This paper provides a new perspective to the problem of reconfiguration of a rolling sphere. It is shown that the motion of a rolling sphere can be characterized by evolute-involute geometry. This characterization, which is a manifestation of our specific selection of Euler angle coordinates and choice of angular velocities in a rotating coordinate frame, allows us to recast the three-dimensional kinematics problem as a problem in planar geometry. This, in turn, allows a variety of optimization problems to be defined and admits infinite solution trajectories. It is shown that logarithmic spirals form a class of solution trajectories and they result in exponential convergence of the configuration variables.

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