Calculate Curvature
Detailed How-To w/ Step-by-Step Examples!

Wouldn’t it be cool if we could use calculus to calculate curvature or measure the “curviness” at various points on a curve?

Well, it’s a good thing we can!

Curvature Definition

Curvature is the measure of how fast the direction changes as we move a small distance along a curve. And we use calculus, namely, the angle of the tangent line with respect to arc length, to measure a curves “curviness.”

Formally, we define the curvature \(\kappa\) (Greek letter Kappa) of a curve as the absolute value of the rate at which the angle between the tangent line and the x-axis changes as a function of arc length.

So, if an object whose position vector at time \(t\) is \(\vec{r}(t)\) has velocity \(\vec{v}(t)=\vec{r}^{\prime}(t)\), then we can record the direction of motion by finding the unit vector:

\begin{equation}
\vec{T}(t)=\frac{\vec{v}(t)}{\|\vec{v}(t)\|}=\frac{\vec{r}^{\prime}(t)}{\left\|\vec{r}^{\prime}(t)\right\|}
\end{equation}

Curvature Formula

And knowing that as the object moves along the curve, the direction of the unit tangent vector \(T\) changes most rapidly when the curve is “curviest.” Therefore, we can find the curvature for any curve in the plane or space by letting \(s\) denote the arc length of a curve, as follows:

\begin{equation}
\kappa=\left\|\frac{d \vec{T}}{d s}\right\|
\end{equation}

How To Calculate Curvature – 3 Ways

But this formula all by itself is a little bit cumbersome to utilize.

Thankfully, we can transform our formula for finding curvature in three different ways, depending on the type of function we are given.

1. If the curve is given in vector form, where \(\vec{r}(t)=\langle x(t), y(t), z(t)\rangle\), then the curvature can be express as follows:

\begin{equation}
\kappa=\frac{\left\|\vec{r}^{\prime}(t) \times \vec{r}^{\prime \prime}(t)\right\|}{\left\|\vec{r}^{\prime}(t)\right\|^{3}}
\end{equation}

2. If the curve is given in the form \(y=f(x)\), then the curvature can be expressed:

\begin{equation}
\boldsymbol{\kappa}=\frac{\left|\frac{d^{2} y}{d x^{2}}\right|}{\left(1+\left(\frac{d y}{d x}\right)^{2}\right)^{\frac{3}{2}}}
\end{equation}

3. And, if the curve is given in the parametric form, where x and y are a function of some parameter t, then:

\begin{equation}
\kappa=\frac{\left|\frac{d^{2} y}{d x^{2}}\right|}{\left(1+\left(\frac{d y}{d x}\right)^{2}\right)^{\frac{3}{2}}} \text { where } \frac{d y}{d x}=\frac{\left(\frac{d y}{d t}\right)}{\left(\frac{d x}{d t}\right)} \text { and } \frac{d^{2} y}{d x^{2}}=\frac{\frac{d}{d t}\left(\frac{d y}{d x}\right)}{\left(\frac{d x}{d t}\right)}
\end{equation}

Example – Find The Curvature Of The Curve r(t)

For instance, suppose we are given \(\vec{r}(t)=\langle 5 t, \sin t, \cos t\rangle\), and we are asked to calculate the curvature.

Well, since we are given the curve in vector form, we will use our first curvature formula of:

\begin{equation}
\kappa=\frac{\left\|\vec{r}^{\prime}(t) \times \vec{r}^{\prime \prime}(t)\right\|}{\left\|\vec{r}^{\prime}(t)\right\|^{3}}
\end{equation}

So, first we will need to calculate \(\vec{r}^{\prime}(t)\) and \(\vec{r}^{\prime \prime}(t)\).

\begin{equation}
\begin{aligned}
&\vec{r}^{\prime}(t)=\langle 5, \cos t,-\sin t\rangle \\
&\vec{r}^{\prime \prime}(t)=\langle 0,-\sin t,-\cos t\rangle
\end{aligned}
\end{equation}

Next, we will compute the cross product \(\vec{r}^{\prime}(t) \times \vec{r}^{\prime \prime}(t)\).

\begin{equation}
\vec{r}^{\prime}(t) \times \vec{r}^{\prime \prime}(t)=\left|\begin{array}{ccc}
\vec{i} & \vec{j} & \vec{k} \\
5 & \cos t & -\sin t \\
0 & -\sin t & -\cos t
\end{array}\right|=\langle-1,5 \cos t,-5 \sin t\rangle
\end{equation}

Now, we will find the magnitude, or length, of this vector \(\left\|\vec{r}^{\prime}(t) \times \vec{r}^{\prime \prime}(t)\right\|\).

\begin{equation}
\begin{aligned}
&\left\|\vec{r}^{\prime}(t) \times \vec{r}^{\prime \prime}(t)\right\|=\sqrt{(-1)^{2}+(5 \cos t)^{2}+(-5 \sin t)^{2}} \\
&\left\|\vec{r}^{\prime}(t) \times \vec{r}^{\prime \prime}(t)\right\|=\sqrt{1+25 \cos ^{2} t+25 \sin ^{2} t} \\
&\left\|\vec{r}^{\prime}(t) \times \vec{r}^{\prime \prime}(t)\right\|=\sqrt{1+25\left(\cos ^{2} t+\sin ^{2} t\right)} \\
&\left\|\vec{r}^{\prime}(t) \times \vec{r}^{\prime \prime}(t)\right\|=\sqrt{26}
\end{aligned}
\end{equation}

And next we will calculate \(\left\|\vec{r}^{\prime}(t)\right\|^{3}\).

\begin{equation}
\begin{aligned}
&\left\|\vec{r}^{\prime}(t)\right\|^{3}=\left(\sqrt{(5)^{2}+(\cos t)^{2}+(-\sin t)^{2}}\right)^{3} \\
&\left\|\vec{r}^{\prime}(t)\right\|^{3}=\left(\sqrt{25+\cos ^{2} t+\sin ^{2} t}\right)^{3} \\
&\left\|\vec{r}^{\prime}(t)\right\|^{3}=\left(\sqrt{25+\left(\cos ^{2} t+\sin ^{2} t\right)}\right)^{3} \\
&\left\|\vec{r}^{\prime}(t)\right\|^{3}=(\sqrt{25+1})^{3}=(\sqrt{26})^{3}=26^{\frac{3}{2}}
\end{aligned}
\end{equation}

Finally, this means the curvature for \(\vec{r}(t)\) is:

\begin{equation}
\kappa=\frac{\sqrt{26}}{(26)^{\frac{3}{2}}}=\frac{26^{\frac{1}{2}}}{26^{\frac{3}{2}}}=\frac{1}{26}
\end{equation}

Notice how our answer is a constant. What does this mean?

It implies that our curve is a circle; thus, \(\boldsymbol{\kappa}=\frac{1}{r}\) where \(r=radius\).

Therefore, the radius of curvature of a curve at a point is the reciprocal of the curvature.

Cool!

Together we will learn how to use all three forms of the curvature formula and also discover some tricks and tips along the way.

There’s a lot to uncover, so let’s dive into the video.

Video Tutorial w/ Full Lesson & Detailed Examples (Video)

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