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An adapted frame along a space curve is an orthonormal vector basis that incorporates the unit curve tangent at each point. Such a frame is said to be rotation-minimizing if its angular velocity maintains a zero component in the direction of the curve tangent, i.e., the normal-plane vectors exhibit no instantaneous rotation about the tangent. Rotation-minimizing frames have useful applications in computer animation, spatial motion design, swept surface constructions, path planning for robotics, and related fields. Recently, the possibility of constructing space curves with exact rational rotation-minimizing frames (RRMF curves), as a proper subset of the spatial Pythagorean-hodograph (PH) curves, has been recognized. The underlying theory and construction of such RRMF curves is presented, and alternative characterizations for them (in terms of the quaternion and Hopf map representations of spatial PH curves) are derived and compared.