Graphics Reference
In-Depth Information
-8
x
+ 6y = -16
(8,8)
-8
x
+ 6y > -16
-8
x
+ 6y < -16
(2,0)
Figure 3.12
The 2D hyperplane −
8
x
+
6
y
=−16 (a line) divides the plane into two halfspaces.
Planes in arbitrary dimensions are referred to as
hyperplanes
: planes with one less
dimension than the space they are in. In 2D, hyperplanes correspond to a line; in 3D,
to a plane. Any hyperplane divides the space it is in into two infinite sets of points on
either side of the plane. These two sets are referred to as
halfspaces
(Figure 3.12). If the
points on the dividing plane are considered included in the halfspace, the halfspace
is
closed
(otherwise, it is called
open
). The
positive halfspace
lies on the side in which
the plane normal points, and the
negative halfspace
on the opposite side of the plane.
A 2D halfspace is also called a
halfplane
.
3.7
Polygons
A
polygon
is a closed figure with
n
sides, defined by an ordered set of three or more
points in the plane in such a way that each point is connected to the next (and the
last to the first) with a line segment. For a set of
n
points, the resulting polygon is also
called an n
-sided polygon
or just n
-gon
. The line segments that make up the polygon
boundary are referred to as the polygon
sides
or
edges
, and the points themselves are
called the polygon
vertices
(singular,
vertex
). Two vertices are
adjacent
if they are joined
by an edge. Figure 3.13 illustrates the components of a polygon.
A polygon is
simple
if no two nonconsecutive edges have a point in common. A
simple polygon partitions the plane into two disjoint parts: the
interior
(the bounded
