Biomedical Engineering Reference
In-Depth Information
where
r
1
=
x
2
+
y
2
+(
z
+
δ
)
2
,
r
2
=
x
2
+
y
2
+(
z
−
δ
)
2
,
(7)
and
δ
is the distance from the origin to either focus. The reverse transformation is
calculated from
x
=
δ
sinh
λ
sin
η
cos
φ,
y
=
δ
sinh
λ
sin
η
sin
φ,
(8)
z
=
δ
cosh
λ
cos
η.
The B-spline basis functions determine the shape of the curve by smoothing, in
a weighted fashion, over the coordinates of the control points. Thus, by changing
the coordinates of the control points from a Cartesian to another description, the
entire curve changes. This distinction is illustrated by the plots in Figure 4. The
Cartesian B-spline curve is shown in Figure 4a. It was plotted on the Cartesian
axes using the formulae
n
i
,
x
(
u
)=
N
i,d
(
u
)
P
(9)
i
=1
n
y
i
,
y
(
u
)=
N
i,d
(
u
)
P
(10)
i
=1
y
i
) is the Cartesian description of the
i
th control point. The non-
Cartesian (polar) B-spline curve is plotted on the same Cartesian axes in Figure 4b.
The curve is produced by the formulae
i
,
where (
P
P
x
(
u
)=
r
(
u
) cos(
φ
(
u
))
n
n
,
φ
i
r
i
=
N
i,d
(
u
)
P
cos
N
i,d
(
u
)
P
(11)
i
=1
i
=1
y
(
u
)=
r
(
u
) sin(
φ
(
u
))
n
n
,
φ
i
r
i
=
N
i,d
(
u
)
P
sin
N
i,d
(
u
)
P
(12)
i
=1
i
=1
φ
i
) is the polar coordinate of the
i
th control point.
r
where (
P
i
,
P
