Many mathematical theorems are clarified by thinking in projective space, where so-called "points at infinity" are added to complete the geometry. Often these points at infinity are left out of drawings, with their relation to the other points and to each other suggested rather than directly illustrated. Yet to an observer living inside of projective space, these points at infinity would be geometrically indistinguishable from any other point. In this work, we present a real-time rendering framework for the intrinsic visualization of 3D real projective space by ray marching directly on the 3-sphere. Leveraging this technique, we observe several properties that are most elegantly stated for projective space. We demonstrate our library's capabilities through visualizing the geometry of symmetric cubic surfaces and families of cubic curves on quartic surfaces, as well as how they manifest in this unbiased, personified view.