Now let us stack these multiple representations along a vertical axis, as in Figure 4 (right), so
that each scaled view Rs intersects the vertical axis at s, and the vertical axis goes through the
origin of each scaled representation. A space-scale diagram (Furnas and Bederson, 1995) is a
conceptual extension of this stacking where every possible scaled representation of the
document is included (in actual practice, of course, just a few will do). A space-scale diagram
is therefore a 3D representation of a 2D document, with two dimensions of space and one
dimension of scale. The main property of a space-scale diagram is that a point (x, y) of the
original document is represented by a great ray—i.e., a semi-line going through the origin and
the point (x, y, 1). All the points on the semi-line are of the form (s.x, s.y, s). This property can
be captured by the following scaling invariant: distance d at scale s is identical to distance d'
at scale s' if and only if
d / s = d' / s'
(3)
Figure 4. Furnas et Bederson’s (1995) space-scale diagram. A document (left) is represented at several
levels of magnification (right). Axes x and y define space, axis s defines scale.
To describe and understand navigation in a multiscale document, a space-scale diagram is
more efficient that the purely spatial representation of Figure 2 because it contains many
possible scaled representations in a single chart. A view of a multiscale document is specified
by a (usually) rectangular window (Figure 5, left), together with the scale at which the
document is viewed. This scale corresponds to the plane that contains the view in the diagram
(Figure 5, middle). Panning consists of moving the view within its plane, i.e. at constant
scale. Zooming consists of moving the view up and down along the scale dimension (Figure 5,
right): zooming-in magnifies the view by moving up the scale dimension, thus revealing
details, while zooming-out minifies the view by moving down the scale dimension, thus
revealing context. Note that the size of the view does not change while zooming: at a larger
scale, the view will show more detail, but this will be at the cost of a reduced selection of the
document.