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15.3. Polynomial Pieces 349
The canonical form does not always have convenient coefficients. For prac-
tical purposes, throughout this chapter, we will find sets of basis functions such
that the coefficients are convenient ways to control the curves represented by the
polynomial functions.
To specify a curve embedded in two dimensions, one can either specify two
polynomials in t: one for how x varies with t and one for how y varies with t;
or specify a single polynomial where each of the a
i
is a 2D point. An analogous
situation exists for any curve in an n-dimensional space.
15.3.2 A Line Segment
To introduce the concepts of piecewise polynomial curve representations, we will
discuss line segments. In practice, line segments are so simple that the mathemat-
ical derivations will seem excessive. However, by understanding this simple case,
things will be easier when we move on to more complicated polynomials.
Consider a line segment that connects point p
0
to p
1
. We could write the
parametric function over the unit domain for this line segment as
f(u)=(1− u)p
0
+ up
1
. (15.6)
By writing this in vector form, we have hidden the dimensionality of the points
and the fact that we are dealing with each dimension separately. For example,
were we working in 2D, we could have created separate equations:
f
x
(u)= (1−u)x
0
+ ux
1
,
f
y
(u)= (1− u)y
0
+ uy
1
.
The line that we specify is determined by the two end points, but from now
on we will stick to vector notation since it is cleaner. We will call the vector of
control parameters, p,thecontrol points, and each element of p,acontrol point.
While describing a line segment by the positions of its endpoints is obvious
and usually convenient, there are other ways to describe a line segment. For
example,
1. the position of the center of the line segment, the orientation, and the length;
2. the position of one endpoint and the position of the second point relative to
the first;
3. the position of the middle of the line segment and one endpoint.