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If the centers of an ordinary line do a hinge, the result is a diamond. The dancers who were centers of the line are still considered the centers of the diamond. Since the ends of the original line didn't move they are still where they were, and are still considered ends, but in a diamond they are also known as "points".
Note that the points of a diamond are "far apart" from each other. They are not touching hands with each other or with anybody else in the diamond. In contrast, the centers are holding hands with each other. To emphasize their distinct positioning, it is traditional for the points to keep their arm raised slightly, "pointing" at the centers.
Note also that, in a diamond, half of the dancers will be facing head walls and half will be facing side walls. Either the centers will be facing head walls and the points will be facing side walls, or the centers will be facing side walls and the points will be facing head walls.
Diamond Circulate means that each dancer in the diamond moves forward into the next position around the diamond. As they do this, they all must turn 1/4 (90 degrees) to have the correct facing direction.
Note also that each center becomes a point, and each point becomes a center. Those who become centers must immediately take hands, to make clear to everybody that they are now the centers.
It is very common for the eight dancers of the square to be arranged to form two diamonds side by side. In this case, the circulate path applies to each diamond separately, with all the dancers staying in their own diamond.
It is also possible for the two diamonds to be next to each other "point to point". The circulate paths are still within each diamond, and the points of each diamond still become centers of that diamond when they circulate.
On a Diamond Circulate, just as with any other kind of circulate, the position of the spots on the floor does not change -- the dancers are just moving from spot to spot.