1. A data visualization system for generating three-dimensional (3D) visualizations of a multidimensional data space, comprising: a display device; and a computing system, comprising: memory containing a 3D data visualization application; and a processing system; wherein the 3D data visualization application directs the processing system to: load a set of data points into a visualization table in the memory, where each data point includes values in multiple data dimensions and an additional visibility value is assigned to each data point in a visibility dimension within the visualization table; create representations of a set of 3D objects corresponding to the set of data points, where each 3D object has a set of visualization attributes that determine the manner in which the 3D object is rendered and the visualization attributes include a location of the 3D object within a virtual space having three spatial dimensions; receive mappings of data dimensions to visualization attributes; determine the visualization attributes of the set of 3D objects based upon the selected mappings of data dimensions to 3D object attributes, where the selected mappings of data dimensions to visualization attributes determine a location for each visible 3D object within the virtual space; update the visibility dimension in the visualization table for each of the plurality of 3D object to reflect the visibility of each 3D object based upon the selected mappings of data dimensions to visualization attributes; and interactively render 3D data visualizations of the 3D objects within the virtual space from viewpoints determined based upon received user input; wherein the 3D data visualization application further directs the processing system to: receive at least one updated mapping of a data dimension to a visualization attribute; determine updated visualization attributes for the set of 3D objects based upon the selected mappings of data dimensions to 3D object attributes, where the updated mappings of data dimensions to visualization attributes determine a location for each visible 3D object within an updated virtual space; generate trajectories for the set of visible 3D objects from their locations in the virtual space to their updated locations in the virtual space; and interactively render animations of the movements of 3D objects along their generated trajectories from their locations in the virtual space to their locations in the updated virtual space from viewpoints determined based upon received user input.