Section 2.4 Curvature

2.4 Curvature

In this section, we learn alternative formulas for curvature, and normal and binormal vectors, two important vectors that have geometry meaning for a curve. in space. We first recall the definition of curvature.

Definition

Let $C$ be a smooth curve in the plane or in space given by $\overrightarrow{r(s)}$, where $s$ is the arc-length parameter. The curvature $\mathit{k}$ at $s$ is $$k=||\frac{\overrightarrow{T(s)}}{ds}||=||\overrightarrow{T^{\prime }(s)}||.$$

Theorem: Alternative Formulas for Curvature

Let $C$ be a smooth curve in the plane or in space given by $\overrightarrow{r(t)}$, where $t$ is the parameter. The curvature $\mathit{k}$ at $t$ is

(a)  $$k=\frac{||\overrightarrow{T^{\prime }(t)}||}{||\overrightarrow{r^{\prime }(t)}||}$$

(b)  $$k=\frac{||\overrightarrow{r^{\prime }(t)}\times \overrightarrow{r”(t)}||}{||\overrightarrow{r^{\prime }(t)}|{|}^3}$$

Proof of (b) use (a) and $\frac{\overrightarrow{r^{\prime }(t)}}{||\overrightarrow{r^{\prime }(t)}||}=\overrightarrow{T(t)}$

Example 1: Find the curvature of $\overrightarrow{r(t)}=<\text{cos}(t),\text{sin}(t),t>$.

Exercise 1: Find the curvature of $\overrightarrow{r(t)}=<t,\text{cos}(t),\text{sin}(t)>$.

Example 2: Find the curvature of $\overrightarrow{r(t)}=<t\text{ln}(t),t,t^2>$ at the point $(0,1,1)$.

Exercise 2: Find the curvature of $\overrightarrow{r(t)}=<e^t,2t-1,e^{2t}>$ at the point $(1,-1,1)$.

When studying motion in three dimensions, two other vectors are useful in describing the motion of a particle along a path in space: the principal unit normal vector and the binormal vector.

Definition: The Normal and Binormal Vectors

Let $\overrightarrow{r(t)}$ be a smooth curve in space and let $\overrightarrow{T(t)}$ be its unit tangent vector, if $\overrightarrow{T^{\prime }(t)}\ne \overrightarrow{0}$, then the unit normal vector at $t$ is defined to be $$\overrightarrow{N(t)}=\frac{\overrightarrow{T^{\prime }(t)}}{||\overrightarrow{T^{\prime }(t)}||}.$$

The binormal vector at $t$ is defined as  $$\overrightarrow{B(t)}=\overrightarrow{T(t)}\times \overrightarrow{N(t)}.$$

Notice that the unit normal vector and the binomial vector are unit vectors. The unit normal vector is a vector that is orthogonal to the tangent vector point to the center of the curve if the curvature is not zero. The binomial vector is a vector that is orthogonal to both $\overrightarrow{T(t)}$ and $\overrightarrow{N(t)}$.

Example 3: Find the unit normal and binormal vectors of $\overrightarrow{r(t)}=<\text{cos}(2t),\text{sin}(2t),t>.$ 

Exercise 3: Find the unit normal and binormal vectors of $\overrightarrow{r(t)}=<t,\text{cos}(3t),\text{sin}(3t)>.$ 

Definition:

The unit normal vector $\overrightarrow{N(t)}$ and the binormal vector $\overrightarrow{B(t)}$ of a curve $\overrightarrow{r(t)}$ form a plane that is perpendicular to the curve at any point on the curve, called the normal plane. Plane determined by the vectors $\overrightarrow{T(t)}$ and $\overrightarrow{N(t)}$ forms the osculating plane of $\overrightarrow{r(t)}$ at any point $P$ on the curve. The circle that lies in the osculating plane of $\overrightarrow{r(t)}$ at $P$, has the same tangent as $\overrightarrow{r(t)}$ at $P$, lies on the concave side of curve, and has radius $\rho =\frac{1}{k}$ (the reciprocal of the curvature) is called the osculating circle (or the circle of curvature) of the curve at $P$.

Notice that the normal plane has $\overrightarrow{T(t)}$ as its normal vector and the osculating plane has $\overrightarrow{B(t)}$ as its normal vector.

Example 4: The equations of normal plane and osculating plane of $\overrightarrow{r(t)}=<\text{cos}(2t),\text{sin}(2t),4t>$ at $P=(-1,0,2\pi )$.

Exercise 4: The equations of normal plane and osculating plane of $\overrightarrow{r(t)}=<6t,\text{cos}(3t),-\text{sin}(3t)>$ at $P=(3\pi ,0,1)$.

Example 5: Find the unit normal and binormal vectors of $\overrightarrow{r(t)}=<2\text{sin}(t),2\sqrt{2}\text{cos}(t),-2\text{sin}(t)>$.

Group work:

1.Find the curvature of $\overrightarrow{r(t)}=<2\text{sin}(t),2\sqrt{2}\text{cos}(t),-2\text{sin}(t)>$

2. Find the unit normal and binormal vectors of $\overrightarrow{r(t)}=<-3\sqrt{2}\text{sin}(t),3\text{cos}(t),3\text{cos}(t)>$. 

3. Find the equations of normal plane and osculating plane of $\overrightarrow{r(t)}=<2\text{cos}(3t),2\text{sin}(3t),t>$

at $P=(-2,0,\pi )$.