AbstractWe present a novel algorithm for generating a planar parameterization of the region surrounding a curve embedded in a 3D surface, which we call a stroke parameterization. The technique, which extends the well‐known Discrete Exponential Map [SGW06], uses the same basic geometric transformations and hence is both efficient and easy‐to‐implement. We also handle self‐intersecting curves, for which a 1–1 map between the original surface and the plane is not possible. Stroke parameterizations provide an ideal coordinate space for solving a variety of computer graphics problems. We present applications including tiling texture and displacement along 3D brush strokes, procedural texturing along 3D paths, and user‐guided crease extraction.