Superthunder bolt did a good Instructable titled Carpenter's Center Finder. He relied on careful measuring and drilling for critical dimensions. What follows automates the process of locating the necessary holes and makes accuracy inevitable.
1 x 2 hardwood
Drill press and drills
Wooden base piece for drilling
Step 1: Mark Center
Step 2: Drill a Pivot Hole
I used the pencil as an axis for the 1 x 2 and set the pencil off an inch and a half or so. I clamped the base piece to the drill press table. Use the drill press to drill a hole to match the dowel you will be using on the line that runs the length of the 1 x 2. I had a 5/16 inch dowel.
Step 3: A Second Hole Near the End of the 1 X 2
Step 4: Add Dowels
Marks on the oak are from dirt on the workbench and are marks I made for another project before I changed my mind.
Step 5: Mark the Center of the Work Piece
This is a no weld version. Originally, I see 1/8 inch strap iron and 1/4 inch rod. The center hold was drilled for the size of a finish nail, and then made conical with a countersink bit. The hole for the finish nail is about right for a pencil lead. I welded the 1/4 inch steel rod into holes drilled in the strap iron.