A Spherical Slide is used to make a controlled, curved surface. It is a modified Cross Slide, and is different from a Curvilinear Slide / Apparatus:
Fundamentally, this works the same way a sphere-cutting jig does.
Image courtesy Carter Products
The video below was given by Peter Gerstel at the 2018 Ornamental Turners International Symposium. It shows his approach which was used to make a dome shape to the end of an object.
The second video, below, was given by Jon Magill at the 2018 Ornamental Turners International Symposium. It shows his approach to making organic curves with ornamental turning. But, what is most intriguing about his approach is the overlay used to allow the ornamental turner to see what the cut will be.