How can I calculate the necessary transformations to construct circular path and maintain the pattern? Can anyone provide me with or reference to relevant resources? Thanks!
You could use a polar grid, and a grid of vertical lines snapped to the edge of each circle, then use the shape builder, delete the outer vertical lines, and fill in the chequered pattern.
Making the grids, and using the shape builder to delete the outer lines
Using the shape builder to create the pattern
I'm afraid there's no single nor 2 click effect available for the wanted effect.
As already said by others it can be got by dividing a set of concentric circles with vertical lines and by filling half of the areas which are formed by the intersection.
Filling the areas with the shape builder may take while, if there's say 30 circles an 60 vertical lines, but there's another approach for faster result. Essentially it also divides concentric circles with vertical lines, but it needs less clicks.
In the left there's a white and a grey rectangle aligned side by side . Both rectangles are exactly 4 mm wide. A duplicate is made and rotated 180 degrees.
Both rectangle pairs are dragged to the swatches panel. It generates 2 fill patterns which are otherwise the same, but white and grey are in the opposite places.
There's a set of concentric circles. The smallest one is 8 mm wide, the next is 8 mm wider, the next is 8 mm wider (etc) and the biggest one is 80 mm wide. As well one could say the circles are concentric, but their radiuses are 4,8,12,16, 20, 24, 28, 32 36 and 40 millimeters. The largest circle is in the bottom of the layering order and the smallest is on the top.
The easiest way to get these circles is the polar grid tool. The grid should have with and height = 80 mm and have 9 concentric dividers.
In Illustrator Blending is the other easy way to get numerous modified copies of a shape, but the polar grid is perfect just in this case. The circles must be ungrouped. It must be done twice to get them all free. The layering order becomes right by default.
The circles which have widths 80, 64, 48, 32 and 16 mm are selected in the same time (hold shift and click he wanted ones). They all got the first fill pattern. The rest of the circles got the second fill pattern. The strokes are removed.
This method has 2 drawbacks:
You must select all circles and move them to the left edge of the artboard to force the fill pattern to start at the edge of the largest circle. Illustrator relates the fill pattern to the artboard. To make the shapes freely movable the fill pattern must be expanded (Object > Expand > Fill)
After expanding the fill this is still a bunch of circles. They must be grouped to keep then surely together. In addition the circles are clipping masks for a big bunch of white and grey rectangles. Many of us will not want to have such burden behind the curtains, because it shows its existence as soon as the mouse hovers over it (an outline ghost) and it's not a path which could be used as a path in other effects.
It looks right, but it has a massive invisible complex behind the visible part. There's no easy way to convert the internal structure of the result to as simple as what's got with the shown shape builder method.
Definitely not asked and not useful in this case, but check the next Illustrator functions 1) Art Brush 2) Pattern Brush 3) Scatter Brush. They all offer ways to make a pattern to follow a curve, but differently. If they happen to be new things for you, you will probably soon find ways to utilize them. I believe even GDSE contains hundreds of them.