A grand tour is a method for visualizing multivariate data through a continuous family of low-dimensional orthogonal projections that is dense
in the Grassmann manifold of subspaces
of the chosen dimension. Asimov (1985) gave the original
published description for the case 
.
Let the vectors 
be the data to be visualized. Fix 
, and let 
be a 
matrix whose columns form
an orthonormal basis for the displayed 
-dimensional plane at time 
. The displayed coordinates
satisfy
 |  |  |
(1)
|
 |  |  |
(2)
|
Here 
denotes the transpose
and 
is the identity
matrix.
Each 
is a point
of the Stiefel manifold 
, while its column space
is a point of the Grassmann
manifold 
.
Multiplying 
on the right by a 
orthogonal matrix changes the orthonormal
basis within the displayed plane, but not the displayed
subspace. The grand tour is therefore naturally a path through the Grassmann
manifold, although it is computed using frames in the Stiefel
manifold.
A common construction repeatedly chooses a target plane and joins it continuously to the current plane, producing an animation in which the displayed data move smoothly between views (Cook et al. 1995, Buja et al. 2005). Unlike principal component analysis, which selects a fixed subspace maximizing retained variance, a grand tour explores many subspaces instead of producing a single reduced representation.
