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

Thus, from the equation of the tangent:
we deduce
or
while for the normal we obtain the equation
1.6 The osculating plane
1.6. The osculating plane 39
Y − y0 = f ′ (x0)(X − x0),
Y − y0 = − F′ x
F ′ (X − x0)
y
(X − x0)F ′ x + (Y − y0)F ′ y = 0,
(X − x0)F ′ y − (Y − y0)F ′ x = 0.
Definition 1.6.1. A parameterized curve r = r(t) is called biregular (or in general position)
at the point t0 if the vectors r ′ (t0) and r ′′ (t0) are not colinear, i.e.
r ′ (t0) × r ′′ (t0) � 0.
The parameterized curve is called biregular if it is biregular at each point 5 .
Remark. It is not difficult to check that the notion of a biregular point is independent
of the parameterization: if a point is biregular for a given parameterized curve, then its
corresponding point through any parameter change is, also, a biregular point.
Definition 1.6.2. Let (I, r) be a parameterized curve and t0 ∈ I – a biregular point. The
osculating plane of the curve at r(t0) is the plane through r(t0), parallel to the vectors
r ′ (t0) and r ′′ (t0), i.e. the equation of the plane is
(R − r(t0), r ′ (t0), r ′′ (t0)) = 0, (1.6.1)
or, expanding the mixed product,
�
�
�
�X
− x0
�
�
�
�
�
Y − y0
�
Z −
�
z0�
�
�
�
� = 0.
�
�
(1.6.2)
x ′ y ′ z ′
x ′′ y ′′ z ′′
5 A biregular curve is also called in some books a complete curve. We find this term a little bit misleading,
since this terms has, usually, other meaning in the global theory of curves (and, especially, surfaces).