Graphics Reference
In-Depth Information
Figure . . (Let) Pair of point clusters representing close planes. (Right) he hexagonal regions
(interior) contain the points π (let) and π for the family of planes with c
=
and
c
[
, .
]
, c
[
, .
]
, c
[
,
]
.Forc varying (c
[
. , .
]
here), the (exterior) regions
are octagonal with two vertical edges
thepoints isnotchaotic.heoutline oftwohexagonal patterns canbediscerned.he
family of planes are “visualizable,” as are variations in several directions. It is possible
to see, estimate and compare errors.
Let's not maintain the suspense any longer: in -D the set of pairs of points rep-
resenting the family of proximate planes form two convex hexagons when c
(an
example is shown in Fig. . , right), and are contained in octagons, each of which
has two vertical edges for varying c . In general, a family of proximate hyperplanes
in N-D is represented by N
=
-agons for
varying c (seeMatskewichetal.( )andInselberg( )formorerecentresults).
hese polygonal regions can be constructed with O
convex N-agons when c
=
or
(
N
+
)
computational complexity.
Upon choosing a point in one of the polygonal regions, an algorithm matches the re-
maining N
(
N
)
pointsfromtheremainingconvexpolygonsthatrepresenthyperplanes
in the family. Each hyperplane in the family can be identified by its N
points.
Earlier, we proposedthat visualization is not about seeing lots of things, but rather
it is about discovering relations among them. While a display of randomly sam-
pled points from a family of proximate hyperplanes is utterly chaotic (the mess in
Fig. . , right, is from points in just one plane), their proximate coplanarity rela-
tion corresponds to a clear and compact pattern. With
-coords, we can focus and
concentrate the relational information rather than wallowing in the details, leading
to the phrase “without loss of information” when referring to
-coords. his is the
methodology's real strength, and where I believe the future lies. Here then is the vi-
sualization challenge. How else can we detect and see proximate coplanarity?
Nonlinear Multivariate Relations: Hypersurfaces
14.5.3
Arelation between two real variables isrepresentedgeometrically byaunique region
in -D.Analogously, a relation between N variables corresponds to a hypersurface in
N-D, hence the need to say something about the representation of hypersurfaces in
-coords. A smooth surface in -D (and also N-D) can be described as the envelope
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