Disclosed are apparatus and methods for generating displays based on a layout. A layout is received that specifies a set of rectangular components within a container rectangle. Each rectangular component has at least one size. The computing device determines a plurality of grid lines from the layout. Each rectangular component of the set of rectangular components is associated with at least two grid lines. The at least two grid lines are based on the at least one size of the rectangular component. The computing device generates a system of constraints, where each constraint is related to at least two grid lines of the plurality of grid lines. The computing device solves the system of constraints to determine a location for each grid line in the plurality of grid lines. The computing device generates a display of the rectangular components based on the locations of the grid lines.