Section 3.3 - Arc Length and Curvature

Section 3.3 - Arc Length and Curvature

Length of a Curve

The length of a curve is a fundamental concept in calculus, allowing us to measure the distance along a path defined by a vector or parametric equation. This section introduces the formulas and methods for calculating arc length for curves in space.

• Definition: Suppose a curve has the vector equation r(t) = [f1(t), f2(t), f3(t)], for a ≤ t ≤ b. The length L of the curve as t increases from a to b is:

$L = \int_a^b \sqrt{\left(\frac{df_1}{dt}\right)^2 + \left(\frac{df_2}{dt}\right)^2 + \left(\frac{df_3}{dt}\right)^2} \, dt$

• Alternate Form: For a curve in the plane, r(t) = [x(t), y(t)]:

$L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$

• Example: Find the length of the arc of the circular helix with vector equation r(t) = cos t \mathbf{i} + sin t \mathbf{j} + t \mathbf{k} from the point (1,0,0) to the point (0,1,2π).

• Note: A single curve can be represented by more than one vector function. For instance, r1(t) = (t, t2, t3) and r2(t) = (et, e2t, e3t) both represent the same space curve (the twisted cubic).

The Arc Length Function

The arc length function measures the distance along a curve from a fixed starting point as a function of the parameter. It is useful for reparametrizing curves by arc length.

• Definition: For a curve with vector function r(t), the arc length function s from a to t is:

$s = \int_a^t |r'(u)| \, du$

• Parameterization by Arc Length: If s is the arc length function, we can reparametrize the curve so that r is a function of s instead of t.

• Example: Find the arc length function of the helix r(t) = cos t \mathbf{i} + sin t \mathbf{j} + t \mathbf{k} measured from (1,0,0) in the direction of increasing t.

Curvature

Curvature quantifies how quickly a curve changes direction at a given point. It is a key concept in differential geometry and calculus, with applications in physics and engineering.

• Definition: The curvature κ of a curve at a point is defined as:

$\kappa = \left| \frac{d\mathbf{T}}{ds} \right|$

• where \mathbf{T} is the unit tangent vector and s is the arc length.

• Unit Tangent Vector: For a vector function r(t):

$\mathbf{T}(t) = \frac{r'(t)}{|r'(t)|}$

• Curvature Formula (in terms of t):

$\kappa(t) = \frac{|r'(t) \times r''(t)|}{|r'(t)|^3}$

• Radius of Curvature: The radius of curvature is 1/κ.

• Example: Show that the curvature of a circle of radius a is 1/a.

• Example: Find the curvature of the twisted cubic r(t) = (t, t2, t3) at a general point and at (0,0,0).

• Plane Curve Formula: For y = f(x), the curvature at a point is:

$\kappa(x) = \frac{|f''(x)|}{[1 + (f'(x))^2]^{3/2}}$

• Example: Find the curvature of the parabola y = x2 at the points (0,0), (1,1), and (2,4).

Unit Normal and Binormal Vectors

The unit normal and binormal vectors are used to describe the orientation of a curve in space. They are essential in the study of the geometry of curves.

• Unit Normal Vector: Indicates the direction in which the curve is turning at each point.

• Definition:

$\mathbf{N}(t) = \frac{\mathbf{T}'(t)}{|\mathbf{T}'(t)|}$

• Binormal Vector: Perpendicular to both \mathbf{T} and \mathbf{N} and is a unit vector.

• Definition:

$\mathbf{B}(t) = \mathbf{T}(t) \times \mathbf{N}(t)$

• Example: Find the unit normal and binormal vectors for the circular helix r(t) = cos t \mathbf{i} + sin t \mathbf{j} + t \mathbf{k} at the point P(t).

Osculating Plane and Circle

The osculating plane and circle provide geometric insight into the local behavior of a curve at a point. The osculating plane contains the tangent and normal vectors, while the osculating circle best approximates the curve near that point.

• Osculating Plane: Determined by the tangent and normal vectors at a point P on a curve.

• Osculating Circle: Lies in the osculating plane and has radius 1/κ. It shares the same tangent, normal, and curvature at P.

• Example: Find equations of the normal plane and osculating plane of the helix at P(0,1,π/2).

Table: Summary of Key Formulas

Concept	Formula	Description
Arc Length (vector function)	$L = \int_a^b |r'(t)| \, dt$	Length of curve from t = a to t = b
Arc Length (parametric)	$L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$	Length for plane curve
Curvature (vector function)	$\kappa(t) = \frac{|r'(t) \times r''(t)|}{|r'(t)|^3}$	Curvature at point t
Curvature (plane curve)	$\kappa(x) = \frac{|f''(x)|}{[1 + (f'(x))^2]^{3/2}}$	Curvature for y = f(x)
Unit Tangent Vector	$\mathbf{T}(t) = \frac{r'(t)}{|r'(t)|}$	Direction of curve at t
Unit Normal Vector	$\mathbf{N}(t) = \frac{\mathbf{T}'(t)}{|\mathbf{T}'(t)|}$	Direction curve is turning
Binormal Vector	$\mathbf{B}(t) = \mathbf{T}(t) \times \mathbf{N}(t)$	Perpendicular to \mathbf{T} and \mathbf{N}
Radius of Curvature	$R = \frac{1}{\kappa}$	Radius of osculating circle

Additional info: Some definitions and examples have been expanded for clarity and completeness. The table summarizes the main formulas for quick reference.