This paper presents a novel robotic hand with a metamorphic palm which changes the traditional structure of a robotic hand. Based on this new hand structure, the paper investigates motion of the robotic fingers with respect to the palm by presenting finger operation planes and by revealing the relationship between finger motion and palm motion. The study presents the normals of the finger operation planes as a function of the input angles of the palm and uses these normals to relate finger motion to palm motion. This leads to the coaxial condition of the finger-palm relationship that is then converted to the coplanar condition of normals of all finger operation planes. The condition is then used to generate a coupler trajectory, and the iterative trajectory fitting and curve approximation are used for synthesis of palm links, leading to differential geometry based synthesis of angular lengths of the metamorphic palm.