Curvature at a Point on a Curve Examples 3
Recall that if $\vec{r}(t) = (x(t), y(t), z(t))$ is a vector-valued function that traces the smooth curve $C$, then the curvature of $C$ at the point $\vec{r}(t)$ can be given by either of the formulas:
- $\kappa (s) = \biggr \| \frac{d\hat{T}(s)}{ds} \biggr \| = \biggr \| \frac{d\hat{T}(s) / dt}{ds / dt} \biggr \|$ if we find an arc length parameterization of $\vec{r}(t)$ as $\vec{r}(s)$.
- $\kappa (t) = \frac{\| \hat{T'}(t) \|}{\| \vec{r'}(t) \|}$ if $\hat{T'}(t)$ is not too difficult to calculate.
- $\kappa (t) = \frac{\| \vec{r'}(t) \times \vec{r''}(t) \|}{\| \vec{r'}(t) \|^3}$.
We will now look at some more examples of finding the curvature of a space curve.
Example 1
Find the curvature of the curve given by the vector-valued function $\vec{r}(t) = (\sin t \cos t, \sin^2 t, \cos t)$.
We have that the derivative of $\vec{r}(t)$ is given by:
(1)The second derivative of $\vec{r}(t)$ is given by:
(2)The cross product of $\vec{r'}(t)$ and $\vec{r''}(t)$ is:
(3)The norm of $\vec{r'}(t) \times \vec{r''}(t)$ is:
(4)Lastly, the norm of $\vec{r'}(t)$ is:
(5)Therefore the curvature $\kappa$ is given by:
(6)Example 2
Prove that if $y = f(x)$ is a twice differentiable real-valued function that traces the curve $C$ on the $xy$-plane, then the curvature at any point $x \in D(f)$ is given by $\frac{ \mid f''(x) \mid}{(1 + (f'(x))^2)^{3/2}}$. What does this formula say about the curvature of straight lines of the form $y = mx + b$?
We will formally prove this statement on the Curvature at a Point on a Single Variable Real Valued Function.
We can easily parameterize a curve $C$ on the $xy$-plane as:
(7)If we differentiate $\vec{r}(t)$ we get:
(8)If we differentiate $\vec{r}(t)$ again, we get that:
(9)Therefore we compute the cross product $\vec{r'}(t) \times \vec{r''}(t)$ as:
(10)Therefore we have that $\| \vec{r'}(t) \times \vec{r''}(t) \| = \sqrt{(f''(x))^2} = \mid f''(x) \mid$.
Now we compute the norm of $\vec{r'}(t)$ as:
(11)Plugging what we have just obtained into the curvature formula, we get that:
(12)