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iii In this dissertation, we study the geometric character of structured Parseval frames, which are families of vectors that provide perfect Hilbert space reconstruc-tion. Equiangular Parseval frames (EPFs) satisfy that the magnitudes of the pairwise inner products between frame vectors are constant. These types of frames are use-ful in many applications. However, EPFs do not always exist and constructing them is often difficult. To address this problem, we consider two generalizations of EPFs, equidis-tributed frames and Grassmannian equal-norm Parseval frames, which include EPFs when they exist. We provide several examples of each type of Parseval frame. To characterize and locate these classes of frames, we develop an optimization pro-gram involving families of real analytic frame potentials, which are real-valued functions of frames. With the help of the Łojasiewicz gradient inequality, we prove that the gradient descent of these functions on the manifold of Gram matrices o