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We introduce a novel method for matching and aligning 3D surfaces that do not have any overlapping surface information. When two matching surfaces do not overlap, all that remains in common between them is a thin strip along their borders. Aligning such fragments is challenging but crucial for various applications, such as reassembly of thin-shell ceramics from their broken pieces. Past work approach this problem by heavily relying on simplistic assumptions about the shape of the object, or its texture. Our method makes no such assumptions; instead, we leverage the geometric and photometric similarity of the matching surfaces along the break-line. We first encode the shape and color of the boundary contour of each fragment at various scales in a novel 2D representation. Reformulating contour matching as 2D image registration based on these scale-space images enables efficient and accurate break-line matching. We then align the fragments by estimating the rotation around the break-line through maximizing the geometric continuity across it with a least-squares minimization. We evaluate our method on real-word colonial artifacts recently excavated in Philadelphia, Pennsylvania. Our system dramatically increases the ease and efficiency at which users reassemble artifacts as we demonstrate on three different vessels.