We will describe an adaptation of the theory of trisections to the setting of properly embedded, smooth, compact surfaces in smooth, compact, orientable four-manifolds with boundary.  The main result is that any such surface can be isotoped to lie in bridge trisected position with respect to a given trisection on the ambient four-manifold.  The trisection of the four-manifold induces an open book decomposition of the boundary three-manifold, and the structure induced on the boundary of the surface is that of a transverse link.  This gives a generalization of Rudolph’s work on braidings of orientable, ribbon surfaces two ways: our theory applies to non-ribbon and non-orientable surfaces in the four-ball, as well as surfaces in any compact four-manifold. As a corollary, we present new diagrammatic descriptions of embedded surfaces: tri-plane diagrams for surfaces in the four-ball and shadow diagrams in the general setting.  This talk will focus on describing surfaces in the four-ball via tri-plane diagrams.