For good hand writing, should your elbow , wrist and hand rest on the desk or should not?

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Definitely rest your elbow and forearm. Wrist should give small detail and forearm should give overall stability to hand writing. I wonder how good painters get so much precision without resting anything.

Elbow and wrist, yes; hand, no. You need arm support to steady your hand, but you need freedom of movement to let the pen flow.