A stroke parameterization engine within a rendering engine is configured to map a texture to the surface of a three-dimensional (3D) model along a stroke defined by an end-user of the rendering engine. The stroke parameterization engine converts the stroke to a polyline, then generates a geodesic trace by sliding a geodesic circle along the polyline and collecting points associated with the 3D model that fall within the geodesic circle. The stroke parameterization engine then parameterizes points associated with the polyline and the geodesic trace using UV coordinates associated with a texture map, thereby generating UV coordinates for each point within the polyline and geodesic trace. The stroke parameterization engine then projects the texture map onto the surface of the 3D model using the UV coordinates generated for the polyline and geodesic trace.