# Project 3D points to a surface

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psprinks on 27 Apr 2017
Edited: DS on 17 Sep 2020
I have fit a surface through a set of x,y,z points. Image attached. I would like to be able to project points that lie within some distance threshold to that surface. Could anyone advise on a method for doing this?
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psprinks on 30 Apr 2017
Yes , I think the minimum euclidean distance would be fine for this case.

John D'Errico on 30 Apr 2017
Edited: John D'Errico on 30 Apr 2017
Minimum Euclidean distance to a general surface is a somewhat nasty problem to solve. With ONE independent variable, IF the function is a polynomial one, there is a solution, of sorts. Not trivial. But a solution. If the surface is non-polynomial, it gets nasty. And in multiple dimensions, things can get messy too. Lets see what happens with a one variable problem.
I'll pick a random cubic polynomial. Not even something where I'll pick the coefficients myself.
format long g
coef = randn(1,4)
coef =
0.318765239858981 -1.30768829630527 -0.433592022305684 0.34262446653865
I'll use these as coefficients of a cubic polynomial, to be evaluated by polyval.
ezplot(@(x) polyval(coef,x),[-3,5])
Now, suppose we choose some arbitrary point in the plane? What is the
xy = randn(1,2)*2
xy =
2.48288696432624 2.97939521557093
hold on
grid on
plot(xy(1),xy(2),'ro')
What is the point of closest approach? Be careful. This might change your mind. :)
axis equal
How would I solve for the projection in terms of minimum Euclidean distance? I would solve for the point that minimizes the function:
(y(x) - y0)^2 + (x-x0)^2
as a function of x. Do that by differentiating, and then searching for the roots of the polynomial. So the square of the Euclidean distance i the (x,y) plane is just:
D2 = (P - xy(2))^2 + (x-xy(1))^2;
xroots = double(solve(diff(D2,x)))
xroots =
0.167082630638436 + 0i
2.91225594186202 + 0i
4.72410209755873 + 0i
-0.48309085270852 - 1.29552403846027i
-0.48309085270852 + 1.29552403846027i
If we ignore the complex roots,
double(subs(D2,xroots))
ans =
12.8937799521223 + 0i
50.8355152126218 + 0i
5.09171123781745 + 0i
6.70892471381755 + 7.41419202614555i
6.70892471381755 - 7.41419202614555i
the real root with the minimum distance (squared) in the (x,y) plane is the third root. (Note that there will ALWAYS be at least one real solution to this problem. I'll leave the simple proof of that claim to the interested student.)
ezplot(@(x) polyval(coef,x),[-3,5])
hold on
grid on
plot([xy(1),xroots(3)],[xy(2),polyval(coef,xroots(3))],'r-o')
axis equal
As you can see, with the axes equal in spacing, the line is orthogonal to the curve, as it must be for a projected point of minimum distance.
So not trivial to solve for the point of minimum Euclidean distance, but not terribly difficult either, at least in one dimension. With very little extra thought, I could have written it as a call to roots, and not gone the symbolic route at all.
With two independent variables though, now we will end up with two multinomial equations, in two unknown. A solution will again always exist for the minimum distance, but we will not have recourse to a tool like roots at all. And there will again be in general multiple solutions, so we would need to find the one with the smallest distance. Solving for a zero of the gradient only yields a stationary point.
So possible, but not perfectly trivial. And you will not be able to compute a solution in a vectorized form.
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DS on 17 Sep 2020
Reply to 2): in case someone is interested in getting the complete set of results above from John's comment:
format long g
coef = [0.318765239858981 -1.30768829630527 -0.433592022305684 0.34262446653865];
ezplot(@(x) polyval(coef,x),[-3,5])
xy = [2.48288696432624 2.97939521557093];
hold on
grid on
plot(xy(1),xy(2),'ro')
axis equal
x = sym('x');
a = sym(coef(1));
b = sym(coef(2));
c = sym(coef(3));
d = sym(coef(4));
y = a*x^3 + b*x^2 + c*x + d;
D2 = (y - xy(2))^2 + (x-xy(1))^2;
xroots = double(solve(diff(D2,x)));
xroots
double(subs(D2,xroots))
ans =
12.8937799521223 + 0i
6.70892471381757 + 7.41419202614556i
6.70892471381757 - 7.41419202614556i
50.8355152126213 + 0i
5.09171123781739 + 0i
hold on
grid on
plot([xy(1),xroots(5)],[xy(2),polyval(coef,xroots(5))],'r-o')
axis equal

Joseph Cheng on 28 Apr 2017
you could try something like this. my curve and fit method is probably different however what you can do is use find() to determine which ones are within some distance/error from the fitted curve for that point.
close all;
f = fit( [x, y], z, 'poly23' )
plot(f, [x,y], z)
withintol = find(f(x,y)-z<0.01);
hold on
ax=gca;
plot3(x(withintol),y(withintol),ax.ZLim(1)*ones(size(withintol)),'.')
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Joseph Cheng on 28 Apr 2017
Oh, i was just too lazy to generate a dummy curve and points to franke was just quickly available. now that i think about it just in the Z direction may not be proper thresholding if we're considering xyz jittering points. hmmm....