Difference between Slope and Curvature

Question

I tried to ask this question couple of times before but got distracted by other projects and now I am back to trying to understand this.

I am trying to analyse a graph and I would like to clarify some of the concepts associated with the process. The example graph included with my question (the green one) represents change in position on (Y).

(1) SLOPE

I now know that the slope calculation for the graph will give me the speed of the motion on the Y axis.

This makes perfect sense. I can see that as the point is about to change its direction, it slows down and the value of the slope graph approaches zero (0).

Now comes the bit that is confusing for me.

(2) CURVATURE?

First of all, in the image below, the pink sections illustrate curvature, is that correct?

If that is correct, what does exactly this bit of information tells us about the graph? Does it possibly describe the rate of change in the direction of the motion? What is the mathematical interoperation of that? By the way, the pink numbers are place holders for now. This is what I will be potentially calculating once I understand the topic. But the values should roughly illustrate the expected values.

Answer 1

One way to view these properties of a curve is as descriptions of increasingly more accurate approximations. The slope of the curve at a point is equal to the slope of the line that best approximates the curve at that point, a.k.a. the tangent line. The curvature, on the other hand, is the inverse of the radius of the circle that best approximates the curve at that point, a.k.a. the osculating circle. What makes for the “best” approximation is given a precise mathematical definition in calculus.

Usually, curvature, like slope, is a signed quantity. The sign tells you whether the curve is turning “left” or “right” at that point.

Answer 2

Short answer: "curvature" is unfortunately a very overloaded word in mathematics, but if you think of the graph as a curve in the plane, then yes, the curvature (which from your diagram appears to be unsigned) is proportional the rate of change of the direction of motion, if you think of a particle "traveling along the curve" at a constant speed.

There is a bit of subtlety here, though, since the direction of motion is a vector in the plane, so its rate of change should also be a vector in the plane. The key insight is that if you're traveling at constant speed, the rate of change of the direction is always perpendicular to the direction you're currently traveling (you can turn to one side or another, but can't accelerate or decelerate), so that the vector-valued rate of change of direction can be represented by a single real number, the rate at which you're turning clockwise or counterclockwise.

That's the intuition, at least... this topic is far too broad for an answer on the site, and for more details I recommend you look at a tutorial on introductory differential geometry of curves.