Biomedical Engineering Reference
In-Depth Information
z
P
e
z
x
=
xe
x
+
ye
y
+
ze
z
y
e
x
O
e
y
x
Figure 7.2
Coordinate system, position of a point P in space.
perpendicular unit vectors. The position of an arbitrary point P in space can be
defined with the position vector
x
, starting at the origin O and with the end point
in P. This position vector can be specified by means of components with respect
to the basis, defined earlier:
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
x
·
e
x
x
·
e
y
x
·
e
z
x
y
z
x
=
x
e
x
+
y
e
y
+
z
e
z
with
∼
=
.
(7.1)
The column
∼
is in fact a representation of the vector
x
with respect to the chosen
basis vectors
e
x
,
e
y
and
e
z
only. Nevertheless, the position vector at hand will
sometimes also be indicated with
∼
. This will not jeopardize uniqueness, because
in the present context only one single set of basis vectors will be used.
The geometry of a curve (a set of points joined together) in three-dimensional
space can be defined by means of a parameter description
x
=
x
(
ξ
), in compo-
nents
∼
=
∼
(
ξ
), where
ξ
is specified within a certain interval, see Fig.
7.3
. The
tangent vector, with unit length, at an arbitrary point of this curve is depicted by
e
, satisfying
dx
d
ξ
dx
d
ξ
1
dx
d
ξ
e
=
with
=
·
(7.2)
and after a transition to component format:
d
∼
d
T
d
∼
d
.
1
d
∼
d
∼
=
with
=
(7.3)
ξ
ξ
ξ
The parameter
equals 1; in that special case the length
of an (infinitesimally small) line piece
dx
(with components
d
∼
) of the curve is
exactly the same as the accompanying change
d
ξ
is distance measuring if
ξ
.
