Biomedical Engineering Reference
In-Depth Information
3
()
(
)
(
)
6
3
3
3
−
6
Dp
=−
−
q
=−
Π
Y
−
Π
/
Y
−
12
/
.
(6.54)
−
−
−
2
−
2
−
In order for equations (6.49) and (6.50) to have all three real solutions,
D
+
and
D
-
have to be positive definitive. If
D
+
and
D
-
are negative, then there exists only
one real root and two complex conjugate roots, which will be unacceptable. Applying
the Tartaglia-Cardan formula to equations (6.49) and (6.50) yields
xuvx
=+
,
=
ε
u vx
+
ε
,
=
ε
u v
+
ε
,
(6.55)
+
1
+
+
+
2
1
+
2
+
+
3
2
+
1
+
and
xuvx
=+
,
=
ε
u vx
+
ε
,
=
ε
u v
+
ε
,
(6.56)
−
1
−
−
−
2
1
−
2
−
−
3
2
−
1
−
where
(
)
(
)
±
(6.57)
ε
12
=−
12
/
3 2
/
,
,
13
/
(
)
12
/
(6.58)
2
3
u
=− +
q
q
+
p
,
+
+
+
+
13
/
(
)
12
/
2
3
(6.59)
v
=− −
q
q
+
p
,
+
+
+
+
13
/
(
)
12
/
2
3
(6.60)
u
=− +
q
q
+
p
,
−
−
−
−
13
/
(
)
12
/
2
3
(6.61)
v
=− −
q
q
+
p
,
−
−
−
−
The possible curvatures with a real value are then calculated as
(
)
=
(
(
)
(
)
)
(
)
−
(
)
−
(
)
κ
=
λ
−
λ
/
2
C
*
1
/
2
C
*
x
−
x
Π
/
Y
Π
/
Y
−
,
(6.62)
E
+
−
+
−
1
+
2
where
(
)
+−
(
)
xx uu
−= −
vv
(6.63)
+
−
+
−
+
−
