This is an exact method for finding the required radius of curvature. We'll actually find the equation of the circle passing through the 3 points.
In general, the x-value for the center of the circle passing through 3 points A (x1, y1), B (x2, y2), C (x3, y3), joined by lines with slopes m1 and m2 is given by:
(The above formula is based on finding the intersection of the perpendicular bisectors of the 2 lines joining the 3 points, as follows.)
So for our given data points, we have:
`x_c` `=((2)(5)(1-8)+(5)(1+2)-(2)(2+3))/(2(5-2))` `=-10.833`
(We found the values for m1 and m2 in Method 2, above.)
Now to find the y-value for center of the circle. The formula for the first perpendicular bisector is given by:
`y_("Perp")` `=- 1/m_1(x-(x_1+x_2)/2)+(y_1+y_2)/2`
So for our data points, we can obtain the y-value for the center of the circle by substituting the known values, as follows:
`y_c` `=-1/2(-10.833-(1+2)/2)+(1+3)/2` `=8.1665`
So the center of the circle passing through the points `(1, 1), (2, 3)` and `(3, 8)` is given by `(-10.83, 8.17)`.
Finally, we can find the radius by simply finding the distance between the center of the circle and any one of the points on the circle. I have chosen `(1,1)`:
[See more on Distance Formula.]
So the radius of curvature for the 3 points `(1, 1), (2, 3)` and `(3, 8)` is `13.83`, when finding the actual circle passing through the 3 points.
Our answer is slightly different to the answers obtained by using a parabolic model and linear approximations.