2.3 Arc Length and Curvature

Arc length and parameterization are crucial concepts in understanding curves in space. They allow us to measure distances along curves and describe their shapes mathematically, providing tools for analyzing motion and geometry in three dimensions.

These concepts build on earlier ideas of vectors and derivatives, extending them to more complex curves. By learning to calculate arc length and reparameterize curves, we gain insights into curvature, motion, and the behavior of objects moving along curved paths.

Arc Length and Parameterization

Arc length of space curves

Top images from around the web for Arc length of space curves

• 

  Arc Length of a Curve and Surface Area · Calculus View original

  Is this image relevant?

• 

  Arc Length of a Curve and Surface Area · Calculus View original

  Is this image relevant?

• 

  Calculus of Parametric Curves · Calculus View original

  Is this image relevant?

• 

  Arc Length of a Curve and Surface Area · Calculus View original

  Is this image relevant?

• 

  Arc Length of a Curve and Surface Area · Calculus View original

  Is this image relevant?

1 of 3

Top images from around the web for Arc length of space curves

• 

  Arc Length of a Curve and Surface Area · Calculus View original

  Is this image relevant?

• 

  Arc Length of a Curve and Surface Area · Calculus View original

  Is this image relevant?

• 

  Calculus of Parametric Curves · Calculus View original

  Is this image relevant?

• 

  Arc Length of a Curve and Surface Area · Calculus View original

  Is this image relevant?

• 

  Arc Length of a Curve and Surface Area · Calculus View original

  Is this image relevant?

1 of 3

• Arc length formula for parametric curves calculates distance traveled along curve $s = \int_{t_1}^{t_2} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2 + (\frac{dz}{dt})^2} dt$
• Calculate arc length by:
  1. Express curve in parametric form (x(t), y(t), z(t))
  2. Compute derivatives $\frac{dx}{dt}$, $\frac{dy}{dt}$, $\frac{dz}{dt}$
  3. Substitute into arc length formula
  4. Evaluate definite integral
• Adapt formula for specific cases:
  • Planar curves omit z component $s = \int_{t_1}^{t_2} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} dt$
  • Functions y = f(x) use $s = \int_{x_1}^{x_2} \sqrt{1 + (f'(x))^2} dx$
• Examples: helix, circular arc, parabola

Arc length parameter for reparameterization

• Arc length parameterization represents curve using distance traveled ensures unit speed
• Derive arc length parameter:
  1. Start with arc length formula
  2. Define $s(t) = \int_{t_0}^t \sqrt{(\frac{dx}{du})^2 + (\frac{dy}{du})^2 + (\frac{dz}{du})^2} du$
  3. Invert function to get t(s)
  4. Substitute t(s) into original parametric equations
• Applications simplify calculations describe motion at constant speed (particle movement, roller coasters)

Concept of curvature

• Curvature measures how quickly curve changes direction inversely proportional to osculating circle radius
• Formula: $\kappa = \frac{|\mathbf{r}'(t) \times \mathbf{r}''(t)|}{|\mathbf{r}'(t)|^3}$
• Geometric interpretation:
  • Straight lines zero curvature
  • Circles constant curvature
  • Smaller radius sharper turns
• Osculating circle best approximates curve at point radius reciprocal of curvature
• Examples: sine wave, spiral, ellipse

Unit vectors of curves

• Unit tangent vector $\mathbf{T}(t) = \frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|}$ points in direction of motion
• Unit normal vector $\mathbf{N}(t) = \frac{\mathbf{T}'(t)}{|\mathbf{T}'(t)|}$ perpendicular to tangent
• Binormal vector $\mathbf{B}(t) = \mathbf{T}(t) \times \mathbf{N}(t)$ perpendicular to both tangent and normal
• Frenet-Serret frame formed by T, N, B vectors provides moving coordinate system
• Vector relationships:
  • $\mathbf{T}' = \kappa \mathbf{N}$
  • $\mathbf{N}' = -\kappa \mathbf{T} + \tau \mathbf{B}$
  • $\mathbf{B}' = -\tau \mathbf{N}$
• Torsion $\tau$ measures curve twisting out of plane
• Applications: robotics, computer graphics, navigation systems