follow a template. For example, the ornamental turner could make a template that looked like a section of a gourd, and then use that template to make a gourd-shaped object on the rose engine lathe.
Lindow Machine Works makes one which replaces the top slide of a Hardinge cross slide.
If you go the route of an MDF rose engine, there are some really inventive people in the community who have made some that work quite well.
The video below shows a number of things, including the use of a curvilinear slide (around 3:50 into the video).
This video is about curvilinear designs. It was given by Martin Strand at the 2018 Ornamental Turners International Symposium.
This video is about a rotary curvilinear design, which is a whole new level of wierd. It was given by Kenneth Newton at the 2018 Ornamental Turners International Symposium.