A unit tangent vector is a vector that is tangent to a curve at a given point and has a magnitude (length) of one. It indicates the direction in which the curve is proceeding at that point.
To find the unit tangent vector using arc length parameterization, you first need to parameterize the curve by its arc length, denoted as \( s \). The unit tangent vector \( \mathbf{T}(s) \) is then given by the derivative of the position vector \( \mathbf{r}(s) \) with respect to \( s \), normalized to have unit length: \( \mathbf{T}(s) = \frac{d\mathbf{r}(s)}{ds} \).
Curvature is a measure of how sharply a curve bends at a given point. For a curve parameterized by arc length \( s \), the curvature \( \kappa(s) \) is defined as the magnitude of the derivative of the unit tangent vector with respect to \( s \): \( \kappa(s) = \left| \frac{d\mathbf{T}(s)}{ds} \right| \).
To compute the curvature using arc length parameterization, you take the derivative of the unit tangent vector \( \mathbf{T}(s) \) with respect to the arc length \( s \) and then find the magnitude of this derivative. Mathematically, it is expressed as \( \kappa(s) = \left| \frac{d\mathbf{T}(s)}{ds} \right| \).
Arc length parameterization is useful for studying curves because it provides a natural and intrinsic way to describe the curve that is independent of the specific coordinate system. It simplifies the computation of geometric properties like the unit tangent vector and curvature, since the arc length \( s \) directly measures distance along the curve.