Published by the Applied Probability Trust
© Applied Probability Trust 2007
119
Circles of Best Fit
GUIDO LASTERS
The problem we consider is how to find a circle of best fit at a given point of a given curve.
Consider, for example, the parabola with equation y = x 2 . We aim to find the circle which
best fits this parabola at the origin; see figure 1. The normal to the parabola at P (0, 0) is the
y -axis, with equation x = 0 . The parabola at the neighbouring point Q = P , with coordinates
(h, h2 ), has slope 2h , so the normal at Q has slope −1/2h and equation
y − h2 = −
1
(x − h).
2h
The normals at P and Q meet when x = 0 in this equation, and so when
y = h2 + 21 ,
i.e. they meet at the point (0, h2 + 21 ) . As Q → P , h → 0 and this point tends to (0, 21 ) .
This will be the centre of the circle of best fit at the origin, so the circle of best fit has
equation
2
x 2 + y − 21 = 41
or
y = x2 + y2.
A similar calculation can be carried out by taking P as the point (1, 1) , again with Q(h, h2 )
as a neighbouring point (not P ). The normal to the parabola at P has equation
y − 1 = − 21 (x − 1)
or
y = − 21 x + 23 ,
y
2
1
−1
1
Figure 1
x