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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