  This paper defines and discusses a set of rectangular all-sky projections
which have no singular points, notably the Tesselated Octahedral Adaptive
Spherical Transformation (or TOAST) developed initially for the WorldWide
Telescope (WWT). These have proven to be useful as intermediate representations
for imaging data where the application transforms dynamically from a
standardized internal format to a specific format (projection, scaling,
orientation, etc.) requested by the user. TOAST is strongly related to the
Hierarchical Triangular Mesh (HTM) pixelization and is particularly well
adapted to the situations where one wishes to traverse a hierarchy of
increasing resolution images. Since it can be recursively computed using a very
simple algorithm it is particularly adaptable to use by graphical processing
units.
