Triple-junction angle in isotropic segmentation (${\theta }_1={\theta }_2={\theta }_3=120^{\circ }$)

At a freely moving regular junction of three smooth image edges with equal isotropic length weights, the endpoint first variation requires the sum of their outward unit tangent vectors to vanish. Three unit vectors with zero sum have pairwise angles of $120$ degrees. Different interface weights instead give a weighted force balance, and constrained or singular junctions require separate hypotheses.