Biomedical Engineering Reference
In-Depth Information
Figure 3.9
Radius of curvature and osculating circle.
xy
��� ���
-
yx
(3.8)
κ
=
3
2
2
2
(
x
� �
+
y
)
where the dot denotes a differentiation with respect to
t
. For a plane curve given
implicitly as
f
(
x
,
y
) = 0, the curvature is
æ
ö
Ñ
f
f
(3.9)
κ
= Ñ
ç
.
÷
Ñ
è
ø
that is, the divergence of the direction of the gradient of
f
. And for an explicit func-
tion
y
=
f
(
x
), the curvature is defined by
2
d y
d x
2
κ
=
(3.10)
3
2
2
æ
ö
æ
d y
d x
ö
1
+
ç
÷
ç
÷
ç
è
ø
÷
è
ø
The situation is more complex for a surface. Any plane containing the vector
normal to the surface intersects the surface along a curve. Each of these curves has
its own curvature. The mean curvature of the surface is defined using the principal
(maximum and minimum) curvatures k
1
and k
2
(Figure 3.10) in the whole set of
curvatures
1
H κ κ
=
+
)
(3.11)
(
1
2
Introducing the curvature radii in (3.11) leads to
1
1
æ
1
1
ö
H
=
(
κ κ
+
)
=
+
(3.12)
ç
÷
1
2
è
ø
2
2
R
R
1
2