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Satellite page for the section Distance approximations (for the problem of distance evaluation from a point to an ellipse or an ellipsoid)


In [1], the following formula is suggested for an approximation of the distance from a point to an ellipse in the plane: $$ d \approx d_{HO} = \sqrt{-B_4/B_3} \, . $$ where $ B_3 $ and $ B_4 $ are the coefficients of the distance equation: $$ \begin{array}{ll} B_3(x_0,y_0)&=-2\,a^2b^2\bigg\{a^2b^2T_0G_0^2 -\bigg[(a^2+b^2)T_0^2+3\,a^2b^2T_0-6\,a^4b^4S_{4,0}\bigg]G_0 +2\,a^2b^2T_0^2S_{4,0}\bigg\} \, , \\ B_4(x_0,y_0)&=a^{4}b^{4}G_0^2 \left(T_0^2+4\,a^2b^2G_0\right) \, . \end{array} $$ and $$ G_0= G(x_0,y_0)=\frac{x_0^2}{a^2}+\frac{y_0^2}{b^2}-1,\ T_0= x_0^2+y_0^2-a^2-b^2,\ S_{4,0}= \frac{x_0^2}{a^4} + \frac{y_0^2}{b^4} \, . $$

Th

Theorem 1 [2]. The radicand in $ d_{HO} $ is negative iff $ B_3(x_0,y_0)>0 $.

Proof follows from an alternative representation for the coefficient $ B_4 $: $$ B_4 \equiv a^4b^4G_0^2\left[(x_0^2-y_0^2+b^2-a^2)^2+4\,x_0^2y_0^2 \right] \, . $$ Therefore, it is nonnegative for any specialization of parameters.

From this theorem it follows that applicability of the approximation $ d_{HO} $ depends crucially on the relative position of the curve $$ B_3(x,y)=0 $$ with respect to the ellipse. Depending on the parameter specializations, this $ 6 $th order curve contains from $ 2 $ to $ 4 $ ovals (branches).

Th

Theorem 2 [2]. For any specialization of parameters $ a $ and $ b $ with $ a >b $, the curve $ B_3(x,y)=0 $ has two ovals lying inside the ellipse. For $ b< a \le \sqrt{2} b $, the curve does not have any other oval. For $ a > \sqrt{2} b $, the curve possesses two extra unclosed ovals lying outside the ellipse.

Ex

Example. For the ellipse $ x^2/324+y^2/25=1 $, the curve $ B_3(x,y)=0 $ is displayed1) in the figure:

The points $ X_0=(x_0,y_0) $ close to the curve $ B_3(x,y)=0 $ break the formula confidence. For instance, the points close to a tiny oval internal to the ellipse yield $$ \begin{array}{c|c|c|c|c|c|c} (x_0,y_0) & (17.20,0) & (17.30,0) & (17.40,0) & (17.48,0 ) & (17.50,0) & (17.60,0) \\ \hline d_{HO} & 0.33 & n/a & n/a & 4.60 & 1.19 & 0.50 \end{array} $$ with the n/a meaning the negative sign of the involved radicand.

References

[1]. Harker M., O'Leary P. First order geometric distance (the myth of Sampsonus), Proc. of the British Machine Vision Conference (BMVC06), I, 2006, Edinburgh, UK, P. 87–96.

[2]. Uteshev A.Yu., Goncharova M.V. Point-to-ellipse and point-to-ellipsoid distance equation analysis. J.Comput. Applied Math., 2018, Vol. 328, P. 232-251

1)
Since the curves possess a symmetry property w.r.t. the axes, they are displayed only in the first quadrant of the Cartesian plane.
matricese/optimize/distancee/apollonius_e/hofault.txt · Последние изменения: 2020/03/11 14:00 (внешнее изменение)