Concepts in Curvature

Step 2: Find the Unit Tangent Vector

The Unit Tangent Vector ($T$) is the tangent of the curve with a uniform length of one. To find $T$, we must first find $\frac{ds}{dt}$, which is the rate of change of a point traveling along the curve with respect to time. In other words, it's the tangent to the curve (you could also think of it as "velocity").

Given that $s(t)$ is defined by the following components,

We can find $\frac{ds}{dt}$ by taking the derivative of each component.

Now that we have $\frac{ds}{dt}$, we can find $T$ by dividing $\frac{ds}{dt}$ by its magnitude, which gives us a vector with a length of one.

The specific math is getting a bit messy at this point, but the important part is that you're able to understand what $\frac{ds}{dt}$ and $T$ are. Try flipping between the visualizations of both of these vectors to get a better understanding of what they represent. Once you're ready, move on to the next step.