Blaga P. Lectures on the differential geometry of - tiera.ru

1.13. The contact between a space curve and a sphere. The osculating sphere 67
the curve with respect to the Frenet frame at M0. Thus, in our case, we have
or
F(x(s), y(s), z(s)) =
�
1
+
− 2b
�
s − 1
6 k(0)χ(0)s3 + o(s 3 )
� 2
6 k2 (0)s 3 + o(s 3 �2 )
− 2a
�
s − 1
�
1
2 k(0)s2 + 1
6 k′ (0)s 3 + o(s 3 �
)
�
1
+
6 k2 (0)s 3 + o(s 3 )
− 2c
2 k(0)s2 + 1
�
−
�
1
6 k(0)χ(0)s3 + o(s 3 �
)
6 k′ (0)s 3 + o(s 3 �2 )
= 0.
+
(1.13.2)
−2as + (1 − bk(0))s 2 + 1
3 (ak2 (0) − bk ′ (0) − ck(0)χ(0))s 3 + o(s 3 ). (1.13.3)
Now, the discussion that follows is, also, similar to that from the case of the contact
between a curve and a plane. Still, here we have more cases to take into consideration.
a) If a � 0, then s = 0 is a simple solution of the intersection equation. Thus, in this
case the sphere and the curve have a contact of order zero (intersection contact).
b) The sphere and the curve have a first order contact (tangency contact) if and only if
the intersection equation has a double zero at the origin. This happens, obviously, iff
a = 0. This means that the first coordinate of the center of the sphere is zero, i.e. this
center lies in the normal plane of the curve at M0. It is easy to see that in this case
the tangent line at M0 of the curve lies in the tangent plane of the sphere at the same
point, which justifies once more the denomination of tangency contact.
c) For an osculation (second order) contact, the coefficient of s 2 in the intersection equation
also has to vanish and we get
b = 1
= R(0), (1.13.4)
k(0)
where R(0) is the curvature radius of the curve at the point M0. Thus, in this case, the
center of the sphere lies on the intersection line of the planes x = 0 (the normal plane)
and y = R(0). This line, which, as one can easily see, is parallel to the binormal of
the curve at M0, is called curvature axis or polar axis of the curve. It intersects the
osculating plane at M0 of the curve at the center of curvature at M0 of the curve.
Thus, any sphere with the center on the curvature axis of the curve has an osculation
contact with this one.