12/30/99
2-1
GEOMETRIC MODELING: A First Course
Copyright © 1995-1999 by Aristides A. G.
Requicha
Permission is hereby granted to copy this document for individual student use at USC,
provided that this notice is included in each copy.
2. Motions and Projections
2.1 Points and Vectors
Imagine a small solid object and let its dimensions decrease indefinitely. The result of this
conceptual experiment is modeled by a mathematical abstraction called a
point
. Modern
mathematics defines rigorously
Euclidean spaces
as sets whose elements, called
points
,
satisfy certain axioms. In everyday language we talk of “being at a point in space”, and in
geometric modeling we use Euclidean points to define mathematically the locations of
objects. In addition, sets of points serve to model more complicated objects, from
trajectories to physical solids.
Consider now a solid object in straight-line motion. The object’s velocity has a direction
and a magnitude, or speed, measured
e.g.
in meters per second. Velocities and other
physical entities such as forces that can be characterized by a direction and a magnitude are
modeled mathematically by
vectors
. Vectors may be added by using the familiar
parallelogram rule of analytical geometry and elementary mechanics. They may also be
multiplied by scalar numbers. Scalar multiplication changes the magnitude of the vector but
not its direction. In modern mathematics a
vector space
is a set of elements, called vectors,
with two operations defined on them—vector addition and multiplication by a scalar—that
have certain algebraic properties defined axiomatically. The vector-space axioms ensure that
the usual Cartesian vectors of analytic geometry are a special case of abstract vectors.
Interestingly, there are many other useful entities that are abstract vectors as well. Examples
include polynomials of degree
≤
n
, the spline functions we will discuss later in this course,
periodic functions with period
T
, continuous functions in a closed interval [
a
,
b
], and so on.
The theory of vector spaces applies equally well to all of these entities. This is a good
example of the power and elegance of abstraction in modern mathematics.
Points and vectors are intimately connected. In principle there are no privileged points or
directions in space, i.e., space is homogeneous and isotropic. But let us pick some
arbitrary point
o
and call it the
origin
. (Typically, the origin is selected for convenience in
solving a specific problem.) Now each point
p
≠
o
plus
o
define a direction and a length.
Therefore, for a fixed origin
o
, each point
p
corresponds to a vector
x
, and conversely.
That is, there is a one-to-one correspondence between points and vectors. We are arguing
intuitively, but the argument can be rigorized by defining
point difference
as an operation
that produces a vector from two point arguments such that
p
–
o
=
x
. We use the notation
p
o
← →
x
to signify that point
p
corresponds to vector