Graphics Reference
In-Depth Information
For a line ℓ in
-D, the three points ℓ
, ℓ
, ℓ
are collinear; we denote this line
by L, and any two represent ℓ.ApolygonallineonalloftheN
points,asgivenby
Eq.
.
or their equivalent, represents a point on the line ℓ. Conversely, two points
determine a line ℓ. Starting with the two polygonal lines representing the points, the
N
−
intersections of their X
i
−
, X
i
portions arethe ℓ
i
−
,i
points fortheline ℓ.Aline
interval in
-D and several of its points is seen in Fig.
.
(right). By the way, it is
essential to index the points ℓ.
−
Planes and Hyperplanes
14.5.2
While a line can be determined from its projections, a plane cannot, even in
-D.
A new approach is called for Eickemeyer (
). Rather than discerning a p-dimen-
sional object from its points, it is described in terms of its (p
)-dimensional subsets
constructed from the points. Let's see how this works. In Fig.
.
(let), polygonal
lines representing a set of coplanar points in
-D are shown. Even the most persistent
pattern-seeker will not detect any clues hinting at a relationship between the points,
muchlessalinear one,based onthis picture.Instead, foreach pairofpolygonal lines,
the line L of the three-point collinearity described above is constructed. he result,
shown on the right, is stunning. All the L lines intersect at a point, which turns out
tobecharacteristic ofcoplanarity, although this isnot enoughinformation tospecify
the plane. Translating the first axis X
to the position X
′
, one unit to the right of the
X
axis, and repeating the construction yields the second point shown in Fig.
.
(let). For a plane given by
−
π
c
x
+
c
x
+
c
x
=
c
,
(
.
)
the two points, in the order they are constructed, are respectively
c
+
c
,
c
S
c
+
c
+
c
,
c
S
π
=
, π
′
=
(
.
)
S
S
Figure
.
.
(Let) he polygonal lines on the first three axes represent a set of coplanar points in
-D.
(Right) Coplanarity! Lines are formed on the plane using the three-point collinearity intersect at a point
