Geoscience Reference
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x-axis coincides with the tangent line at point P. Together with the y-axis, the right-
handed coordinate system is constituted (see Fig.
5.10
). Obviously, in the new
coordinate system, the equation of the normal section plane is y
0. Put the
equation of the normal section plane together with the equation of the ellipsoid to
form simultaneous equations. The solutions to the simultaneous equations cannot
be found unless the equation of the ellipsoid in the new coordinate system is
obtained. In this case, the equation of the normal section will easily be solved,
and the radius of curvature of the normal section can also be obtained.
To conclude, derivations of the formulae can be broken down into the following
three steps:
ﾼ
1. Find the equation of the ellipsoid in the coordinate system P-xyz.
2. Find the equation of the normal section in any arbitrary direction.
3. Find the radius of curvature of the normal section in an arbitrary direction.
In formula derivations we need the formula for the coordinate transformation by
rotating the systems and the formula for the radius of curvature of a plane curve, as
given below.
We transform the right-handed Cartesian coordinate system by rotating the
coordinate system through a counterclockwise angle
ʸ
z
about the Z-axis (
ʸ
z
is
positive) according to right-hand rule; then:
2
4
3
5
2
4
3
5
2
4
3
5
2
4
3
5
X
Y
Z
cos
ʸ
z
sin
ʸ
z
0
X
Y
Z
X
Y
Z
new
ﾼ
sin
ʸ
z
cos
ʸ
z
0
old
ﾼ
R
Z
ʸðÞ
old
:
0
0
1
R
Z
is the rotation matrix. By the same token, we can obtain the transformation
formulae of the coordinate system by rotating it about the X-axis and the Y-axis, and
the rotation matrixes R
X
and R
Y
are given by:
2
4
3
5
,
1
0
0
R
X
ʸðÞﾼ
0
cos ʸ
x
sin ʸ
x
0
sin
ʸ
x
cos
ʸ
x
2
4
3
5
,
cos
ʸ
y
0
sin
ʸ
y
ʸ
y
ﾼ
R
Y
0
1
0
sin
ʸ
y
0
cos
ʸ
y
The rotation matrix satisfies the orthogonality condition (orthogonal matrix).
According to higher mathematics, the formula for the radius of curvature at point
x
0
on the plane curve y
ﾼ
f(x) can be written as:
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