GEOMETRIC MODELING: A First Course
Copyright © 1995-1999 by Aristides A. G.
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
mathematics defines rigorously
as sets whose elements, called
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
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 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
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
, the spline functions we will discuss later in this course,
periodic functions with period
, continuous functions in a closed interval [
], 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
and call it the
. (Typically, the origin is selected for convenience in
solving a specific problem.) Now each point
define a direction and a length.
Therefore, for a fixed origin
, each point
corresponds to a vector
, 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
as an operation
that produces a vector from two point arguments such that
. We use the notation
to signify that point
corresponds to vector