Differential Geometry of Curves - Definitions

Definitions

Let n be a natural number, r a natural number or ∞, I be a non-empty interval of real numbers and t in I. A vector-valued function

of class Cr (i.e. γ is r times continuously differentiable) is called a parametric curve of class Cr or a Cr parametrization of the curve γ. t is called the parameter of the curve γ. γ(I) is called the image of the curve. It is important to distinguish between a curve γ and the image of a curve γ(I) because a given image can be described by several different Cr curves.

One may think of the parameter t as representing time and the curve γ(t) as the trajectory of a moving particle in space.

If I is a closed interval, we call γ(a) the starting point and γ(b) the endpoint of the curve γ.

If γ(a) = γ(b), we say γ is closed or a loop. Furthermore, we call γ a closed Cr-curve if γ(k)(a) = γ(k)(b) for all k ≤ r.

If γ:(a,b) → Rn is injective, we call the curve simple.

If γ is a parametric curve which can be locally described as a power series, we call the curve analytic or of class .

We write -γ to say the curve is traversed in opposite direction.

A Ck-curve

is called regular of order m if for any t in interval I

are linearly independent in Rn.

In particular, a C1-curve γ is regular if

for any .