How to bend one side of a planar mesh fade the bending effect towards the other side I couldn't figure out how to bend one side of a planar mesh like the edges on that side follow a circle. I also want this bending effect to fade smoothly towards the other edge. It is like square(projected) piece cut out from a sphere but I want to control the fading effect and I also want one side of the piece to be a line. I thought, this is a primitive thing, it cannot be considered as "do the modeling for me" request. I'm sorry, If I'm not in the right place to ask such a question. I tried different modifiers but I couldn't converge to what I was willing to see. What is the easiest way of doing this in blender?

Edit: I tried to sketch it:
In this sketch, side view shows that from B to C it is bended. This is the final thing that I want to model. Then, I will repeat this round a cylinder. That part is easy. But, initially, I have to figure out how to bend one side of the mesh from B to C like in the first figure. The edges connected to the corner A are just lines. And the shape is symmetrical.
Here is how to do it. Doing it with modifiers isn't possible as far as I know. If you need more control, comment.


Start by adding a circle. 
rotate the circle by 90 degrees on the X axis(RX90). 
Delete the bottom half of it.
Take the bottom to vertices and duplicate the by 2 units on the y axis (Shift-DY2). 

fill the two vertices.
hit (Ctrl-R).
add 1 less loop cuts then half the number of vertices in your circle you can use the scroll wheel to do this.
Join up all the faces (Ctrl-E>Bridge Edgeloops).
add some loop cuts until your faces are about square.
tweak the shape using proportional editing.