A Spherical Slide is used to make a controlled, curved surface. It is a modified Cross Slide, and is very different from a Curvilinear Slide / Apparatus.
With a spherical slide, the cutting frame (or drilling frame) is rotated around a vertical "line" which is parallel to the Y axis.
Fundamentally, a spherical slide works the same way a sphere-cutting jig does. (The picture to the left shows the Perfect Sphere from Carter Products.)
Conversely, on a curvilinear slide, the cutting frame (or drilling frame) is moved in an out (in the direction of the X axis) whilst moving along the Z axis.
The spherical slide rest shown in the pictures below was designed by Al Collins (the "A" in the MADE Ornamental Rose Engine), and it shows a very fine example of one for a rose engine lathe.
You can click on any of the 3 pictures to see a bigger version.
This video 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.
This video 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.