13.3 Arc Length and Curvature

Arc Length

• Same as Arc Length but slightly diffrent as this is now 13.1 Vector Functions and Space Curves

  • $L={\int }_a^b\sqrt{(f^{\prime }(t){)}^2+(g^{\prime }(t){)}^2+(h^{\prime }(t){)}^2}dt$

  • Can be rewritten as $L={\int }_a^b|r^{\prime }(t)|dt$

    • this is important for later!

• Arc length Function

  • Lets call $S$ as the length function, to get this we can see that $\frac{ds}{dt}=|r^{\prime }(t)|$ because $L$ is the magnitude of $s$ at $t$

    • Using The Fundamental Theorem Of Calculus we can integrate to respect to length and use t as our upper bound to make a function of t

      • $s(t)={\int }_0^t|r^{\prime }(u)du$
      • u and t are the same but for the sake of brevity we rewrite

Curvature

• Curvature of a function is defined by the rate of change of the tangent vector 13.2 Integrals and Derivatives in Vector Functions in a length $ds$

  • Let $\kappa$ be curvature

    • $\kappa =|\frac{dT}{ds}|$
      • where T is the unit tangent vector

    • The curvature is easier to compute if its in terms of the parameter so using chain rule

      • $\kappa =|\frac{\frac{dT}{dt}}{\frac{ds}{dt}}|⟹\kappa (t)=\frac{|T^{\prime }(t)|}{|r^{\prime }(t)|}$

Normal and Bi normal Vectors

• What is does this mean???

  • A bi normal vector is a vector that is normal to two vectors

• Definition

  • At a given point on a smooth space ruve r(t). there are many vectors that are orthoganal to the unit tangent vector T(t).

    • We can single these vectors out, because |T(t)|=1 of all t
  • $N(t)=\frac{T^{\prime }(t)}{|T^{\prime }(t)|}$

Frenet-Serret Frame

• What is this

  • Lets say there is a particle that is moving through space, the speed that its changing direction is teh curvature

• Definition

  • Let C be a piece- wise ssmooth curve given by a vector function $\vec{r}(t)$for $a\le t\le b$ we define two more unit vectors $\vec{N},\vec{B}$: for $\mid {\vec{T}}^{\prime }(t)\mid$ =! 0

    • $\vec{N}(t)=\frac{{\vec{T}}^{\prime }(t)}{|{\vec{T}}^{\prime }(t)|}$ $B=NxP$
    • this creates an ocilating plane that is binormal to B
  • 3

• Ocilating circles

  • circles show that the osc