Graphics Reference
In-Depth Information
3.2
Coordinate Systems and Points
A
point
is a position in space, the location of which is described in terms of a
coordinate
system
, given by a reference point, called the
origin
, and a number of
coordinate axes
.
Points in an
n
-dimensional coordinate system are each specified by an
n
-tuple of real
numbers (
x
1
,
x
2
,
...
,
x
n
). The
n
-tuple is called the
coordinate
of the point. The point
described by the
n
-tuple is the one reached by starting at the origin and moving
x
1
units along the first coordinate axis,
x
2
units along the second coordinate axis, and so
on for all given numbers. The origin is the point with all zero components, (0, 0,
...
, 0).
A coordinate system may be given relative to a parent coordinate system, in which
case the origin of the subordinate coordinate system may correspond to any point in
the parent coordinate system.
Of primary interest is the
Cartesian
(or
rectangular
) coordinate system, where the
coordinate axes are perpendicular to each other. For a 2D space, the two coordinate
axes are conventionally denoted the
x
axis and the
y
axis. In a 3D space, the third
coordinate axis is called the
z
axis.
The
coordinate space
is the set of points the coordinate system can specify. The
coordinate system is said to
span
this space. A given set of coordinate axes spanning
a space is called the
frame of reference
,or
basis
, for the space. There are infinitely many
frames of reference for a given coordinate space.
In this topic, points are denoted by uppercase letters set in italics (for example,
P
,
Q
, and
R
). Points are closely related to vectors, as discussed in the next section.
3.3
Vectors
Abstractly,
vectors
are defined as members of
vector spaces
. A vector space is defined in
terms of a set of elements (the vectors) that support the operations of
vector addition
and
scalar multiplication
, elements and operations all obeying a number of axioms.
In this abstract sense,
m
n
matrices of real numbers may, for example, be elements
(vectors) of a vector space. However, for the practical purposes of this topic vectors
typically belong to the vector space
×
n
, whose elements are
n
-tuples of numbers from
the domain of real numbers. A vector
v
is thus given as
R
v
=
(
v
1
,
v
2
,
...
,
v
n
).
The number terms
v
1
,
v
2
,
...
,
v
n
are called the
components
of
v
. In fact, in this topic
vectors are predominantly restricted to the special cases
2
and
3
of
n
, thus being
R
R
R
given as tuples of two and three real numbers, respectively.
Informally, vectors are often described as entities having both direction and magni-
tude. They are therefore usually represented graphically by an arrow, or geometrically
as a directed line segment, pointing in a given direction and of a given length.Vectors
are denoted by boldface lowercase letters (for example,
u
,
v
, and
w
).
