Create a node in the space between two nodes (A and B) and following the direction normal to a node (N)

Hi. I need to create a node in the space between two nodes (A and B) and following the direction normal to a node (N) as in the figure.

Is there a function that can do this? The result are the coordinates of node C.

A = [-25.5530   -185.3199   -211.6502];
B = [-25.4769   -185.6468   -211.2523];
N = [-25.4602   -185.4676   -211.6694];
normal_plane = [0.2437  -0.62123    0.7447];

figure
plot3(N(:,1),N(:,2),N(:,3),'r.','Markersize',25);
hold on
plot3(A(:,1),A(:,2),A(:,3),'k.','Markersize',25);
plot3(B(:,1),B(:,2),B(:,3),'k.','Markersize',25);
hold off
axis equal
grid off

[William Rose]

Yes there is a way D=(A+B)/2; % the location E=cross((A-B),(N-D)); % the direction

[William Rose]

Sorry about the formatting, doing it on phone

[Catalytic]

@Alberto Acri

You've asked a million questions on coordinate goemetry on this forum. Surely you now by now how to find the equation of a plane and its intersection with a line.

Answer 1

[Matt J]

I might be misinterpreting your question somehow, but from your figure illustration, it doesn't make sense that you would be supplying both N and normal_plane independently. The normal plane there will be a function of A,B, and N.

A = [-25.5530   -185.3199   -211.6502 ,1]'; %work in homogeneous coordinates
B = [-25.4769   -185.6468   -211.2523 ,1]';
N = [-25.4602   -185.4676   -211.6694 ,1]';

n=B-A; %normal
normal_plane=[n(1:3);-n'*N]; %plane
L=(A*B'- B*A'); %The line [A,B]

C=L*normal_plane; C=C/C(4)

C = 4×1

-25.5396 -185.3775 -211.5802 1.0000

normal_plane'*C  %Check that it lies in the plane

ans = -1.4211e-14

dot(A-B,N-C)  %Check normality

ans = -1.5446e-14

f=@(z) z(1:3);
norm(cross(f(C-A),f(B-A)))  %Check co-linearity of C with [A,B]

ans = 4.9595e-12

[Alberto Acri]

Hi @Matt J! Regarding your answer, I would like to ask you why, using the attached data, node 'C' is on the line of nodes 'A' and 'B' but is not passing through the yellow plane. I would like node 'C' to be located as in the figure (see node 'Cc'). Can you help me?

%%
A = importdata("A_test.mat");
B = importdata("B_test.mat");
N = importdata("G_test.mat");
nodes_test = importdata("nodes_test.mat");


    %%
    A = [A, 1]';
    B = [B, 1]';
    N = [N, 1]';

    n = B-A; %normal
    normal_plane = [n(1:3);-n'*N]; %plane
    L = (A*B'- B*A'); %The line [A,B]
    C = L*normal_plane; 
    C = C/C(4);
    C = C';
    C = C(1,1:3);   

    A = A';
    A = A(1,1:3);
    B = B';
    B = B(1,1:3);  

    % =================

    figure
    plot3(N(1),N(2),N(3),'r.','Markersize',20)
    hold on

    plot3(nodes_test(:,1), nodes_test(:,2), nodes_test(:,3), 'y.', 'Markersize', 30);
    patch(nodes_test(:,1), nodes_test(:,2), nodes_test(:,3), 'y'); % Plotting the plane  

    plot3(A(:,1),A(:,2),A(:,3),'c.','Markersize',20)
    plot3(B(:,1),B(:,2),B(:,3),'m.','Markersize',20)
    plot3(C(:,1),C(:,2),C(:,3),'g.','Markersize',20)

    hold off
    axis equal
    grid off

[Matt J]

node 'C' is on the line of nodes 'A' and 'B' but is not passing through the yellow plane

It is not clear what defines the "yellow plane". It doesn't even look like a plane, but rather 3 colinear yellow dots. In your original post, the requirements on the plane were that it be normal to [A,B] and contain the point N. You can see below that that is still true with this new set of data.

%%
A = importdata("A_test.mat");
B = importdata("B_test.mat");
N = importdata("G_test.mat");


    %%
    A = [A, 1]';
    B = [B, 1]';
    N = [N, 1]';

    n = B-A; %normal
    normal_plane = [n(1:3);-n'*N]; %plane
    L = (A*B'- B*A'); %The line [A,B]
    C = L*normal_plane; 
    C = C/C(4);

    % =================

    figure
    plot3(N(1),N(2),N(3),'r.','Markersize',20)
    xlabel x; ylabel y; zlabel z; axis equal
    hold on


    plot3(A(1),A(2),A(3),'c.','Markersize',20)
    plot3(B(1),B(2),B(3),'m.','Markersize',20)
    plot3(C(1),C(2),C(3),'g.','Markersize',20)
     D=N+2*(C-N);
    plot3(D(1),D(2),D(3),'.','MarkerFaceColor','none');

    ax=axis;
    fsurf(@(x,y) -n(3)\(n(1)*x+n(2)*y + normal_plane(4)),...
        'FaceColor','b','FaceAlpha',0.3,'EdgeColor','none');
    axis(ax);


    grid off

    hold off; view(85,20)

[Alberto Acri]

Sorry Matt, yellow plan (not plane ahaha). Yellow plan passing through 3 yellow nodes.

The 'problem' is that the green node C I also want it on the plane and not just between nodes A and B!

[Matt J]

So the requirement that the plane be perpendicular to [A,B] is gone (or was never there)? And if the plane is determined entirely by the 3 nodes_test data points, what is the purpose of the additional given point N?

[Alberto Acri]

The condition that point C must lie between A and B is valid.

Node N may indeed be ignored (it lies on the yellow plan).

[Matt J]

In that case, there isn't much that needs changing. Just use planarFit(), which you are familiar with, to fit a plane to the nodes_test data,

P=planarFit(nodes_test);

normal_plane=[P.normal, -P.d];

and the rest of the code should work as before.

[Alberto Acri]

Could you tell me where to use these lines of code within the code you provided?

I had replaced these two lines of code with yours but not work

n = B-A; %normal
normal_plane = [n(1:3);-n'*N]; %plane

[Matt J]

%%
A = importdata("A_test.mat");
B = importdata("B_test.mat");

    %%
    A = [A, 1]';
    B = [B, 1]';
    
    P=planarFit(nodes_test);
    n=P.normal;
    normal_plane=[P.normal, -P.d];
    L = (A*B'- B*A'); %The line [A,B]
    C = L*normal_plane; 
    C = C/C(4);

    % =================

    figure
    

    plot3(A(1),A(2),A(3),'c.','Markersize',20)
     xlabel x; ylabel y; zlabel z; axis equal
     
    hold on

    plot3(B(1),B(2),B(3),'m.','Markersize',20)
    plot3(C(1),C(2),C(3),'g.','Markersize',20)
    
    ax=axis;
    fsurf(@(x,y) -n(3)\(n(1)*x+n(2)*y + normal_plane(4)),...
        'FaceColor','b','FaceAlpha',0.3,'EdgeColor','none');
    axis(ax);


    grid off

    hold off; view(85,20)

[Alberto Acri]

thank you, but I get an error on the line

C = L*normal_plane;

[Matt J]

Probably because normal_plane is a row vector and should be a column vector.

[Alberto Acri]

I changed the vector to an column vector but I do not get the desired result.

[Alberto Acri]

With this code I obtained this result. The green node C is in an incorrect position. How come?

A = importdata("A_test.mat");
B = importdata("B_test.mat");
N = importdata("G_test.mat");
nodes_test = importdata("nodes_test.mat");
   

    %% DATA
    A = [A, 1]';
    B = [B, 1]';
    
    P = planarFit(nodes_test);
    n = P.normal;
    normal_plane = [P.normal, -P.d];
    normal_plane = normal_plane';
    L = (A*B'- B*A'); %The line [A,B]
    C = L*normal_plane; 
    C = C/C(4);
    % =================
    figure
    
    plot3(A(1),A(2),A(3),'c.','Markersize',20)
     xlabel x; ylabel y; zlabel z; axis equal
     
    hold on
    plot3(B(1),B(2),B(3),'m.','Markersize',20)
    plot3(C(1),C(2),C(3),'g.','Markersize',20)

    plot3(nodes_test(:,1), nodes_test(:,2), nodes_test(:,3), 'y.', 'Markersize', 30);
    patch(nodes_test(:,1), nodes_test(:,2), nodes_test(:,3), 'y'); % Plotting the plane  

    
    ax=axis;
    fsurf(@(x,y) -n(3)\(n(1)*x+n(2)*y + normal_plane(4)),...
        'FaceColor','b','FaceAlpha',0.3,'EdgeColor','none');
    axis(ax);
    grid off
    hold off; view(85,20)

[Matt J]

A = importdata("A_test.mat");
B = importdata("B_test.mat");
nodes_test = importdata("nodes_test.mat");
   

    %% DATA
    A = [A, 1]';
    B = [B, 1]';
    
    P = planarFit(nodes_test');  %<--------------- Note transpose
    n = P.normal;
    normal_plane = [P.normal, -P.d];
    normal_plane = normal_plane';
    L = (A*B'- B*A'); %The line [A,B]
    C = L*normal_plane; 
    C = C/C(4);
    % =================
    figure
    
    plot3(A(1),A(2),A(3),'c.','Markersize',20)
     xlabel x; ylabel y; zlabel z; axis equal
     
    hold on
    plot3(B(1),B(2),B(3),'m.','Markersize',20)
    plot3(C(1),C(2),C(3),'g.','Markersize',20)

    plot3(nodes_test(:,1), nodes_test(:,2), nodes_test(:,3), 'y.', 'Markersize', 30);
    patch(nodes_test(:,1), nodes_test(:,2), nodes_test(:,3), 'y'); % Plotting the plane  

    
    ax=axis;
    fsurf(@(x,y) -n(3)\(n(1)*x+n(2)*y + normal_plane(4)),...
        'FaceColor','b','FaceAlpha',0.3,'EdgeColor','none');
    axis(ax);
    grid off
    hold off; view(85,20)

Answer 2

[William Rose]

@Alberto Acri

D=(A+B)/2; % the location 
E=cross((A-B),(N-D)); % the direction

good luck

Answer 3

[Catalytic]

A = [-25.5530   -185.3199   -211.6502];
B = [-25.4769   -185.6468   -211.2523];
N = [-25.4602   -185.4676   -211.6694];

C=(B-A)'\(N-A)'*(B-A)+A

C = 1×3

-25.5396 -185.3775 -211.5802

Answer 4

[Shubham]

Hi Alberto,

To create a node C in the space between two nodes A and B, following the direction normal to a node N, you can follow these steps:

1. Calculate the midpoint (M) between nodes A and B. This gives you a point that lies directly between them.
2. Determine the desired distance (d) you want node C to be from the midpoint M along the normal direction.
3. Scale the normal vector (normal_plane) to have the desired length (d).
4. Add the scaled normal vector to the midpoint M to find the coordinates of node C.

Here is how you can implement this in MATLAB:

A = [-25.5530, -185.3199, -211.6502];
B = [-25.4769, -185.6468, -211.2523];
N = [-25.4602, -185.4676, -211.6694];
normal_plane = [0.2437, -0.62123, 0.7447];

% Calculate the midpoint between A and B
M = (A + B) / 2;

% Determine the desired distance d for node C from the midpoint M
% This value is for illustration; you'll need to choose an appropriate value.
d = 10; % Replace with the desired distance

% Normalize the normal vector
normal_plane_unit = normal_plane / norm(normal_plane);

% Scale the normal vector to have the desired length d
normal_plane_scaled = normal_plane_unit * d;

% Calculate the coordinates of node C
C = M + normal_plane_scaled;

% Plotting
figure
plot3(N(1), N(2), N(3), 'r.', 'Markersize', 25); hold on;
plot3(A(1), A(2), A(3), 'k.', 'Markersize', 25);
plot3(B(1), B(2), B(3), 'k.', 'Markersize', 25);
plot3(C(1), C(2), C(3), 'b.', 'Markersize', 25); % Node C in blue
hold off;
axis equal;
grid on;

In the code above, d is a placeholder for the distance you want node C to be from the midpoint M along the normal direction. You'll need to define d based on your specific requirements. The normal vector is also scaled to this distance and then added to the midpoint M to find the coordinates of node C.

Node C is plotted in blue for distinction. Make sure to adjust the value of d as needed for your specific use case. If you want node C to be placed in a different way related to node N, you would need to adjust the calculation accordingly.