We present a robust method for tracing a curve that is represented as the contour of a function in Euclidean space of any dimension. The method proceeds locally by following the intersections of the contour with the facets of a triangulation of space. The algorithm is robust in the presence of high curvature of the contour, and gives reasonable results when the curve is self-intersecting. It accumulates essentially no round-off error, and has a well-defined integer test for detecting a loop. In developing the algorithm we explore the nature of a particular class of triangulations of Euclidean space, namely, those generated by reflections. (Revised November 1987)