Introduction
What is Curvature? Curvature is the value that is determined by the difference between a straight line and a curve. In the case of a surface, curvature is the measure of the difference of a curved plane to a plane. Another way to put curvature is the rate of change of direction of the curve.

Curvature, in mathematics, the rate of change of direction of a curve with respect to distance along the curve. At every point on a circle, the curvature is the reciprocal of the radius; for other curves (and straight lines, which can be regarded as circles of infinite radius), the curvature is the reciprocal of the radius of the circle that most closely conforms to the curve at the given point.
Example of Curvature:

*I don’t know why there are so many circles but I believe the one that is accurate to the curvature is the biggest circle formed.
Curvature, at a point of a curve, can determine a circle and the radius of the circle, with the value of curvature.
Example:Curvature Computation
Curvature is denoted with the variable, kappa (κ).
One of the ways to find curvature would be finding the magnitude of the derivative of a unit tangent vector function with respect to arc length, given by the equation:


As we compute this formula, we come to understand that the tangent vector (dT) is the direction that the curve is traveling in while the rate of change of the curve is determined by ds, indicating how quickly the curve is turning.
To start off the computation of curvature, we would have to find the tangent vector first.
With the formula of curvature as:

We can also rewrite the equation as:

As we work to evaluate this equation, we start off with the vector s(t), finding the derivative ds/dt. Once we determine the value of ds/dt, by plugging in a value for t, we can determine the velocity of the particle at the location of the function. Furthermore, once we determine what ds/dt is, we can find the magnitude of the vector by getting the squares of the sum of the two values of the vector.

Once we have determined the derivative of the vector s(t) and the magnitude, || s’(t) ||, We can find the value of T(t) with this equation:

After finding the function T(t), we would have to determine the value of dT/dt.
Lastly, with each required component, we can determine the curvature given by the formula.

Step by step process given by Khan Academy:


Through other scenarios, curvature for each scenario can be determined by:

Now let’s talk about curvature in 2 and 3 space. While I do not currently have a good example of curvature in 3-space, let’s go back to our class session on 12/7.
Recall that we were talking about a cone and how if we were to remove the base and treat the lateral(?) side of the cone as a label, cutting it down one side, we would get a pacman shape. However, just because we got a pacman shape does not mean we are limited to that shape alone. Based on how the cone is shaped, we can get the small sector that is missing from the pacman shape or a half circle. This, I found, has a correlation to curvature in a way and believe that it could help to explain curvature. Now in the case of the cone we had in class, it formed a pacman shape due to a low curvature, with the cone turning at not a very sharp turn. As the curvature increases though, the base of the cone would be smaller and in turn, would form a circle, curve, sector, etc. (not sure what word to use here exactly). In the case of forming a small circle for the base, we would get a stronger/higher curvature of the edge of the circle base, resulting in a label that seems like a small slice of (literally) any round object. With a greater base though, we would get a greater curvature in which the base is larger as the circle is formed with the low curve (The label here looks like a pacman you could say). In the case of a label forming a half circle, it’s just a situation where the base and the curvature just happens to conveniently be at a value where the label cut forms a half circle.

* Totally not a compressed cone (These were the best I could find)

Conclusion
So let’s go over Curvature. The simplest way to remember Curvature is to think of it as a value that determines how sharply a curve turns at a specific point of a function. Let’s put it in the perspective of a car drive. Once we turn the steering wheel, the car will be turning based on how much the steering wheel is turned. If we leave the steering wheel in that state and don’t turn it in any direction, we’ll end up driving a full circle (given enough space). How much the distance we travel in that one full circle will be dependent on our curvature, with a lower curvature resulting in a bigger circle/longer drive and a higher curvature resulting in a short and quick rotation.