1.4 Arc length and curvature

Arc length and curvature are key concepts in understanding vector-valued functions. They help us measure the distance along a curve and how much it bends. These ideas are crucial for analyzing the geometry of curves in space.

By studying arc length and curvature, we can describe the shape and behavior of curves more precisely. This knowledge is essential for applications in physics, engineering, and computer graphics, where accurate representations of curved paths are needed.

Arc Length and Parameterization

Calculating Arc Length

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• Arc length measures the distance traveled along a curve $C$ between two points $P(a)$ and $P(b)$
• Formula for arc length: $L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2} dt$
  • Requires the curve to be parameterized by a variable $t$
  • $\frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt}$ represent the derivatives of the coordinate functions with respect to $t$
• Example: Find the arc length of the curve $\vec{r}(t) = (t, t^2, t^3)$ for $0 \leq t \leq 1$
  • $\frac{dx}{dt} = 1, \frac{dy}{dt} = 2t, \frac{dz}{dt} = 3t^2$
  • $L = \int_{0}^{1} \sqrt{1 + 4t^2 + 9t^4} dt$

Arc Length Parameterization

• Arc length parameterization reparameterizes a curve using the arc length $s$ as the parameter
• Allows for the curve to be traversed at a constant speed of 1 unit per unit of parameter $s$
• To find the arc length parameterization, solve the equation $s = \int_{t_0}^{t} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2} dt$ for $t$ in terms of $s$
• Example: Find the arc length parameterization of the curve $\vec{r}(t) = (\cos t, \sin t)$ for $0 \leq t \leq 2\pi$
  • $\frac{dx}{dt} = -\sin t, \frac{dy}{dt} = \cos t$
  • $s = \int_{0}^{t} \sqrt{\sin^2 t + \cos^2 t} dt = \int_{0}^{t} dt = t$
  • The arc length parameterization is $\vec{r}(s) = (\cos s, \sin s)$ for $0 \leq s \leq 2\pi$

Natural Parameterization

• Natural parameterization, also known as unit speed parameterization, is a special case of arc length parameterization
• The curve is parameterized such that the magnitude of the velocity vector $\vec{r}'(s)$ is always 1
• Useful for simplifying calculations involving curvature and torsion
• Example: The curve $\vec{r}(s) = (\cos s, \sin s)$ is naturally parameterized because $\|\vec{r}'(s)\| = \sqrt{(-\sin s)^2 + (\cos s)^2} = 1$

Curvature

Measuring Curvature

• Curvature measures how much a curve deviates from a straight line at a given point
• Denoted by $\kappa$ and is defined as the magnitude of the rate of change of the unit tangent vector $\vec{T}$ with respect to arc length $s$
• Formula for curvature: $\kappa = \left\|\frac{d\vec{T}}{ds}\right\| = \frac{\|\vec{r}'(s) \times \vec{r}''(s)\|}{\|\vec{r}'(s)\|^3}$
  • $\vec{r}'(s)$ and $\vec{r}''(s)$ are the first and second derivatives of the position vector with respect to arc length $s$
• Example: Find the curvature of the curve $\vec{r}(t) = (t, t^2, t^3)$ at $t = 1$
  • $\vec{r}'(t) = (1, 2t, 3t^2), \vec{r}''(t) = (0, 2, 6t)$
  • At $t = 1$, $\vec{r}'(1) = (1, 2, 3), \vec{r}''(1) = (0, 2, 6)$
  • $\kappa = \frac{\|(1, 2, 3) \times (0, 2, 6)\|}{\|(1, 2, 3)\|^3} = \frac{\sqrt{14}}{\sqrt{14}^3} = \frac{1}{\sqrt{14}}$

Radius of Curvature

• The radius of curvature $\rho$ is the reciprocal of the curvature $\kappa$
• Represents the radius of the osculating circle at a given point on the curve
• Formula for radius of curvature: $\rho = \frac{1}{\kappa} = \frac{\|\vec{r}'(s)\|^3}{\|\vec{r}'(s) \times \vec{r}''(s)\|}$
• Example: Find the radius of curvature of the curve $\vec{r}(t) = (t, t^2, t^3)$ at $t = 1$
  • From the previous example, $\kappa = \frac{1}{\sqrt{14}}$
  • $\rho = \frac{1}{\kappa} = \sqrt{14}$

Osculating Circle

• The osculating circle is the circle that best approximates the curve at a given point
• Its center lies on the normal plane to the curve at that point
• The radius of the osculating circle is equal to the radius of curvature $\rho$
• The osculating circle shares the same tangent line and curvature as the curve at the point of contact
• Example: Find the equation of the osculating circle for the curve $\vec{r}(t) = (t, t^2)$ at $t = 1$
  • $\vec{r}'(t) = (1, 2t), \vec{r}''(t) = (0, 2)$
  • At $t = 1$, $\vec{r}(1) = (1, 1), \vec{r}'(1) = (1, 2), \vec{r}''(1) = (0, 2)$
  • $\kappa = \frac{\|(1, 2) \times (0, 2)\|}{\|(1, 2)\|^3} = \frac{2}{5\sqrt{5}}, \rho = \frac{5\sqrt{5}}{2}$
  • The center of the osculating circle is $\vec{r}(1) + \rho \vec{N}(1) = (1, 1) + \frac{5\sqrt{5}}{2}\left(-\frac{2}{\sqrt{5}}, \frac{1}{\sqrt{5}}\right) = \left(1-5, 1+\frac{5}{2}\right)$
  • The equation of the osculating circle is $(x-(1-5))^2 + (y-(1+\frac{5}{2}))^2 = \left(\frac{5\sqrt{5}}{2}\right)^2$

Torsion

Understanding Torsion

• Torsion measures how much a curve twists out of the plane of curvature
• Denoted by $\tau$ and is defined as the rate of change of the binormal vector $\vec{B}$ with respect to arc length $s$
• Formula for torsion: $\tau = -\frac{d\vec{B}}{ds} \cdot \vec{N} = -\frac{(\vec{r}'(s) \times \vec{r}''(s)) \cdot \vec{r}'''(s)}{\|\vec{r}'(s) \times \vec{r}''(s)\|^2}$
  • $\vec{N}$ is the principal normal vector, which is perpendicular to both the tangent vector $\vec{T}$ and the binormal vector $\vec{B}$
  • $\vec{r}'(s), \vec{r}''(s), \vec{r}'''(s)$ are the first, second, and third derivatives of the position vector with respect to arc length $s$
• Torsion is positive if the curve twists counterclockwise and negative if it twists clockwise
• A curve with zero torsion at every point is called a planar curve and lies entirely in a single plane
• Example: Find the torsion of the curve $\vec{r}(t) = (\cos t, \sin t, t)$ at $t = \frac{\pi}{2}$
  • $\vec{r}'(t) = (-\sin t, \cos t, 1), \vec{r}''(t) = (-\cos t, -\sin t, 0), \vec{r}'''(t) = (\sin t, -\cos t, 0)$
  • At $t = \frac{\pi}{2}$, $\vec{r}'(\frac{\pi}{2}) = (-1, 0, 1), \vec{r}''(\frac{\pi}{2}) = (0, -1, 0), \vec{r}'''(\frac{\pi}{2}) = (1, 0, 0)$
  • $\tau = -\frac{(-1, 0, 1) \times (0, -1, 0) \cdot (1, 0, 0)}{\|(-1, 0, 1) \times (0, -1, 0)\|^2} = -\frac{1}{2}$