Finding closest point to implicitly defined curve

Question

I'm facing the following problem: Given an implicit curve given by $f(x,y)=0$ and an initial guess $(x_0,y_0)$ which is hopefully close to the curve, I want to find the point $(x,y)$ on the curve which is closest to $(x_0,y_0)$.

My attempt:

I thought of using FindRoot with the equation repeated twice, i.e.

FindRoot[{f[x,y],f[x,y]},{{x,x0},{y,y0}}]

But this fails spectacularly with $f(x,y)=y-x^2$ and initial points $(0,y_0)$ with $y_0$ large.

Is there a good way to solve this problem?

Thanks!

NMinimize[{EuclideanDistance[{x, y}, {x0, y0}], f[x, y] == 0}, {x, y}]? — , Jul 08 '15 at 06:36
"closest" - that kind of screams for the use of Minimize[], no? — J. M.'s missing motivation, Jul 08 '15 at 06:37
Better yet, why not answer your own question instead, so you can share what you've learned? — J. M.'s missing motivation, Jul 08 '15 at 07:00
It might be better to minimize an EuclideanDistance square (a square of the euclidean norm instead of the norm.) — Igor, Jul 08 '15 at 11:55
Related: http://mathematica.stackexchange.com/questions/34204/finding-a-point-from-a-set-of-contours-such-that-it-is-nearest-to-a-given-point — Michael E2, Jul 08 '15 at 15:35

score 12 · Accepted Answer · answered Jul 08 '15 at 07:02

12

RegionNearest will do this automatically for many cases, including $y-x^2=0$:

RegionNearest[ImplicitRegion[y - x^2 == 0, {x, y}], {0, y0}]

Mathematica graphics

answered Jul 08 '15 at 07:02

Szabolcs

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Finding closest point to implicitly defined curve

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