We can approximate the value of `(dy)/(dx)` at the middle point `(2,3)` of our 3 points as follows.

The slope of the line joining `(1, 1)` and `(2, 3)` is given by:

`m_1=(Deltay)/(Deltax)=2/1=2`

The slope of the line joining `(2, 3)` and `(3,8)` is given by:

`m_2=(Deltay)/(Deltax)=5/1=5`

We "average" these slopes to find a crude value for `(dy)/(dx)`:

`"Average slope"=(2+5)/2=3.5~~(dy)/(dx)`

Now for the slope of the slope (i.e. the second derivative). We find the change in slope m divided by the change in x for the interval x = 1.5 to x = 2.5 (which are the mid-points of our 2 lines joining the 3 given points):

`"Slope of slope"=(Deltam)/(Deltax)=(5-2)/1=3~~(d^2y)/(dx^2)`

Interestingly, these approximate values are exactly the same as our parabola approximation above.

Substituting into our radius of curvature formula, we obtain the same value as for Method 1:

`[1+(dy/dx)^2]^(3text(/)2)/(|(d^2y)/(dx^2)|)=[1+(3.5)^2]^(3//2)/3`

`=16.08`

Checking our answer, we can see that the (dark red) approximating circle (center D, with radius `16.08`) passes quite closely through our data points. It may be possible to improve on this by taking weighted averages to get more appropriate values for the first and second derivatives.

radius of curvature

Of course, my reader would need to do the same process for every set of 3 adjacent data points. He was writing a computer program to do this, so it would not be so tedious.

Get the Daily Math Tweet!
IntMath on Twitter