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in their topic [142]. For quadratic periodic B-spline curves, we have
m
=3
and it can be shown that the starting and end points are given by
1
P
s
=
2
(
V
0
+
V
1
)
1
P
e
=
2
(
V
n
1
+
V
n
)
,
and for cubic curves (
m
=4)
1
6
(
V
0
+4
V
1
+
V
2
)
P
s
=
1
6
(
V
n
2
+4
V
n−
1
+
V
n
)
.
P
e
=
The first derivative at these points for
m
=3is
P
s
=
V
1
−
V
0
P
e
=
V
n
−
V
n−
1
,
while for cubic periodic B-spline (
m
= 4), first and second order derivatives
are
P
s
=
P
s
V
0
−
1
2
(
V
2
−
V
0
)
2
V
1
+
V
2
P
e
=
1
2
(
V
n
−
V
n−
2
P
e
V
n−
2
−
2
V
n−
1
+
V
n
.
Multiple coincident vertices at one end of a periodic B-spline curve pulls the
starting and end points of the curve nearer to the vertex. For
m
1 multiple
coincident vertices, the end point of the curve coincides with the vertices and
the tangent vector of the curve.
−
5.5 Rational B-Spline Curve
Adams and Rogers nicely explained the rational B-spline curves in their topic
[142]. We shall slightly review it so that the image processing and machine
vision community can examine the possibility of using it in their area. Before
we explain rational B-spline curve, we would like to explain homogeneous
coordinates in some detail.
5.5.1 Homogeneous Coordinates
In order to study the geometric relationships of figures under perspective
transformations, projective planes were introduced by geometers. The two
dimensional projective plane is defined as follows:
In a three dimensional Cartesian space, consider the set of all lines through
the origin and the set of all panes through the origin. In the projective plane,
a line through the origin is called a point, while a plane through the origin
is called a line of the projective plane. This is so, because if we consider the
perspective projection onto the plane
z
= 1 using the origin as the center of
projection, then a line through the origin always projects onto a point on the
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