Traditional perspective painting projects the world from the eye-point onto a single rectangular screen. In contrast, this article works with spherical images, created by projecting the world onto a closed sphere surrounding the eye-point, the viewable sphere. We focus on the use of stereographic projection of the viewable sphere to produce flat images which are conformal and include arbitrarily large fractions of the viewable sphere. We describe a practical, real-time implementation of this projection embedded in a general-purpose visualisation system. We then apply the method to provide a new way of visualising tessellations of three-dimensional euclidean (and non-euclidean) space, and give a preliminary discussion of the results.