In this paper we present an algorithm with a new trace-terminating conditionfor tracing along surface-surface intersection curves on surface boundaries, while several tracingmethods and embedding methods that include tracing scheme may cause false termination witha traditional trace-terminating condition: tracing stops when the surface-domain’s boundaryis reached. And we also suggest another iterative method to calculate intersection points onsurface boundaries with parallel surface normal. Some numerical examples with these two ideaseffectiveness of our algorithm.