Graphics Reference
In-Depth Information
a wobbly platform. Planar rotations are discussed in detail in Asimov (
), more
simply in Buja and Aasimov (
a), and very technically in Buja et al. (
), and
in Asimov and Buja (
), and Buja et al. (
b) as well.
Differences in the method of selecting the target plane yield different types of
tours.hegrandtourusesarandomselectionoftargetplane.heguidedtourquanti-
fiesthe structurepresent ineachprojection andusesthis toguidethe choiceoftarget
plane. Manual controls let the user choose the target direction by manipulating the
values of projection coe
cients.
Terminology: Plane, Basis, Frame, Projection
2.2.1
It is conventional to use a p-dimensional orthonormal basis:
...
p
p
todescribe p-dimensional Euclideanspace.A d-dimensional planein p-spacecan be
defined by an infinite number of d-dimensional orthonormal frames. For example,
consider the d
=
-dimensional frames:
both of which describe the same
-D plane. We conventionally use
A
as the frame
describing the
-Dplane, but we could just as validly use
A
.Figure
.
illustrates the
two frames, which result in the same but rotated projections of the data.
In GGobi tours, we generate a new target basis and use this to define the target
plane.Buttheactualbasisusedtocreatethedata projectionisawithin-plane rotation
of the target basis that matches the basis describing the starting plane.
−
a
a
a
p
a
a
a
p
A
and
A
=
=
=
=
Interpolating Between Projections: Making a Movie
2.2.2
Amovieofdata projections iscreated byinterpolating along ageodesic path fromthe
current (starting) plane to the new target plane. he algorithm follows these steps:
. Given a starting p
d projection
A
a
, describing the starting plane, create a new
target projection
A
z
, describing the target plane. It is important to check that
A
a
and
A
z
describe different planes, and generate a new
A
z
if not. To find the
optimal rotation of the starting plane into the target plane we need to find the
frames in each plane that are the closest.
