Abstract Solving multi-manifolds clustering problems that include delineating and resolving multiple intersections is a very challenging problem. In this paper we propose a novel procedure for clustering intersecting multi-manifolds and delineating junctions in high dimensional spaces. We propose to explicitly and directly resolve ambiguities near the intersections by using 2 properties: One is the position of the data points in the vicinity of the detected intersection; the other is the reliable estimation of the tangent spaces away from the intersections. We experiment with our method on a wide range of geometrically complex settings of convoluted intersecting manifolds, on which we demonstrate higher clustering performance than the state of the art. This includes tackling challenging geometric structures such as when the tangent spaces at the intersections points are not orthogonal.