The disclosure describes one or more embodiments of systems, methods, and non-transitory computer-readable media that modify vertex positions of a 3D-object mesh to align the vertices with discontinuities indicated by a sharpness map for a displacement map. For example, the disclosed systems access a sharpness map that encodes discontinuities of a displacement map to preserve sharp features. During sharp tessellation, for a given vertex of the 3D-object mesh, the disclosed systems iterate over neighboring vertices to search for a nearest intersection between one of the incident edges of the vertex and neighboring vertices and a discontinuity line from the sharpness map. Then, the vertex is moved within a threshold distance of the nearest discontinuity line intersection along the incident edge. The disclosed systems similarly reposition other vertices of the 3D-object mesh to generate a modified 3D-object mesh that includes vertices that align with discontinuities present in a displacement map.