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We present GyVe, an interactive visualization tool for understanding structure in sparse three-dimensional (3D) point data. The scientific goal driving the tool's development is to determine the presence of filaments and voids as defined by inferred 3D galaxy positions within the Horologium-Reticulum supercluster (HRS). GyVe provides visualization techniques tailored to examine structures defined by the intercluster galaxies. Specific techniques include: interactive user control to move between a global overview and local viewpoints, labelled axes and curved drop lines to indicate positions in the astronomical RA-DEC-cz coordinate system, torsional rocking and stereo to enhance 3D perception, and geometrically distinct glyphs to show potential correlation between intercluster galaxies and known clusters. We discuss the rationale for each design decision and review the success of the techniques in accomplishing the scientific goals. In practice, GyVe has been useful for gaining intuition about structures that were difficult to perceive with 2D projection techniques alone. For example, during their initial session with GyVe, our collaborators quickly confirmed scientific conclusions regarding the large-scale structure of the HRS previously obtained over months of study with 2D projections and statistical techniques. Further use of GyVe revealed the spherical shape of voids and showed that a presumed filament was actually two disconnected structures.