Differential geometry notes 2
Definition 9.1: Let be an allowable parametric representation of an arc with initial point and terminal point . Denote by the length of a broken line of chords whose end points lie on and correspond to the values . Let and . If , then is called rectifiable and is called the length of .
Theorem 9.1: Let be an allowable parametric representation of an arc of a curve of class ( times differentiable). Then has length:
and is independent of the choice of the allowable parametric representation.
Arc length: The function is called the arc length of .
The arc length may be used as the parameter in a parametric representation of a curve: is an allowable parametric representation. is called the natural parameter. The choice of as the parametric representation simplifies many investigations.
Unit tangent vector: The vector is called the unit tangent vector to the curve at point .
The straight line passing through a point of in the direction of the corresponding unit tangent vector is called the tangent to the curve at . The tangent can be represented in the form . Thus, is the point of contact between and the tangent.
The totality of all vectors bound at a point of which are orthogonal to the corresponding unit tangent vector lie in a plane. This plane is called the normal plane to at .