Not my most impressive project, but I've been tinkering a lot and I think this is probably the only one I've made as of late that I think is worth sharing. Drag the control points around the warp the surface of the plane, some of the ways you can bend it remind me of a hyperbolic paraboloid.
Do tell me if you have any other observations you'd like to comment on.
They all start out coplanar, judging by the rectangular grid. But when you move them around, they seem to escape the plane. How does that work? We are after all dragging the mouse around in a plane.
Very nice - works well for me 
The plane stops being convex and I think that's where the 'bending' illusion occurs, the way the grid is rendered may be the cause of why it appears the way it does.
EDIT: The opposite point of another point on the plane should move the opposite way, moving just one point bends the plane. like pulling one corner of a piece of paper instead of pushing & pulling on two opposite corners respectively to rotate it.
If you could somehow project an image to the plane, that would be amazing!
I have an idea of how you would do it, by remapping each pixel of a costume to a point on the plane using the pen, maybe I will create a new project as a proof of concept, it may not run well though.