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Geometric data, however, routinely contains sharp features like edges, corners, self\u2010intersections, branching junctions, and other singularities, rendering standard methods ill\u2010defined at these points. To bring geometry processing to these and other singular spaces, we introduce the \u201ctangent blow\u2010up,\u201d a representation inspired by algebraic geometry that restores structure at singularities by lifting to the product of the ambient space and the Grassmannian of tangent planes. After iterating this construction, points that coincide in position but differ in tangent direction, curvature, or higher\u2010order contact become well\u2010separated. We equip the tangent blow\u2010up with a product metric and define discretized differential operators, such as the gradient, divergence, and Laplacian, directly in the lifted domain. 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