In the boundary condition capture method for the immersion interface, a rigid solid in a fluid is replaced by the fluid in the same rigid motion as the solid, and the boundary conditions at the solid are captured by necessary jump conditions across the interface that separate the fluid in the rigid motion and the fluid around. In this paper, we describe how to compute the necessary jump conditions using a triangular mesh representation of an interface in 3D. In contrast to a global parametrization, a triangular mesh is simple and robust to represent a complex interface that can also be non-smooth.