Graphics Reference
In-Depth Information
(1) by choosing from a collection of well-known predefined manifolds that topo-
logically are homeomorphic to spaces such as D 3 , S 3 , or P 3 ,
(2) by gluing two parts of the boundary of an already constructed manifold
together,
(3) by attaching one manifold to another along regions in their boundary,
(4) by a connected sum operation on two existing manifolds,
(5) via a spherical modification (surgery) operation on an existing manifold or by
adding a handle to the boundary, and
(6) as a quotient of D 3 by the action of certain transformation groups.
Visualization.
One can visualize a manifold M using one of two views.
The “global” view:
This is the intrinsic view where the observer is moving
through M .
This shows a perspective view of the individual cells in R 3
that were used in the construction of M . There is an associ-
ated orthographic view, which is simply the orthographic pro-
jection of the local view onto the x-y plane.
The “local” view:
One can think of these views as corresponding to two worlds - a global and local
world. Looking at this another way, there is an abstract cell structure which is the
same in the global and local world but their point data is different (the two worlds
correspond to different realizations of the abstract cells). Basically, the observer is
inside the manifold in the global world and outside looking at its parts in the local
world. Users can move the cells in the local view without changing the topological
type of the manifold. The local view has a mode where the identification of bound-
ary polygons is indicated. Both the global and perspective local view can be seen
simultaneously. The observer is indicated by means of a cone in the perspective and
orthographic local view. The observer can be moved in various ways and all the
representations of the observer in the different views move in parallel.
Initially, when a manifold is defined, the boundaries of the cells that make it
up are shown when one is in the global view. It is convenient to think of the bound-
ary polygons as “walls.” At this point the user can start collapsing the cell structure,
either by piercing individual walls or by letting the program automatically collapse
all the walls as far as possible to a “reduced” state. The observer would then see
only walls that are left and that cannot be collapsed any further. One can make
these also invisible, but in that case, assuming that the manifold did not have any
boundary, there would be nothing to see (actually the user can specify how far one
can look in terms of the number of cells that one can look through). To make things
more interesting one can define (and move) objects in the cells of the manifold,
so that the topology of the manifold will then influence the way and the number of
times that one sees them as one looks straight ahead. One can also make marks on
the walls.
Figures 16.1-16.3 (the color version of these figures are .GIF files that can be found
on the accompanying CD) are examples of what one can see in the SPACE program.
Figure 16.1 shows the (2 1 / 2 )-dimensional (global) view from inside the surface S 2 . In
fact, we see how the entire screen on the computer monitor would look. The top of
the screen shows some status information. On the right are menu items. Immediately
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