An electrostatic discharge machining fixture includes a fixture body, two or more electrically conductive face contacts seated in the fixture body, and two or more electrically resistive point contacts seated in the fixture body. The electrically conductive face contacts and the electrically resistive point contacts define a 3-2-1 alignment system to locate an additively manufactured article relative to the fixture body during an electrostatic discharge machining operation. Electrostatic discharge machining arrangements and methods of supporting additively manufactured workpieces during electrostatic discharge machining operations are also described.