making a connectivity array for an ellipse based of rectangles and traingles

I have the following code:

clear all
close all
clc
n = 3; % number of divisions on the boundary
x1 = linspace(0,2,n);
y1_x1 = sqrt(1-( (x1.^2)/4 ) );
y2_x1 = -sqrt(1-( (x1.^2)/4 ) );
x2 = linspace(0,-2,n);
y1_x2 = sqrt(1-( (x2.^2)/4 ) );
y2_x2 = -sqrt(1-( (x2.^2)/4 ) );
X1 = repelem(x1,[n:-1:1]);
X2 = repelem(x2,[n:-1:1]);
Y1_X1 = [];
for i = 1:n
    Y1_X1 =  [Y1_X1, y1_x1(i:end)]; % concatenate
end
Y2_X1 = [];
for i = 1:n
    Y2_X1 =  [Y2_X1, y2_x1(i:end)]; % concatenate
end
Y1_X2 = [];
for i = 1:n
    Y1_X2 =  [Y1_X2, y1_x2(i:end)]; % concatenate
end
Y2_X2 = [];
for i = 1:n
    Y2_X2 =  [Y2_X2, y2_x2(i:end)]; % concatenate
end

x = [X1 X1 X2 X2];
y = [Y1_X1 Y2_X1 Y1_X2 Y2_X2];
figure()
scatter(x,y) %shows all points of mesh

Now, I want to assign a number to each point in this ellipse such that a 4xN matrix is formed that has either 4 values (for square elements that are on the inside) and 3 values (for all traingles formed from 3 points on the outside line) with one NaN. For this example, at n = 3, the output connected matrix should look like this:

[1 3 2 NaN
1 4 3 NaN
2 6 5 NaN
2 3 7 6
3 4 8 7
4 9 8 NaN
5 6 10 NaN
6 7 11 10
7 8 12 11
8 9 12 NaN
10 11 13 NaN
11 12 13 NaN]

Here, each row represents a shape, and the numbers in the shape represent the number assigned to its coordinates.

The logic behind numbering the coordinate points is it starts at the lowest point then goes above, but from left end to right end.

And the assembly of the array is such that it starts from the left bottom shape, goes to the right and takes one step upwards (but starts at the left everytime), and the assignment of numebrs within the array is counterclockwise for that shape. The assignment code should be such that it works for any value of n (for example more points would be generated if n is increased from 3 to any larger number in line 4 of initial code).

Can this be done by a loop?

I am attaching a sketch where I show the shapes with encircled numbers and numbers assigned to each point without the encircling.

Hope this helps.

Thanks a lot in advance

Feel free to ask questions.

[Matt J]

Now, I want to assign a number to each point in this ellipse such that a 4xN matrix is formed that has either 4 values (for square elements that are on the inside) and 3 values (for all traingles formed from 3 points on the outside line) with one NaN.

Wouldn't a cell array be better?

Answer 1

[Matt J]

Using spatialgraph2D from this FEX download,

https://www.mathworks.com/matlabcentral/fileexchange/73630-spatialgraph2d

load xymesh

xy=sortrows( unique([x(:),y(:)],'rows'),[2,1] );
x=xy(:,1); y=xy(:,2);

DT=delaunayTriangulation(x,y);

E=DT.edges;
Eb=DT.freeBoundary;
Eint=setdiff(E,Eb,'rows');

fn=@(z) abs(diff(z(Eint),1,2));

keep=xor(fn(x)<1e-6,fn(y)<1e-6);

E=[Eb;Eint(keep,:)];

G=graph(table(E,'Var',{'EndNodes'}));

obj=spatialgraph2D(G,x,y);

mosaic(obj); 

[p,IDs]=obj.polyshape;

[~,is]=sortrows( cell2mat( arrayfun(@(z) {max(z.Vertices)},p(:)) ) ,[2,1]);

p=p(is); IDs=IDs(is);

N=numel(p);

out=nan(N,4);

for n=1:N

    id=IDs{n}; m=length(id);
    out(n,1:m)=id;

end

out=sort(out,2)

out = 12×4

1 2 3 NaN 1 3 4 NaN 2 5 6 NaN 2 3 6 7 3 4 7 8 4 8 9 NaN 5 6 10 NaN 6 7 10 11 7 8 11 12 8 9 12 NaN

[Saim Ehtesham]

Dear Matt

I do not know where to save the xymesh.m file.

Could you please guide me?

[Saim Ehtesham]

Dear Matt

Please help, I am unable to run your code. What steps do I take to incorporate xymesh and spatial2Dgraph in to my MATLAB software?

[Matt J]

xymesh.mat is not necessary. It was just my way of providing input x,y points for demonstration purposes.

[Saim Ehtesham]

Hi Matt

Thanks for the help

However, the assignment of point values in the variable out is not an exact match. How can I do that?

[Matt J]

I don't know what you mean. In what way are they not an exact match?

[Saim Ehtesham]

This was the intended output:

[1 3 2 NaN
1 4 3 NaN
2 6 5 NaN
2 3 7 6
3 4 8 7
4 9 8 NaN
5 6 10 NaN
6 7 11 10
7 8 12 11
8 9 12 NaN
10 11 13 NaN
11 12 13 NaN]

this is your code's ouptut:

1     2     3   NaN
1     3     4   NaN
2     5     6   NaN
2     3     6     7
3     4     7     8
4     8     9   NaN
5     6    10   NaN
6     7    10    11
7     8    11    12
8     9    12   NaN

The column entries in every row don't match. I apologize if I am being unreasonable.

[Matt J]

Each row contains the same numbers in both cases, but in a different order. You have not explained by what rule the vertices of each triangle/rectangle should be ordered. You could of course loop through the rows and sort them according to your preferred rule.

[Saim Ehtesham]

My apologies. The preferred rule is to look at the shape, start at the lower left corner and go counter cockwise. For example, traingle 1 (row 1) starts at 1 and goes counterclockwise from that point, so row one becomes [1 3 2]. and for the 5th row (a rectangle), it starts at lower left edge of shape that is 3, goes counterclockwise and the 5th row becomes [3 4 8 7]. I hope this gives a good explanation

[Matt J]

There are ways to do that reordering, but what is the eventual purpose of this? Note that the output of the section of the code below gives you a polyshape vector p. With the triangles and rectangles in polyshape form, you can analyze the shapes much more easily than with the matrix you are pursuing.

[p,IDs]=obj.polyshape;

[~,is]=sortrows( cell2mat( arrayfun(@(z) {max(z.Vertices)},p(:)) ) ,[2,1]);

p=p(is); 

[Saim Ehtesham]

Dear Matt, thanks a lot for your help.

I rearranged the "out" variable by realizing a few similarities and using those as defining indexing rules.

Can I just ask one more thing please? For the mosaic figure, is there a way to modify the number shown at each point?

Can we hide the numbers or change the size or opacity or anything else?

I have tried finding myself but apparently your "mosaic" is a part of the spatial2Dgraph which I do not want to edit as it does the most important job for me

[Matt J]

You can get a handle to the label text with findobj. Then, you can modify it or delete it as you wish, e.g.,

mosaic(obj); 

ht=findobj(gca,'type','text');
set(ht,'FontSize',12);

[Saim Ehtesham]

Thanks a lot for all the help Matt. As you asked, the purpose of that specific arrangement is to do Finite element analysis on the elliptic area. My partner and I are making the connectivity matrix to multiply the shape functions from a rectangular or square master element but the shape functions are stored counter clockwise, starting from lower left corner to keep a homogenous approach throughout the solution.

Again, thanks for all the help.