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

16 Chapter 1. Space curves
Definition 1.2.2. We shall say that a compact parameterized curve r : [a, b] → R 3 is
piecewise C k if there exists a finite subdivision (a = a0, a1, . . . , an = b) of the interval
[a, b] such that the restriction of r to each compact interval [ai−1, ai] is of class C k , where
i ∈ {1, . . . , n}.
Remark. It is not difficult to show that a parameterized curve r : [a, b] → R 3 is piecewise
C k iff the following conditions are simultaneously fulfilled:
(i) The set
is finite.
(ii) f (k) is continuous on [a, b] \ S .
S = � t ∈ [a, b] | f (k) does not exist �
(iii) f (k) has a finite left and right lateral limit at each point of S .
Hereafter we shall suppose, all the time, that the order of smoothness k is high
enough and we will not mention it any longer (with some exceptions, however), using
the generic term of smooth parameterized curve (meaning, thus, at least C 1 and, in
any particular case, if kth order derivatives are involved, at least C k ).
The image r(I) ⊂ R 3 of the interval I through the mapping (1.2.1) is termed the
support of the path (I, r).
If r(t0) = a, we shall say that the parameterized curve passes through the point
a for t = t0. Sometimes, for short, we shall refer to this point as the point t0 of the
parameterized curve.
Examples
1. Let r0 ∈ R 3 be an arbitrary point, and a ∈ R 3 a vector, a � 0, while I = R. The
parameterized curve R → R 3 , t → r0 + ta is called a straight line. Its support is
the straight line passing through r0 (for t = 0) and having the direction given by
the vector a.
2. I = R, r(t) = r0 + t 3 a. The support of this path is the same straight line.
3. I = R, r(t) = (a cos t, a sin t, bt), a, b ∈ R. The support of this parameterized curve
is called a circular cylindrical helix (see figure 1.2).
4. I = [0, 2π], r(t) = (cos t, sin t, 0). The support of the path is the unit circle, lying
in the xOy-plane, having the center at the origin of the coordinates.