This paper presents a novel and unified analytic formulation for kinematics, statics, and shape restoration of multiple-backbone continuum robots. These robots achieve actuation redundancy by independently pulling and pushing three backbones to carry out a bending motion of two-degrees-of-freedom (DoF). A solution framework based on constraints of geometric compatibility and static equilibrium is derived using elliptic integrals. This framework allows the investigation of the effects of different external loads and actuation redundancy resolutions on the shape variations in these continuum robots. The simulation and experimental validation results show that these continuum robots bend into an exact circular shape for one particular actuation resolution. This provides a proof to the ubiquitously accepted circular-shape assumption in deriving kinematics for continuum robots. The shape variations due to various actuation redundancy resolutions are also investigated. The simulation results show that these continuum robots have the ability to redistribute loads among their backbones without introducing significant shape variations. A strategy for partially restoring the shape of the externally loaded continuum robots is proposed. The simulation results show that either the tip orientation or the tip position can be successfully restored.