Problem with mesh grid and intersection of two surfaces

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Zac Lee 2025-8-1

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评论： Zac Lee 2025-8-2

Hello everyone,

I’m trying to plot the intersection points between two second-order polynomial equations. These equations depend on two variables, a and b, which I define as vectors.

To evaluate the solutions, I use meshgrid to create all combinations of a and b. However, this leads to an issue: for a single root (solution), the same value of a appears repeatedly in the result. This causes my plot to show vertical lines instead of individual points, since multiple identical a values are associated with different b values (or vice versa).

Ideally, for each unique pair (a, b), I want to obtain a unique solution. But using meshgrid evaluates all possible combinations, including repeated values for a or b.

I’d appreciate any advice on how to better approach this problem, particularly in terms of how to structure the evaluation and plotting to avoid this repetition and get cleaner, more accurate plots.

Thanks in advance!

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Zac Lee 2025-8-1

编辑：Matt J 2025-8-1

在 MATLAB Online 中打开

clear all; clc; close all

pas = 0.001;

N11list = 0:pas:0.01;

N31list = 0:pas:0.01;

[N11vec, N31vec] = meshgrid(N11list, N31list);

N = [N11vec(:), N31vec(:)];

racine = [];

N11sol = [];

N31sol = [];

% Stockage des valeurs déjà rencontrées (pour chaque variable séparément)

valeurs_n11 = [];

valeurs_n31 = [];

valeurs_racine = [];

tol = 1e-8;

for i = 1:length(N)

n11 = N(i, 1);

n31 = N(i, 2);

%% Polynôme 1

A1 = n11^2;

B1 = 2*n11 * n31^2;

C1 = (5 * n11^2 + 3 * n31);

%% Polynôme 2

A2 = n31^2;

B2 = n31 + n11 * n31^3;

C2 = n31 * n11^2 + n31 * n11^3+1;

%% Intersection

A = A1 - A2;

B = B1 - B2;

C = C1 - C2;

if A == 0 && B == 0

continue

end

roots_poly = roots([A, B, C]);

for k = 1:length(roots_poly)

x = roots_poly(k);

if isreal(x)

% Vérifier que la racine n’a pas déjà été vue

if any(abs(valeurs_racine - x) < tol)

continue

end

% Tout est unique : on stocke

racine(end+1) = x;

N11sol(end+1) = n11;

N31sol(end+1) = n31;

valeurs_racine(end+1) = x;

valeurs_n11(end+1) = n11;

valeurs_n31(end+1) = n31;

end

% Affichage

scatter3(racine,N11sol,N31sol,'.b');

xlabel('ROOT');

ylabel('N11');

zlabel('N31');

Catalytic 2025-8-1

编辑：Catalytic 2025-8-1

My ptoblem, is that, i'm having multiple times the same value ffor a and b for the same root.

Are you trying to develop a mapping from (a,b) to the roots, or the other way around? If you are mapping from (a,b) to the roots then it shouldn't matter if there are multiple (a,b) that map to the same root.

If you are mapping from the roots back to (a,b) then you first have to map the roots to a set of polynomial coefficients and then map the coefficients to (a,b). Bear in mind though that polynomial coefficients are unique only up to a scale factor - so you need to decide on some way to remove that ambiguity, before you can map the coefficients to a unique (a,b).

Zac Lee 2025-8-2

Dear @Catalytic, thank you verymuch for your answer. Effectively, I want to develop a mapping from (a, b) to the roots. The main problem is that I have several repeated instances of the same (a, b) pairs, which leads to the issue shown in the picture. I think I’ll need to apply some kind of filtering to ensure that each (a, b) pair appears only once. That way, I can be sure to avoid this problem — or at least reduce it significantly. I think i have to approach the problem in another way

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Answer 1

Catalytic 2025-8-2

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编辑：Catalytic 2025-8-2

在 MATLAB Online 中打开

Effectively, I want to develop a mapping from (a, b) to the roots

If so, then @Matt J already told you what you are doing wrong. (a,b), or I guess (N11,N13), should be on the xy axis -

pas = 0.001;

[N11, N31] = meshgrid(0:pas:0.01);

%% Polynôme 1

A1 = N11.^2;

B1 = 2 .* N11 .* N31.^2;

C1 = (5 .* N11.^2 + 3 .* N31);

%% Polynôme 2

A2 = N31.^2;

B2 = N31 + N11 .* N31.^3;

C2 = N31 .* N11.^2 + N31 .* N11.^3 + 1;

%% Intersection

A = A1 - A2;

B = B1 - B2; %normalize leading coefficient

C = C1 - C2; %normalize leading coefficient

D=B.^2-4.*A.*C; %Discriminant

R1=(-B-sqrt(D))./2./A; %First root

R2=(-B+sqrt(D))./2./A; %Second root

k=D>0 & (A|B);

R1=R1(k); R2=R2(k); N11=N11(k); N31=N31(k);

scatter3(N11,N31,R1,'.b'); hold on

scatter3(N11,N31,R2,'.r'); hold off

xlabel('N11');

ylabel('N12');

zlabel('ROOT');

legend ROOT1 ROOT2

●

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Answer 2

Matt J 2025-8-1

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https://ww2.mathworks.cn/matlabcentral/answers/2178911-problem-with-mesh-grid-and-intersection-of-two-surfaces#answer_1568683

编辑：Matt J 2025-8-1

在 MATLAB Online 中打开

Perhaps as follows:

tol = 1e-8;

pas = 0.001;

[N11, N31] = meshgrid(0:pas:0.01); N11=N11(:); N31=N31(:);

%% Polynôme 1

A1 = N11.^2;

B1 = 2 .* N11 .* N31.^2;

C1 = (5 .* N11.^2 + 3 .* N31);

%% Polynôme 2

A2 = N31.^2;

B2 = N31 + N11 .* N31.^3;

C2 = N31 .* N11.^2 + N31 .* N11.^3 + 1;

%% Intersection

A = A1 - A2;

B = (B1 - B2)./A; %normalize leading coefficient

C = (C1 - C2)./A; %normalize leading coefficient

D=B.^2-4*C; %Discriminant

R1=(-B-sqrt(D))/2; %First root

R2=(-B+sqrt(D))/2; %Second root

%Post-process: discard redundant or illegal solutions

T=[R1,R2,N11,N31];

T( any(~isfinite(T),2) | D<0 , : )=[]; %discard non-finite or complex solutions

[~,Iu]=uniquetol(T(:,1:2),tol,'ByRows',true); %discard repeat solutions (within tolerance)

T=num2cell(T(Iu,:),1);

[R1,R2,N11,N13]=deal(T{:});

figure

scatter3(R1,R2,N11,'.b');

xlabel('ROOT #1');

ylabel('ROOT #2');

zlabel('N11');

figure

scatter3(R1,R2,N13,'.b');

xlabel('ROOT #1');

ylabel('ROOT #2');

zlabel('N13');

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Zac Lee 2025-8-2

Dear @Matt J and @Catalytic, it seems working npow. Thank you verymuch for your help.

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