Curves

Vector Functions → Curves

• Curve = any continuous deformation of a line segment
• A curve $\Leftarrow$ continuous vector function
  • Not a one-to-one mapping because we can parameterize a curve into multiple vector-functions
  • That’s why we say “the curve is traced by this vector function”

Smooth Curves

• A curve is smooth if it has continuous unit tangent vectors
• The derivative shortcuts are the same
  • Product rule applies to both cross and dot product
• Checking if a given curve is smooth
  1. Find $\vec{r}(t)$ and $\vec{r}\prime (t)$ for the curve
  2. Check for values of t that $\vec{r}(t)=\vec{0}$ or DNE
  3. Check ${\lim }_{t\to a^{-}}\hat{T}(t)={\lim }_{t\to a^{+}}\hat{T}(t)$ at those points

Arc Length

• Given
  • $\vec{r}(t),t\in [a,b]$ is a simple parameterization of C
    • Simple = at least one component is an injective (one-to-one) function
  • $\vec{r}(t)$ is continuous on $[a,b]$
• $L_C={\int }_a^b\Vert {\vec{r}}^{\prime }(t)\Vert \text{d}t$

Natural Parameterization

• You can label every point on a curve by the arc length between a point on the curve and a point of interest
• Steps
  1. Get a relation between s and t ($s=s(t)$ or $t=t(s)$)
     1. $s(t)={\int }_a^t\Vert {\vec{r}}^{\prime }(\tau )\Vert \text{d}\tau$ for smooth curves
     2. Must determine that $s(t)$ strictly monotonic so it is able to be turned into $t(s)$
  2. $\vec{R}(s)=\vec{r}(t(s))$