Recently, we have studied visualizations of the 4-space in the double orthogonal projection onto two mutually perpendicular 3-spaces. A studied object is projected onto its two conjugated images in the modeling 3-space - 3D graphics on the computer screen.
In this contribution, we use this method of projection to construct spherical, circular and point intersections of a 3-sphere with a 3-space, plane, and line. The provided step-by-step constructions are created in the interactive software GeoGebra 3D.