The graph connecting the generic data points `(1, 1)`, `(2, 3)` and `(3, 8)` is as follows.
One way of finding the radius of curvature is to find the (unique) parabola passing through these 3 points. Parabolas are excellent for approximating curves in a local region. (See more on Parabolas.)
We proceed as follows. In general, a parabola has the form:
y = ax2 + bx + c
Substituting our given points (1, 1), (2, 3) and (3, 8) into the general form of the parabola gives us 3 equations in 3 unknowns:
1 = a + b + c
3 = 4a + 2b + c
8 = 9a + 3b + c
Solving this set of 3 simultaneous equations gives us:
a = 1.5
b = −2.5
c = 2
So the parabola passing through the points `(1, 1), (2, 3)` and `(3, 8)` is
y = 1.5x2 − 2.5x + 2
We see this parabola passes through each of the 3 points:
Using this as our function (which we can only do in this local region for the original data points), we apply the formula for the radius of curvature:
`text(Radius of curvature)` `=([1+((dy)/(dx))^2]^(3//2))/|(d^2y)/(dx^2)|`
We need to find the first and second derivatives and evaluate them at the center point `(2, 3)`.
At `x = 2` (the middle of our 3 points), `dy/dx = 3.5`
Now for the second derivative:
So the radius of curvature at the middle point (2, 3) is:
So we've found a parabola that approximates our function for the local area near our data points. Then we found the radius of curvature - that is, the radius of a circle that "fits" our curve near our data points.
Here are the 3 data points, the parabola we found, and the circle indicating the curvature for the given 3 points. It has radius `16.08`.