Here I am showing the effect of changing the weight at the corner's control points using the four segments circle previously illustrated.
Top right: w = 0.7071. The curve describe a perfect arc circle. (note the with a value of 1, the curve is pulled slightly toward the point, this is why the non-rational bezier3 cannot represent a circle exactly).
Top left: w = 0. This result in eliminating the influence of that point on the curve and therefore we get a line.
Bottom left, w = 10. This result in pulling the curve toward the point creating sort of a rounded corner.
Bottom right: w = -0.5. This result in pushing the curve away from the point creating an inside arc.
Now imagine you have an object with all the points represented with the x,y,z,w you could use the w to influence the object points and sculpt details using say ...... a weight map!
Now is this a novel idea ....