Problem 43642. Euclidean distance from a point to a polynomial

Created by John D'Errico

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A not uncommon problem in the area of computational geometry is to find the closest point to a straight line from a given point, or the distance from a point to a line. As you might expect, there is a simple formula for those things.As an extension, I decided one day to write a tool that would compute the distance from a point to a general polynomial function in the (x,y) plane. That is your problem here:Given a point (x0,y0), and a polynomial in the form y=P(x) where the function P is defined by the coefficients of a polynomial, you need to compute the minimum <a href = "https://en.wikipedia.org/wiki/Euclidean_distance">Euclidean distance</a> to that polynomial. So you need to find and return the minimum distance in the (x,y) plane between the point (x0,y0), and the function y=P(x).The function P will be passed in as the coefficients of a polynomial in standard MATLAB form, thus with the highest order coefficient first in a vector, like that generated by polyfit, and used by polyval. (P might be as simple as a constant function.) The point in question will be passed in as a vector of length 2, thus [x0,y0].As test case for you to check your code, the distance from the point (-2,-5) to the curve y=x^2/2+3*x-5 should be:<pre class="language-matlab">x0y0 = [-2 -5];
P = [0.5 3 -5];
D = distance2polynomial(P,xy)
D =
 1.89013819497707
</pre>(Be careful plotting these curves in case you want to plot your solution. The command "axis equal" is a good idea.)The symbolic TB tells me the distance is 1.8901381949770695260066523338279..., but I'll allow some slop in your solution, since you may have chosen a different algorithm than the one I chose. You should expect to provide at least 13 correct significant digits in the solution.Disclaimer: I'm not really sure why anyone needs such a code, which is why I've not posted my solution on the FEX. Anyway, my solution is a pretty one that I thought might make a fun Cody problem, and I wanted to see how others might approach the problem. I expect that my reference solution will score poorly for Cody purposes, since it is carefully coded, complete with error checks, and returns more than just the minimum distance.

A not uncommon problem in the area of computational geometry is to find the closest point to a straight line from a given point, or the distance from a point to a line. As you might expect, there is a simple formula for those things.

As an extension, I decided one day to write a tool that would compute the distance from a point to a general polynomial function in the (x,y) plane. That is your problem here:

Given a point (x0,y0), and a polynomial in the form y=P(x) where the function P is defined by the coefficients of a polynomial, you need to compute the minimum <https://en.wikipedia.org/wiki/Euclidean_distance Euclidean distance> to that polynomial. So you need to find and return the minimum distance in the (x,y) plane between the point (x0,y0), and the function y=P(x).

The function P will be passed in as the coefficients of a polynomial in standard MATLAB form, thus with the highest order coefficient first in a vector, like that generated by polyfit, and used by polyval. (P might be as simple as a constant function.) The point in question will be passed in as a vector of length 2, thus [x0,y0].

As test case for you to check your code, the distance from the point (-2,-5) to the curve y=x^2/2+3*x-5 should be:

x0y0 = [-2 -5];
  P = [0.5 3 -5];
  D = distance2polynomial(P,xy)
  D =
          1.89013819497707

(Be careful plotting these curves in case you want to plot your solution. The command "axis equal" is a good idea.)

The symbolic TB tells me the distance is 1.8901381949770695260066523338279..., but I'll allow some slop in your solution, since you may have chosen a different algorithm than the one I chose. You should expect to provide at least 13 correct significant digits in the solution.

Disclaimer: I'm not really sure why anyone needs such a code, which is why I've not posted my solution on the FEX. Anyway, my solution is a pretty one that I thought might make a fun Cody problem, and I wanted to see how others might approach the problem. I expect that my reference solution will score poorly for Cody purposes, since it is carefully coded, complete with error checks, and returns more than just the minimum distance.

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Last Solution submitted on Jan 22, 2023

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John D'Errico on 30 Oct 2016

I'm enjoying these creative solutions.

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Solution 1040753

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James on 31 Oct 2016

John, I just tried your file exchange solution out of pure curiosity. Your guess is correct; it does score poorly. I also got the error "You may not use the command(s) builtin in your code" for the test cases. All I did was change only the first line to call the function and the variable names with no other checking. I'll do some real work on it now. :-)

Solution 1038560

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2 Comments

Zhengyi on 2 Feb 2018

Will not solution like this be sooo much slower than simple loops?! Short yes, efficient no.

Zhengyi on 2 Feb 2018

well, I think I was wrong...This solution is slower if the degree of the polynomial is small but much faster if the degree gets higher...solving polynomials of high degree is really slow with ROOTS.

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Problem 43642. Euclidean distance from a point to a polynomial

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Problem 43642. Euclidean distance from a point to a polynomial

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