In 1969, at the USAF Machinist's Course at Chanute AFB, IL, the grinding shop had a center lapping machine. It was set up like a drill press, with some sort of chuck holding a rod shaped stone in line with a static center on the table below. There was a rickety appendage that could be swung into place to move a diamond tip in a 30 degree angle to the vertical chucked stone. Thus, was put a 60 degree cone point on the working end.
After 'truing' the stone point, you'd settle one end of the workpiece into the lower static center and run the turning stone point into the top center. All this story telling is by way of suggesting how you might set up your own center lapping device.